Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can (...) be also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

Consider the following. The first is a one-premise argument; the second has two premises. The question sign marks the conclusions as such. -/- Matthew, Mark, Luke, and John wrote Greek. ? Every evangelist wrote Greek. -/- Matthew, Mark, Luke, and John wrote Greek. Every evangelist is Matthew, Mark, Luke, or John. ? Every evangelist wrote Greek. -/- The above pair of premise-conclusion arguments is of a sort familiar to logicians and philosophers of science. In each case the first premise is (...) logically equivalent to the set of four atomic propositions: “Matthew wrote Greek”, “Mark wrote Greek”, “Luke wrote Greek”, and “John wrote Greek”. The universe of discourse is the set of evangelists. We presuppose standard first-order logic. -/- As many logic texts teach, the first of these two premise-conclusion arguments—sometimes called a complete enumerative induction— is invalid in the sense that its conclusion does not follow from its premises. To get a counterargument, replace ‘Matthew’, ‘Mark’, ‘Luke’, and ‘John’ by ‘two’,’four’, ‘six’ and ‘eight’; replace ‘wrote Greek’ by ‘are even’; and replace ‘evangelist’ by ‘number’. This replacement converts the first argument into one having true premises and false conclusion. -/- But the same replacement performed on the second argument does no such thing: it converts the second premise into the falsehood “Every number is two, four, six, or eight”. As many logic texts teach, there is no replacement that converts the second argument into one with all true premises and false conclusion. The second is valid; its conclusion is deducible from its two premises using an instructive natural deduction. -/- This paper “does the math” behind the above examples. The theorem could be stated informally: the above examples are typical. (shrink)

Elementary Applied Symbolic Logic was first published by Prentice-Hall in 1976. It went through two editions with them, then had a successful classroom run of 25 years by various publishers, before it finally went out of print in 2001.I am reviving it here, because during its run it acquired a reputation as an outstanding textbook for getting students to understand symbolic logic.I immodestly believe it is the best textbook ever written on the subject.------------This is a book on applied (...) symbolic logic. It provides the bridge between statements and arguments in English, and their formal counterparts in symbolic logic. Extensive exercises are given, illustrating how different natural-language concepts can correspond to the same symbolism, and how English sentences may be translated into formulae. Translation is heavily emphasized.It is intended to make learning symbolic logic (relatively) easy, by starting out with very basics and progressing from there a step at a time, building on what came before. I tried to make it as close to a self-teaching text as I could manage. It has two major divisions: Propositional Logic and Quantifier Logic.The first starts with propositions and truth-values, then truth-tables for evaluating the status of statements and arguments. It then moves to natural deduction, with rules for making inferences and transformations. Procedures are given for proving both validity and invalidity.Exercises increase in complexity as things move along. Solutions to selected exercises are included at the back of the book.Quantifier Logic starts with Monadic predicate logic, involving only single-place predicates ("properties"). It starts with singular statements and propositional functions, then moves to statements containing a single universal or existential quantifier, then to statements and arguments involving multiple quantifiers. It covers inferences using quantificational inference and transformation rules, and gives methods of invalidity proof.Its second half goes into polyadic predicates ("relations") of various degrees, moves on to identity, and finally to definite descriptions.Appendices on various related and supplementary topics are included at the end. The original appendix on Completeness and Consistency was complicated and confusing. It has been deleted, and replaced with an addendum at the end. (shrink)

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then (...) treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. (shrink)

It is argued that the goal of Hilbert's program was to prove the model-theoretical consistency of different axiom systems. This Hilbert proposed to do by proving the deductive consistency of the relevant systems. In the extended independence-friendly logic there is a complete proof method for the contradictory negations of independence-friendly sentences, so the existence of a single proposition that is not disprovable from arithmetic axioms can be shown formally in the extended independence-friendly logic. It can also (...) be proved by means of independence-friendly logic that proof-theoretical consistency of a sentence S implies the existence of a model in which S is not false. Hence the consistency of the axioms of arithmetic in the sense of being not-false in a model can be proved. (shrink)

This account of propositional logic concentrates on the algorithmic translation of important methods, especially of decision procedures for (subclasses of) propositional logic. Important classical results and a series of new results taken from the fields of normal forms, satisfiability and deduction methods are arranged in a uniform and complete theoretic framework. The algorithms presented can be applied to VLSI design, deductive databases and other areas. After introducing the subject the authors discuss satisfiability problems and (...) satisfiability algorithms with complexity considerations, the resolution calculus with different refinements, and special features and procedures for Horn formulas. Then, a selection of further calculi and some results on the complexity of proof procedures are presented. The last chapter is devoted to quantified boolean formulas. The algorithmic approach will make this book attractive to computer scientists and graduate students in areas such as automated reasoning, logic programming, complexity theory and pure and applied logic. (shrink)

A term td is called a ternary deductive term for a variety of algebras V if the identity td≈r holds in V and ∈θ yields td≈td for any A∈V and any principal congruence θ on A. A connective f is called td-distributive if td)≈ f,…,td). If L is a propositional logic and V is a corresponding variety that has a TD term td, then any admissible in L rule, the premises of which contain only td-distributive operations, is (...) derivable, and the substitution r↦td is a projective L-unifier for any formula containing only td-distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal to prove structural completeness of the implication fragment of intuitionistic propositional logic. (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the (...) deductive sciences, and to computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

Suszko's Thesis maintains that many-valued logics do not exist at all. In order to support it, R. Suszko offered a method for providing any structural abstract logic with a complete set of bivaluations. G. Malinowski challenged Suszko's Thesis by constructing a new class of logics (called q-logics by him) for which Suszko's method fails. He argued that the key for logical two-valuedness was the "bivalent" partition of the Lindenbaum bundle associated with all structural abstract logics, while his (...) q-logics were generated by "trivalent" matrices. This paper will show that contrary to these intuitions, logical two-valuedness has more to do with the geometrical properties of the deduction relation of a logical structure than with the algebraic properties embedded on it. (shrink)

Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic \. A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system \ is structurally complete if any rule admissible for the logic defined (...) by \ is derivable in \. It is known that any logic can be defined by a structurally complete deductive system—its structural completion. The main goal of the paper is to study the following problem: given a superintuitionistic logic L, is the structural completion of L hereditarily structurally complete? It is shown that, on the one hand, there is continuum many of such logics, including \, and many of its standard extensions. On the other hand, there is continuum many superintutitionistic logics structural completion of which is not hereditarily structurally complete. It is observed that the class of hereditarily structurally complete superintuitionistic consequence relations does not have the smallest element and it contains continuum many members lacking the finite model property. The following statement is instrumental in obtaining negative results: if a Lindenbaum algebra of formulas on one variable is finite and has more than 15 elements, then a structural completion of such a logic is not hereditarily structurally complete. (shrink)

JOHN CORCORAN AND WAGNER SANZ, Disbelief Logic Complements Belief Logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 USA E-mail: corcoran@buffalo.edu Filosofia, Universidade Federal de Goiás, Goiás, GO 74001-970 Brazil E-mail: sanz@fchf.ufg.br -/- Consider two doxastic states belief and disbelief. Belief is taking a proposition to be true and disbelief taking it to be false. Judging also dichotomizes: accepting a proposition results in belief and rejecting in disbelief. Stating follows suit: asserting a proposition conveys belief and denying conveys disbelief. (...) Traditional logic implicitly focused on logical relations and processes needed in expanding and organizing systems of beliefs. Deducing a conclusion from beliefs results in belief of the conclusion. Deduction presupposes consequence: one proposition is a consequence of a set of a propositions if the latter logically implies the former. The role of consequence depends on its being truth-preserving: every consequence of a set of truths is true. This paper, which builds on previous work by the second author, explores roles of logic in expanding and organizing systems of disbeliefs. Aducing a conclusion from disbeliefs results in disbelief of the conclusion. Aduction presupposes contrequence: one proposition is a contrequence of a set of propositions if the set of negations or contradictory opposites of the latter logically implies that of the former. The role of contrequence depends on its being falsity-preserving: every contrequence of a set of falsehoods is false. A system of aductions that includes, for every contrequence of a given set, an aduction of the contrequence from the set is said to be complete. Historical and philosophical discussion is illustrated and enriched by presenting complete systems of aductions constructed by the second author. One such, a natural aduction system for Aristotelian categorical propositions, is based on a natural deduction system attributed to Aristotle by the first author and others. ADDED NOTE: Wagner Sanz reconstructed Aristotle’s logic the way it would have been had Aristole focused on constructing “anti-sciences” instead of sciences: more generally, on systems of disbeliefs. (shrink)

Ken Pledger devised a one-sorted approach to the incidence relation of plane geometries, using structures that also support models of propositional modal logic. He introduced a modal system 12g that is valid in one-sorted projective planes, proved that it has finitely many non-equivalent modalities, and identified all possible modality patterns of its extensions. One of these extensions 8f is valid in elliptic planes. These results were presented in his 1980 doctoral dissertation, which is reprinted in this issue of (...) the Australasian Journal of Logic. Here we show that 12g and 8f are strongly complete for validity in their intended one-sorted geometrical interpretations, and have the finite model property. The proofs apply standard technology of modal logic together with a step-by-step procedure introduced by Yde Venema for constructing two-sorted projective planes. (shrink)

In his “The Foundations of Mathematics”, Ramsey attempted to marry the Tractarian idea that all logical truths are tautologies and vice versa, and the logicism of the Principia. In order to complete his project, Ramsey was forced to introduce propositional functions in extension (PFEs): given Ramsey's definitions of 1 and 2, without PFEs even the quantifier-free arithmetical truth that 1 ≠ 2 is not a tautology. However, a number of commentators have argued that the notion of PFEs (...) is incoherent. This response was first given by Wittgenstein but has been best developed by Sullivan. While I agree with Wittgenstein and Sullivan's common conclusion, I believe that even the most compelling of Sullivan's arguments is importantly mistaken and that Wittgenstein's remarks are too opaque to be left as the end of the matter. In this article I uncover the fault in Sullivan's argument and present an alternative criticism of PFEs which is Wittgensteinian in spirit without being too mystifying. (shrink)

: Muḥammad ibn Yūsuf al-Sanūsī, who was one of the theologian of later Muslim Asʿharī theologians, is one of the scholar came to the fore in Maghrib. Although al-Sanūsī shapes his thoughts within the Asʿharī’s kalam system, he presents new contributions to this system with his own unique perspective. In particular, his effort to give importance to reason in maʿrifat Allāh is remarkable. According to him, maʿrifat Allāh can only be reached with reflection. Also the reflection is waʿẓ or contexture (...) process done in the reason. When a method is followed, such an reflection leads to the maʿrifatullāh. In particular, the ruling wisdom adds a form to the functioning of the reason through its perceived and compassionate perceptions. For a believer to know the necessary, the permissible and the impossible is to use the reason. In this context al-Sanūsī imposes a central role in the concept of judgment. If the human can operates its reason in this way, necessarily will comprehend that he/she is created. In the process of acquiring the knowledge, al-Sanūsī draws attention to the heart as much as the reason and that shows that he is in the effort of combining these two. Summary: Muḥammad ibn Yūsuf al-Sanūsī is a scholar raised at the Maghreb geography. Although al-Sanūsī, living in the 9th/15th century, gave a lot of work in different fields, his theologian identity became front-line. By standing against the innovation that he had seen during his era, he wrote many beliefs works. Among these works, the relation of maʿrifat Allāh-reason attracts attention. al-Sanūsī thinks that the most important means of delivering to maʿrifat Allāh is deduciton. According to him, the deduction is “to reveal the known to reach the desired thing or to arrange two or more works”. If we evaluate this definition in terms of Asʿharī tradition, we will see that al-Sanūsī is in the effort to emending the definition of his al-Qāḍī al-Bayḍāwī’s definition who is his contemporary. al-Bayḍāwī described deduction as ‘to organize things that are not known to be known’. al-Sanūsī made a contribution to this recognition. He has adopted the arrange part completely and also add the waʿẓ expression into that. The reason why al-Sanūsī goes to such a separation is the reason for trying to explain the functioning of the reason in a logical form. In the logical sense, the thought consists of envision and confirmation. From these concepts that constitute thought, envision refers to waʿẓ, and confirmation refers to arrange. If the definition of al-Bayḍāwī is accepted as valid, the envision in reason is incomplete. This shows the importance that al-Sanūsī gives to the functioning of the reason in terms of the achievement of the maʿrifat Allāh. The confirmation made without envision can lead a person to imitation. Therefore, the deduction made by waʿẓ must be taken into account. It can be imagined groundlessly that in the definition of deduciton which al-Sanūsī made, he had an understanding of reason that doesn’t base on Allah. However, al-Sanūsī has never broken the Asʿharī tradition in this matter. According to al-Sanūsī, both the deduction and the knowledge formed as a result of this, is Allah’s creation without means. There is no influence of deduction on knowledge and also the knowledge on deduction. Also, the human has no influence neither in inference nor knowledge. It seems to be contradictory to the definition of such a discourse. However, he made his interpretations within the framework of the understanding of the omnipotence that the Asʿharites had drawn. It is unthinkable that Allah, who created everything and even man’s actions, can’t be imagine that doesn’t create thought. Intellect produces knowledge, but Allah creates both intellect and the knowledge produced by intellect. al-Sanūsī, while describing deduction, said that it could be produced with waʿẓ in the sense of envision. However, there is another very important concept that creates a conception from his point of view, which is “judgment”. According to him, judgment is “the proof or negation of something”. The phrases “the universe is created” or “Allah is not created” are the examples of a proof or negation of an issue. See, this is a judgment. In this sense, envision is a road to judgment. That is, the information obtained by envision reaches a judgment in the forms of the reason. al-Sanūsī classify the judgment in to three: lawful, ordinary, intellectual. The lawful judgment is the salutation of Allah with the demand related to the actions of the amenable, not being good deed or sin or waʿẓ. Ordinary judgment is the link between the existence or absence of a state and the existence or absence of another state. The most important of these three judgments is the intellectual judgment. As for the intellectual judgment, it is the proof or the negation of a work without waʿẓ and repetition. It is separated from lawful judgment without being waʿẓ and separated from ordinary judgment with being not repeated. In this sense, if the mind needs the examination or repetition while deciding for a work’s certainty or negation, this kind of a judgment is not lawful or ordinary but it is intellectual judgment. The reason judges either a work’s certainty or absence together, or accepts only the certainty or accepts the negation. According to al-Sanūsī, having the right to comment on belief is pertain to intellectual judgment in the first place. al-Sanūsī takes the maʿrifat Allāh to the center in matters related to the place of reason and examines the relation of the maʿrifat Allāh with these three concepts. What al-Sanūsī tends in doing so is that to show that there is an inherent meaning of reason as well as having relevance of knowing some things. As a result, according to al-Sanūsī, knowing these three parts of the intellectual judgment is the reason itself. Anybody who does not know these, is the person without reason. al-Sanūsī thinks that it is necessary for the amenable to know the intellectual judgments and its parts. This necessity is to make the amenable separate the things about Allah or his prophets related to necessity, permissible and impossible attributes; but also, the things exist within the limits of intellectual judgment or not. The amenable who know these has evidence and thereby can defeat the doubts that may have formed or may will form in him. al-Sanūsī also emphasizes the deduction of reason to reach the maʿrifat Allāh with reason. According to him, it is the best deduction method in terms of the evidence that Allah exists is that the deduction from causing agent to cause and the deduction from one correlative to another The acceptance of the possible entity has agent who wills its existence, oppose to the Fakhr al-Dīn al-Rāzī, that this proposition is whether imperative or not. According to him, these propositions are not imperative, they need deduction. In this respect, deduction is necessary for the maʿrifat Allāh. The reason which is considered in the sense of logic in al-Sanūsī is not a structure which merely functions as a analogy and a deduction. On the contrary, according to al-Sanūsī, there is a direction going towards the heart from the point of reason and it seems that the heart is used in meaning of reason. This is an example of al-Sanūsī’s attempt to combine mystical knowledge with reason and legal injunction. This is a remediation project in terms of his time. (shrink)

This textbook is a logic manual which includes an elementary course and an advanced course. It covers more than most introductory logic textbooks, while maintaining a comfortable pace that students can follow. The technical exposition is clear, precise and follows a paced increase in complexity, allowing the reader to get comfortable with previous definitions and procedures before facing more difficult material. The book also presents an interesting overall balance between formal and philosophical discussion, making it suitable for both (...) philosophy and more formal/science oriented students. This textbook is of great use to undergraduate philosophy students, graduate philosophy students, logic teachers, undergraduates and graduates in mathematics, computer science or related fields in which logic is required. (shrink)

In this book, I will discuss three main topics: the roots and aims of scientific knowledge, scientific knowledge in society, and science and values I understand scientific knowledge as being a planned and continuous production of the general and common knowledge of scientific communities. I begin my discussion with a brief analysis of the main differences between sciences, on the one hand, and everyday experience, philosophies, religions, and ideologies, on the other. I define the concept of science as a set (...) of family resemblances (in Wittgenstein's sense). The sciences can be distinguished along three lines, in terms of the difference between empirical and non-empirical sciences, differences in degrees of theoretical and methodological structure among individual sciences and scientific disciplines. Scientific knowledge is based upon some grounds, paradigmatic knowledge bases. Scientific knowledge is founded on a unified paradigmatic knowledge base. The latter includes fundamental theories, basic facts, generally accepted empirical regularities (laws), scientific research methods, scientific explanations, and established relations to other sciences. Scientific knowledge in general is defined well in classical terms by the definition that knowledge is a true and rationally-justified belief. In this connection, scientific truth is relative, depending on the given paradigmatic knowledge base and on the precision and accessibility of the measurements and observations involved. In the book's first part, I consider the production and testing of scientific hypotheses, scientific explanations, the structure of scientific theories, and the problem of scientific realism. I consider explanation an integral part of a wider nexus of heuristic and theoretical activities in science like giving reasons, understanding, clarification, and founding. Continuous production of scientific explanations is a necessary condition for the systematic reproduction of scientific knowledge. Accordingly, it does not suffice to describe them in terms of the logical structure of the arguments on which they are based, but rather, scientific explanations ought to be given complete with an epistemic and pragmatic context of the explanation. I consider especially deductive-nomological, inductive-statistical, dispositional, teleological-functional, and genetic explanations. I emphasize the ever-increasing importance of explanations by analogy and by model. Orthogonal to these distinctions lies the distinction between causal and epistemic explanation. Epistemic explanations relate scientific discoveries to epistemic progress, and causal explanations are these epistemic explanations that relate scientific discoveries to reality too. Causal explanations inform us that certain changes occur in the world as agents of other changes, or rather, that certain changes are real consequences of other changes. Thus, by having explanations of causes, we can intervene in the world to induce by design certain changes in some course of phenomena. I consider scientific theories as logically and epistemically ordered wholeness of scientific knowledge in a realm of research that enables us to construct useful models of reality. After the discussion of different 'statement' views on theories and different interpretations of the theory-experience relation, I introduce the 'model' theoretical (non-statement) views on theories, especially the structural theory of science. The model views on theories transcend the realism-antirealism dualism by introducing models between theory and reality. Therefore, the question of scientific realism is not merely whether and, if so, which element(s), of scientific theories represent reality, but rather which conditions suffice to infer from a theory that one of its models truly expresses reality. In other words, under which conditions do models of a theory represent models of reality? Generally, models are just as much images of reality in a theory as they are images of a theory in reality. The actual problem of scientific realism doesn't originate in the observable-unobservable distinction (even if this distinction is otherwise theoretically loaded), but rather in ontological quandaries, of the sort especially encountered in modern physical theories such as quantum mechanics, relativity theory, and cosmological theories, although they are known to arise in other sciences as well. Scientific theories may be real, even if they describe nothing about the world. However, such theories can be given this status only if their formal, mathematical postulates exhibit some formal homology between reality and mathematical forms. The very fact that it is possible to describe reality with the help of mathematics even in cases, where every (representational, conceptual) language and thought fall short, would indicate that even mathematics itself is, in some sense, a part of reality, although 'outside' of the empirical world. It follows we have to distinguish the non-empirical reality and the empirical world. Formal objectivity and non-arbitrariness of theories do not emerge from either our language or thoughts, or from the world of facts, causes, and effects, but rather from non-empirical reality, which frames the world as some kind of objective transcendental condition for the possibility of the existence of the world, thoughts, and language. In the second part of the book, I consider the incorporation of scientific work and scientific knowledge into modern society and its relations to values and ethics. Production of scientific knowledge goes on as a parallel distributed process of a special kind of social (shared) cognition. The central part of this cognition is the presentation of parts of the world by scientific (theoretical) models which are warranted by the shared and public practice of justification, and in reflective acting in the world as it would. I consider various formal concepts of collective propositional knowledge (distributed, mutual, common knowledge). Scientific knowledge is an achievement to be ascribed to a scientific community or, indirectly, to all those who apply or expand it, rather than to individuals per se. It presents the continuous and partially planned production of non-trivial common knowledge of the research communities in modern societies. However, besides common knowledge, shared dispositional knowledge (knowledge how of scientific teams) and shared knowledge by acquiring (knowledge of fundamental similarities and differences in scientific teams) are parts of the real body of scientific knowledge. Scientific common knowledge implies the projection into idealized universal knowledge of all real or potentially rational and well-informed individuals. The respect of scientific knowledge thus presents the norm of rationality. Science as the intensively shared and socially distributed kind of cognition enters into modern society as a special kind of work (general work), as a productive force (social invention drive), and as a form of capital (human, social, cultural, informational, etc.). However any of these categories can adequately represent the socio-economical role of science in modern societies, and we are still waiting for a more adequate general concept that would theoretically cover the 'scientification' of modern societies. My thesis is that the most important general effect of scientification is (and will be) the growth of social processes that structurally resemble parallel distributed shared social cognition of science, and not so much the rise of the use of scientific knowledge and scientifically based information in all kinds of production. Scientification of modern societies inevitably includes a much stronger need for moral and social responsibility of scientists for their work and its effects in their local society and the global world. This need comes in contradiction with the quite large accepted idea (and even the 'norm') on value-free research. I discuss to some degree different arguments for the absence of non-epistemic values and ethical reasoning from science and find them non-convincing. The sound idea of value-free research lies in the need for the absence of categorical value statements from scientific explanations and theoretical deductions. Values and ethical judgment have, and sometimes they even have to play an important role in another part of research and its application in science or outside it. Not the value-loudness of research is a danger for science but its blind, non-reflective use. It leads to the ideologization and politicization of science. A similar danger lies in the purely value-neutrality of research. It leads to the instrumentalization of science to the technical or bureaucratic means for 'any' aims. (shrink)

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language (...) and the semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study. (shrink)

This paper introduces an axiomatisation for equational hybrid logic based on previous axiomatizations and natural deduction systems for propositional and first-order hybrid logic. Its soundness and completeness is discussed. This work is part of a broader research project on the development a general proof calculus for hybrid logics.

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in (...) the language of the system is deducible by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

continent. 1.2 (2011): 78-91. This article consists of three parts. First, I will review the major themes of Quentin Meillassoux’s After Finitude . Since some of my readers will have read this book and others not, I will try to strike a balance between clear summary and fresh critique. Second, I discuss an unpublished book by Meillassoux unfamiliar to all readers of this article, except those scant few that may have gone digging in the microfilm archives of the École normale (...) supérieure. The book in question is Meillassoux’s revised doctoral dissertation L’Inexistence divine (or The Divine Inexistence ), with its seemingly bizarre vision of a God who does not yet exist but might exist in the future. Without literally accepting this view, I will claim that it is philosophically interesting in ways that even a hardened sceptic might be able to appreciate. Third and finally, I will speculate on the possible future of Meillassoux’s speculative materialism itself. And here I mean its future development not by Meillassoux, but by those readers who might be inspired by his book. Plato could never have predicted the emergence of Aristotle’s philosophy, despite the obvious debt of the latter to the former. Nor could Descartes have predicted Spinoza and Leibniz, nor Kant the German Idealists, and neither could Husserl in 1901 have foreseen the later emergence of Heidegger. How are the works of interesting philosophers transformed by later thinkers of comparable importance? While it may seem that there are countless ways to do this, I think there are only two basic ways in which this happens: you can radicalize your predecessors, or you can reverse them. I will close this article with a few words about these two methods, and try to imagine how Meillassoux might be radicalized or reversed by some future admirer. My view is that the more important thinkers are, the easier they are to radicalize or reverse. This helps explain why the great philosophers of the West have so often appeared in clusters, succeeding one another at relatively brief intervals during periods of especial ferment. 1. After Finitude After Finitude is unusually short for such an influential book of philosophy: running to just 178 pages in the original French, and an even more compact 128 pages in the English version, despite the introduction of roughly eight pages of new material for the English edition. Rather than summarizing Meillassoux’s book in the order he intended, I will focus on six points that strike me as the pillars of his debut book. Along the way, I will offer a few criticisms as well. The first pillar of the book is Meillassoux’s own term “correlationism.”1 Although he introduces this term as the name for an enemy, it is striking that Meillassoux remains impressed by correlationism much more than his fellow speculative realists are. This continued appreciation for his great enemy influences the shape of his own ontology. Is there a world outside our thinking of it, or does the world consist entirely in being thought? Traditionally, this dispute between realism and idealism has been dismissed in continental philosophy as a “pseudo-problem,” in a strategy pioneered by Husserl and extended by Heidegger. We cannot be realists, since following Kant we have no direct access to things-in-themselves. But neither are we idealists, since the human being is always already outside itself, aiming at objects in intentional experience, deeply engaged with practical implements, or stationed in some particular world-disclosing mood. The centuries-old dispute between realism and idealism is dissolved by saying that we cannot think either real or ideal in isolation from the other. There is neither human without world nor world without human, but only a primordial correlation or rapport between the two. This is what “correlationism” means: philosophy trapped in a permanent meditation on the human-world correlate, trying to find the best model of the correlate: is it language, intentionality, embodiment, or some other form of correlation between human and world? Among other problems, this generates some friction between philosophy and the literal meaning of science. When cosmologists say that the universe originated 13.5 billion years ago, they do not mean “13.5 billion years ago for us ,” but literally 13.5 billion years ago, well before conscious life existed, and thus at a time when there was no such thing as a correlate. Meillassoux also coins the term “ancestrality” (10) for the reality that predated the correlate, and later expands this term to “dia-chronicity,” (112) to refer to events occurring after the extinction of human beings no less than to those occurring before we existed. Up to this point, Meillassoux’s focus on ancestral entities existing prior to consciousness might seem like a straightforward realist who wants to unmask correlationism as just another form of idealism. Yet Meillassoux also admires the correlationist maneuver, which can obviously be traced back to Kant. Unlike a thinker such as Whitehead, Meillassoux feels no nostalgia for the pre-critical realism that came before Kant: “we cannot but be heirs of Kantianism,” he says (29). What impresses Meillassoux about correlationism is something both simple and familiar. If we attempt to think a tree outside thought, this is itself a thought . Any form of realism which thinks it can simply and directly address the world the way it is fails to escape the correlational circle, since the attempt to think something outside of thought is itself nothing other than a thought, and thereby collapses back into the very human-world correlate that it pretends to escape. For Meillassoux this step, suggested by Kant but first refined by the ensuing figures of German Idealism, marks decisive forward progress in the history of philosophy that must not be abolished. Any attempt to break free from the correlate must first acknowledge its mighty intellectual power. Realist though he may seem, Meillassoux’s works are filled with praise of such figures as Fichte and Hegel, not of so-called “naïve realists.” It is also the case that for Meillassoux, not all correlationisms are the same. The second pillar of his book is a distinction between various positions that I have termed “Meillassoux’s Spectrum,” though of course he is never so immodest as to name it after himself. He distinguishes between at least six different possible positions, and perhaps we could add even subtler variations if we wished. But in its simplest form, Meillassoux’s Spectrum allows for just four basic outlooks on the question of realism vs. anti-realism. Three of these are easy to understand, since we have already been discussing them. At one extreme is so-called “naïve realism,” which holds that a world exists outside the mind, and that we can know this world. Meillassoux rejects this naïve realism as having been overthrown by Kant’s critical philosophy. At the other extreme is subjective idealism, in which nothing exists outside the mind. For to think a dog outside thought immediately turns it into a thought, and therefore there cannot be anything outside; the very notion is meaningless. In between these two is what we have called correlationism. And here comes a crucial moment for Meillassoux, since he distinguishes between the two forms of “weak” and “strong” correlationism, and chooses the strong form as the launching pad for his own philosophy. Weak correlationism is easy to explain, since we all know it from the philosophy of Kant. The things-in-themselves can be thought but not known. They certainly must exist, since there cannot be appearances without something that appears. And we can think about them, which idealism holds to be impossible. They are simply unknowable due to the finitude of human thought. Strong correlationism is the new position introduced by Meillassoux (though he sees it at work in numerous twentieth century thinkers), midway between weak correlationism and subjective idealism. The major difference between the three positions is as follows. Weak correlationism says: “The things-in-themselves exist, but we cannot know them.” The subjective idealist says: “This is a contradiction in terms, since when we think the things-in-themselves, we already turn them into thoughts.” But the strong correlationist says: “Just because ‘things-in-themselves’ is a meaningless notion does not mean that they cannot exist. No one has ever traveled to the world-in-itself and come back to make a report on it. Thus, the fact that we cannot think things-in-themselves without contradiction does not prove that they do not exist anyway. There may be things-in-themselves, we simply are not capable of thinking them without contradiction form within the correlational circle.” This step is crucial for Meillassoux, since strong correlationism is the position he attempts to radicalize into his own new standpoint: speculative materialism. As I see it, this step of the argument fails. Strong correlationism cannot avoid collapsing into subjective idealism, since the statements of the strong correlationist are rendered meaningless from within. All three of the other positions in the Spectrum make perfectly good sense even for those who disagree with them. The naïve realist says that things-in-themselves exist and we can know them; the meaning of this statement is clear. The weak correlationist can say that things-in-themselves exist but lie forever beyond our grasp; this too makes perfect sense, even though the German Idealists try to show a contradiction at work here. We can also understand the claim of the subjective idealist that to think anything outside thought turns it into a thought, and that for this reason we cannot think the unthought. The strong correlationist, alone among the four, speaks nonsense . This person says “I cannot think the unthought without turning it into a thought, and yet the unthought might exist anyway.” But notice that the final phrase “the unthought might exist anyway” is fruitless for this purpose. For we have already heard that to think any unthought turns it into a thought. But now the strong correlationist wants to do two incompatible things simultaneously with this unthought. On the one hand, he neutralizes the unthought by showing that it instantly changes into just another thought. But on the other hand, he wants to appeal to the unthought as a haunting residue that might exist outside thought, thereby undercutting the absolute status of the human-world correlate found in idealism. But this is impossible. If you accept the argument that thinking the unthought turns it into a thought, you cannot also add “but maybe there is something outside that prevents this conversion from being absolutely true,” because this “something outside” is immediately converted into nothing but a thought for us. In short, Meillassoux here seems to be offering a kind of Zen koan: his “strong correlationism” is reminiscent of the gateless gate, or the sound of one hand clapping, or the command to punch Hegel in the jaw when meeting him on the road. We cannot at the same time both destroy the realist challenge of the things-in-themselves in order to undercut realism and reintroduce that very realist sense in order to undercut idealism. In a world where everything is instantly converted into thought, we cannot claim that there might be something extra-mental anyway, because this “might be something” is itself converted into a thought by the same rules that condemned dogs, trees, and houses to the idealist prison. This brings us to the third pillar of Meillassoux’s argument, which is the key to all the rest: the necessity of contingency. His strategy is to transform our supposed ignorance of things-in-themselves into an absolute knowledge that they exist without reason, and that the laws of nature can change at any time for no reason at all. In this way the cautious agnosticism of Kantian philosophies is avoided, but so is the collapse of reality into thought as found in German Idealism. Meillassoux does try to prove the existence of things-in-themselves existing outside thought; he simply holds that they must be proven after passing through the rigors of the correlationist challenge, not just arbitrarily decreed to exist in the manner of naïve realism. As he puts it, “Everything could actually collapse: from trees to stars, from stars to laws, from physical laws to logical laws; and this not by virtue of some superior law whereby everything is destined to perish, but by virtue of the absence of any superior law capable of preserving anything, no matter what, from perishing” (53). If idealism thinks that the human-world correlate is absolute, for Meillassoux it is the facticity of the correlate that is absolute. He tries to show this with a nice brief dialogue between five separate characters (55-59) which is covered in detail in my forthcoming book,2 but which I will simplify here for reasons of time. In this simplified version, we first imagine a dogmatic realist arguing with a dogmatic idealist. The realist says that we can know the truth about the things-in-themselves; the idealist counters that we can only the truth about thought, since all statements about reality must be turned into statements concerning our thoughts about reality. Here the correlationist enters and proclaims that both of these positions are equally dogmatic. For although we have access to nothing but thoughts, we cannot be sure that these thoughts are all that exist; there could be a reality outside thought, there is simply no way to know for sure. And this latter position is the one that Meillassoux attempts to transform from an agnostic, skeptical point into an ontological claim about the contingency of everything. Consider it this way. How does the correlationist defeat the idealist? The idealist holds that the existence of anything outside thought is impossible. The correlationist, by contrast, holds that something might exist outside the human-world correlate. But this “something might” has to be an absolute possibility. It cannot mean that “something outside thought might exist for thought ,” because that is what the idealist already says. No, the correlationist must mean that something might exist outside thought quite independently of thought. In other words, the correlationist says that idealism might be wrong, and this means it is absolutely true that idealism might be wrong. Thus, correlationism is no longer just a skeptical position. It holds that all the possibilities of the world are absolute possibilities. We have absolute knowledge that any of the possibilities about the existence or non-existence of things-in-themselves might be true, and this means that correlationism flips into Meillassoux’s own position: speculative materialism. As Meillassoux sees it, there are only two options here. Option A is to absolutize the human-world correlate, which is what the idealist does: there absolutely cannot be anything outside thought. Option B, by contrast, is to absolutize the facticity of the correlate: its character of simply being given to us, without any inherent necessity. The correlationist cannot have it both ways by saying: “there absolutely might be something outside thought, yet maybe this is absolutely impossible.” In other words, once we escape dogmatism we can only be idealists or speculative materialists, not correlationists. The human-world correlate is merely a fact, not an absolute necessity. But this facticity itself cannot be merely factical: it must be absolute. Here Meillassoux coins the French neologism factualité , which has been suitably translated into the English neologism “factiality.” (7, 122-3) Factialty means that for everything that exists, it is absolutely possible that it might be otherwise, not just that we cannot know whether or not it might be otherwise. Just as Kant transformed philosophy into a meditation on the categories governing human finitude, Meillassoux wishes to turn philosophy into a meditation on the necessary conditions of factiality, which he calls “figures”—a new technical term for him. (80) One such figure is that the law of non-contradiction must be true, and for an unusual reason. Since everything is proven to be contingent, nothing that exists can be contradictory, for whatever is contradictory has no opposite into which it might be transformed, and thus contingency would be impossible.3 Another such figure is that there must be something rather than nothing: for since contingency exists, something must exist in order to be contingent. It is a daring high-wire act, one that sacrifices realism to the correlational circle in order to rebuild it from out of its own ashes. Some might conclude that the lack of reason in things is a byproduct of the ignorance of finite humans, Meillassoux is making precisely the opposite point. For in fact, the doctrine of finitude usually leads directly to belief in a hidden reason. The fact that it lies beyond human comprehension merely increases our belief in this arbitrarily chosen concealed ground. By defending anew the concept of absolute knowledge Meillassoux evacuates the world of everything hidden. The reason for things having no reason is not that the reason is hidden, but that no reason exists. Thus, even while insisting on the necessity of non-contradiction, he rejects the other Leibnizian principle: sufficient reason. Everything simply is what it is, in purely immanent form, without deeply hidden causes. Or as Meillassoux puts it: “There is nothing beneath or beyond the manifest gratuitousness of the given—nothing but the limitless and lawless power of its destruction, emergence, or persistence” (63).The world is a “hyper-chaos”(64). But this is not the same thing as flux. For the chaos of the world is such that stability might occur just as easily as constant, turbulent change. Let’s now digress a bit, and return to the question of ancestrality, which Meillassoux transforms later in the book into “dia-chronicity.” Correlationism holds that all talk of a world outside the correlate is immediately recuperated by the correlate. The phrase “13.5 billion years ago” becomes “13.5 billion years ago for us ,” and the phrase “the universe following the extinction of humans” becomes “the universe following the extinction of humans for humans .” But notice that whether we talk about the world before or after humans, in both cases it is time that is used to challenge the correlate. Meillassoux has no interest in challenges that might be posed by space. For example, what about a vase in a lonely country house that topples to the floor and smashes when no one is there to watch it? Isn’t this also a challenge to correlationism, no less than the Big Bang or the heat death of the universe long after humans have vanished? In an eight-page supplement to the English translation of After Finitude ,4 possibly in response to my own 2007 review of the French original,5 Meillassoux bluntly denies that space is of any relevance to the question. Spatial distance is a merely harmless challenge to the human-world correlate. After all, even though no one is there in the lonely country house to witness the shattering of the vase, we can say that had there been an observer , that observer would have witnessed the toppling and destruction of the vase. For this event still occurs in a world in which the human-world correlate already exists, whereas the diachronicity of events both before and after the existence of humans makes it impossible to say that had there been an observer they would have witnessed the Big Bang occurring in such and such a fashion. However, it seems to me that Meillassoux merely asserts that the temporal simultaneity of our existence with that of the vase in the lonely country house is enough to render it harmless. It is true that the house does not exist prior to the correlate, but nonetheless it exists outside the correlate, and that is enough to make the same challenge. It is difficult to see why the “had there been an observer” maneuver succeeds in the case of a vase in the countryside in April 2011 but fails in the case of the Big Bang. This is not just a matter of nitpicking Meillassoux’s argumentative style: the fact that he bases his argument on time has at least two important consequences for his position. For in the first place, even though Meillassoux insists that the laws of nature are absolutely contingent, this turns out to be true only in a temporal sense. That is to say, it is a paradoxical feature of Meillassoux’s philosophy that he does allow for the existence of laws of nature, and simply believes that they can change at any moment without reason. Within any given moment, laws of nature do exist. He never suggests that different parts of the universe can have different laws at the same time, nor does he have any interest in the laws of part/whole composition that take place within any given instant. Could it be the case that rather than being made of gold atoms, a small chunk of gold could be made of silver atoms, cotton, horses, or that this same small piece of gold could be made of gigantic vaults filled with even more gold? These are not topics that draw Meillassoux’s attention, since he is focused solely on how the laws of nature might change or endure from one moment to the next . Another implication for Meillassoux’s system is that his concept of things-in-themselves turns out to be to be inadequate. For when he proves that things-in-themselves can exist without humans, this turns out to be true only in a temporal sense as well. Namely, things-in-themselves existed ten billion years ago, and they will continue to exist after all humans have succeeded in exterminating themselves. However, being able to exist before our births and after our deaths is just one small part of what it means to be a thing-in-itself. The more important part is that even if a thing is sitting on a table right now, in front of me, even if I stroke it lovingly or press my face up against it directly, I am still dealing only with a phenomenal version of the thing; the thing-in-itself continues to withdraw from all access. Yet no such thing is acknowledged by Meillassoux. For him finitude is a disaster, and absolute knowledge is in fact possible. Meillassoux’s thing-in-itself exists in independence only of the human lifespan , not of human knowledge. The fifth pillar of Meillassoux’s argument is his use of Cantor’s transfinite mathematics to show that even if the laws of nature are contingent, they need not be unstable, and thus we cannot use the apparent stability of nature to disprove his metaphysics of absolute contingency. What Cantor showed is that there are different sizes of infinity, and that all these infinities cannot be totalized in a single infinite number of infinities. Meillassoux sees this as crucial, since it allows him to discredit any “probabilistic” argument against his theory. The probabilistic argument (as defended quite clearly by Jean-René Vernes)6 would say this: given that the laws of nature seem so stable, it it is extremely improbable that there is no hidden reason for their remaining so stable. As Meillassoux sees it, probability is of value only when we can index an accessible total of cases. These can even be infinite: for example, there are an infinite number of points where a rope can break when stretched tight, but this does not stop us from calculating probabilities for various sections of the rope to break. By contrast, there is no way to sum up the number of possible laws of nature. For here there is no way to totalize; we cannot stand outside of nature and calculate the possible number of laws so as to determine the probabilities that any one of them might change. Therefore, although we can speak of probability when dealing with intraworldly events such as elections, horse races, and coin-flips, we cannot use the words “probable” or “improbable” when describing alterations at the level of nature as a whole. Rather than commenting on the validity of this argument and its use of Cantor, let me simply note that it once again creates a dualistic ontology. We already saw that Meillassoux treats time differently from space. In analogous fashion, he now treats the level of world differently from that of intraworldly events. The emergence of worlds is purely contingent and virtual and governed by no probability at all, while events within the world necessarily follow laws (even if these laws can change at any moment without reason), and thus their probabilities can be calculated. It is a strategy deeply reminiscent of Badiou’s (2005) own dualism between the normal “state of the situation” and the rare and intermittent “event.” The sixth and final pillar of Meillassoux’s book can be dealt with briefly, since we have already touched on it elsewhere. It comes at the very beginning of the book, when Meillassoux says that we must revive the distinction between primary and secondary qualities, and that the primary qualities are the ones that can be mathematized. He admits that he has not yet published a proof of this idea, though in fact it is already known as one of his primary doctrines. And here we encounter the familiar problem with Meillassoux’s inadequate conception of things-in-themselves. “Primary qualities” refers to those qualities that a thing has independently of its relations with us or anything else. But if the primary qualities can be mathematized, this means that they are not entirely independent of us, since our knowledge can get right to the bottom of them. The mathematized qualities of things are independent of us only in Meillassoux’s sense that they will still have those qualities even when all humans are dead. But to repeat, autonomy from the human lifespan is not the same as autonomy from human access. Here once more Meillassoux is concerned only with independence from the human-world correlate across time, not in any given instant. 2. L’Inexistence divine In 1997, the same year in which he turned thirty years old, Meillassoux earned his doctorate at the École normale supériuere with a brazen dissertation entitled L’Inexistence divine ( The Divine Inexistence ). The work was substantially revised in 2003. But even then, with typical fastidiousness, Meillassoux decided that the work was not yet ready for press. It has now been scrapped in favor of some future, multi-volume work bearing the same title. While writing my book on Meillassoux for Edinburgh University Press, I was permitted to translate excerpts from this unpublished work for use as an appendix in my own book; in total, the appendix contains approximately twenty percent of Meillassoux’s 2003 manuscript, the first time any of it will be published in any language. Nonetheless, a portion of the argument was already tested in the article “Spectral Dilemma,” published in English in the journal Collapse (2008: 261-75). There the philosophical motives for the virtual God are already made clear. What troubles us most are early deaths, brutal deaths, deaths of especial injustice– the sorts of deaths in which the brutal twentieth century was so abundant. And here, neither the atheist nor the believer can help us. The atheist can offer nothing but a sad and cynical resignation when reflecting on the victims of these terrible crimes. The believer does little better, being unable to explain how God could have allowed such things to happen, due to the famous intractability of the problem of evil. The solution offered to this dilemma by Meillassoux is bold, and all the more so given that he emerges from such a deeply Leftist, materialist, and unreligious background. His solution is that God does not yet exist, and therefore is not blameworthy for these catastrophes. Given that everything is contingent in Meillassoux’s philosophy, this God and divine justice might never exist, but they can at least exist as an object of hope. Let’s begin by jumping to the end of L’Inexistence divine , where the alternatives are laid out so nicely. There are four basic attitudes that humans can have towards God, Meillassoux says. First, we can believe in God because he exists. This is the classical theist attitude, rejected for the simple reason that it would be amoral and blasphemous to believe in a God who allows children to be eaten by dogs, to use Dostoevsky’s example. Second, we can disbelieve in God because he does not exist: the classical atheist attitude. But this leads to sadness, cynicism, and a sneering contempt for the greatness of human capacity. The third option, rather more complex, is to disbelieve in God because he does exist: in other words, to exist in rebellion against God as the one who must be blamed for the evils of the earth. The examples here might range from Lucifer himself, to the more human figure of Captain Ahab in Melville’s Moby-Dick , to Werner Herzog’s even more recent catchphrase, “Every man for himself, and God against all.” That leaves only the fourth option: believing in God because he does not exist. Meillassoux closes his book by saying that the fourth option has now been tried (namely, in the course of his own book), and that now that all four have been specified, we must choose. The first reaction to this theory of the inexistent God will be laughter. Few readers will ever be literally convinced by it, and probably none will immediately be convinced by it. But if we ask ourselves why we laugh, the answer is because it sounds so improbable that an inexistent God might suddenly emerge and resurrect the dead. It obviously sounds more like a gullible theology than a rigorous piece of philosophical work. Yet two things need to be kept in mind. First, Meillassoux’s theories are hardly more unlikely than those of great philosophers of the past such as Plato, Plotinus, Avicenna, Malebranche, Spinoza, Leibniz, Nietzsche, or Whitehead. We read the great philosophers not because their systems are plausible in commonsense terms that can be measured by the laws of probability. Instead, we read them precisely because they shatter the existing framework of common sense and open up new window on the universe. Second, and even more importantly, Meillassoux has already rejected probability as a valid measuring stick in philosophy. Or rather, he accepts probability in the intra-worldly realm (where it is linked with potentiality), and rejects it at the level of the world itself (where potentiality is replaced with what he calls virtuality). The virtual God can appear at any moment for no reason at all, just as any other new configuration of laws of nature can appear: in a manner that the laws of probability cannot calculate. Responding to those who might ridicule the idea of a sudden emergence of God and a resurrection of the dead, Meillassoux cites Pascal, who asserts that the resurrection of the dead would be far less incredible than the fact that we were born in the first place. This shifts philosophy onto new ground. Rather than concerning ourselves with what is likely to happen in the world as we know it, we focus instead on the most important things that could happen. For this reason, the expected objection that a virtual God is no more likely to appear than a virtual unicorn or a virtual flying spaghetti monster misses the point. Unicorns and spaghetti monsters could also appear, just like any other non-contradictory thing. But these would just be novel bizarre entities among others, not the heralds of completely new worlds. For Meillassoux, the emergence of matter, life, and thought have been the three truly amazing advents of the world so far, each of them dependent on the advent(s) preceding them. As he sees it, there can be no greater intraworldly entity than the human beings who already exist, since nothing in the world is better than the absolute knowledge of which humans alone are capable. This means that the next great advent must be something that perfects human beings rather than superseding them. And this can only be the world of justice, in which the dead are resurrected and their horrible deaths partially cancelled (Meillassoux never considers the possibility of a God who would literally erase the pre-divine past so that it never happened at all). The only immortality worth having is an immortality of this life, not an existence in some ill-defined afterworld. Human existence, he holds, must always be governed by a “symbol” that gives us the “immanent and comprehensible inscription of values in a world.” And just as cosmic history made the three great contingent leaps of matter, life, and thought, with a leap to justice as the only one still to come, a similar structure occurs within human culture and its symbols, which consist so far of the cosmological, naturalistic, and historical symbols, with a “factial” symbol still to come. We can review each of these symbols briefly. The cosmological symbol refers to the ancient dualism between the terrestrial and celestial spheres. Here below everything is conflict, corruption, and decay; but in the heavens nothing is perishable, all movement is circular, and everything is arranged in mutual harmony. This symbol is ended by modern physics when Galileo discovers such blemishes as sunspots and craters on the moon, and when Newton integrates both celestial and terrestrial movement into a single gravitational law. Next comes the naturalistic or romantic symbol, in which perfection comes not from the sky but from nature itself. The world is filled with pretty flowers (Meillassoux claims that the ancients never discussed the beauty of flowers until Plotinus in the third century) and with living creatures naturally moved by pity, at least until society corrupts them. This symbol collapses in the face of reality as we know it, since pity is no more common than war, corruption, and violence. This brings us to the historical symbol, which only now is passing away. Bad things may happen, but history has an inner logic of its own, such that everything works out in the end. The ultimate form of the historical symbol is the economic symbol, whether in a Marxist or neo-liberal form. Just as the Marxist holds that the inner economic logic of the capitalists will inexorably lead them to self-destruction, the neo-liberal assumes that the sum total of individual selfish actions will lead, in the long run, to the greatest possible good. We worship the economy and let it guide history for us, just as the ancients worshipped celestial bodies and held them to be free from blemish. The final remaining symbol is the factial symbol, which Meillassoux hopes will now emerge. Factiality, we recall, is his term for the absolute contingency of everything that exists. Once we have grasped this absolute contingency, we are free to expect the dramatic advent of the coming fourth World: the world of justice, inaugurated by a virtual God and even mediated by a messianic human figure. There is the added feature, however, that this messiah must abandon all claims to special status once the messianic realm of justice is achieved. The messianic figure will then be no more special than any person on the street, since a reign of human equality will have arisen. Although this focus on human being might seem like a return to standard humanism, Meillassoux holds that human pre-eminence has never truly been maintained. Previously, humans have been treated as special only because they contemplate the Good, because they resemble their omnipotent creator, or because they happen to be the temporary victors in a cruel Darwinian death-match between millions of living species. For Meillassoux, by contrast, humans have value because they know the eternal. But it is not the eternal that is important, since this merely represents the blind, anonymous contingency of each thing. What is important is not knowledge of the eternal , but knowledge of the eternal. We should not admire Prometheus for stealing fire from the gods; Prometheus is simply as bad as all the gods, no matter how much he increased our power. Feuerbach and Marx were wrong to say that God is a projection of the human essence, since for Meillassoux the usual concept of God represents the degradation of the human essence. If the traditional God was allowed to inflict plagues and tsunamis on the human race, the Promethean human of the twentieth century simply assumes the right to inflict death camps and atomic fireballs instead. In this respect, we have simply begun to imitate the degradation of humanity that was formerly invested in an omnipotent and arbitrary God. In response to charges that absolute contingency might lead to political quietism, Meillassoux counters that the World of justice would mean nothing unless we had already hoped for it beforehand. A World of justice that came along at random would merely be an improved third World of thought: indeed, a perfect one. But it would have satisfied no craving, and would therefore have no redemptive power. For this reason, we must actively hope for the fourth World of justice for such a fourth World ever to arise. Not only justice, but beauty is dependent on such hope: for Meillassoux, who is here somewhat dependent on Kant, beauty means an accord between our human symbolization and the actual world, which could never be present in a World of the blessed any more than justice could. And just as a messianic figure is needed to incarnate our hope and then abandon power once the World of justice is realized, it is the figure of the child whose fragile contingency shows us a dignity and a demand for justice beyond all power. 3. Meillassoux Radicalized or Reversed Given the promising reception of Meillassoux’s first book, it would not be groundless to engage in early speculation about what it might take to earn him a place in the history of philosophy. Maybe this will never happen—who knows?—but quite possibly it will: his lucid argumentative methods and sheer philosophical imagination at least make him a good candidate to be read well into the future, especially following further elaboration in print of his mature system. Philosophy is often practiced as thought it were nothing more than the amassing of “knockdown arguments.” But this is no more insightful than saying that good architecture is the amassing of steal beams. It is true that poorly constructed building cannot stand for long, but sound construction is merely the first, indispensable step in building. In fact, I am inclined to say that what really makes a philosopher important is not being right, but being wrong . I mean this in a very specific sense. I once heard the interesting remark about twentieth century culture that “you have to remember that the sixties really happened in the seventies.” That is to say, it was in the 1970’s rather than the more honored 1960’s that civil rights, free love, long hair, and the rock and roll drug culture really took root. With respect to the history of philosophy, we might just as easily say: “you have to remember that Plato really happened in Aristotle,” that “Kant really happened in Hegel” or “Hume really happened in Kant,” or that “Husserl’s phenomenology first achieved its truth in Heidegger.” One becomes an important philosopher not by being right, but by attracting rebellious admirers who tell you that you are wrong , even as their own careers silently orbit around your own. To recruit faithful disciples may be comforting and flattering, but the greatest thinkers have generally had to experience refutation at the hands of their most talented heirs. For this reason, I would propose that we size up the magnitude of living thinkers not by deciding how many times they are right and wrong, but by asking instead: who would take the trouble to refute this author? For this reason I do not ask: “Is Meillassoux right?”, since I do not believe in the virtual God myself, nor am I convinced by any important aspect of Meillassoux’s philosophy. Instead, I ask if there are interesting ways to overturn him. Only by being overturned, by no longer remaining a contemporary, does one become a classic. Let’s begin with a simple model of refutation, which can be refined further at a later date once the basic point is established. One kind of refutation simply consists in saying: “This author is a complete idiot.” The refuter now walks away in celebration, and no link between the present and the future is built; all is reduced to rubble. But this sort of mediocre triumphalism is generally practiced by those who achieve little of their own, and is not especially interesting. Much more interesting is the sort of refutation that does not take its target to be a complete idiot. I would like to suggest that there are just two basic ways in which this can be done: radicalization and reversal . It has not escaped my notice that this is a fairly good match for the Deleuzian distinction between irony and humor. Whereas irony critiques and adopts the opposite principle of what it attacks, humor accepts what it confronts but pushes it into highly exaggerated form. The ironist is like the worker who sows chaos by rebeling and contradicting the boss, while the humorist is like the worker who follows orders to an absurdly literal degree, with equally chaotic results. Let’s start with a few examples. In Aristotle’s treatment of Plato, and Heidegger’s of Husserl, we find reversal. Plato’s eidei are transformed by Aristotle into mere secondary substances, and the individual worldly things despised by Plato become what is primary. For Husserl what is primary is whatever is present to consciousness, while for Heidegger this is precisely what is secondary, since the primary stuff of the world withdraws from any form of presence at all. As for radicalization, it is most easily found in the transformation of Kant by German Idealism: “Kant was right to wall off the things-in-themselves from human access, and simply should have realized that the thought of the Ding an sich is also a thought, and thereby the noumena are just special cases of the phenomena,” with much following from this discovery. It would also be easy to read Spinoza as a radicalizer of Descartes, and Berkeley and Hume as radicalized versions of Locke. Perhaps the distinction is now sufficiently clear. Admiring refutations are not those that say “Professor X is an idiot,” which is merely the flip side of the eager disciple’s fruitless “Professor X got everything right.” Instead, it will be some variant of one of the following two options: “Professor X is important, but got it backwards,” or “Professor X is important, but didn’t push things far enough.” In the history of philosophy these two latter cases have often been painful in purely human terms: Aristotle expresses sadness at refuting Plato, Kant is openly annoyed at Fichte, and Husserl feels betrayed and used by Heidegger. Rude handling from later figures almost seems to be the sine qua non of being a great philosopher. Now, it has already been claimed that Meillassoux is an emerging philosopher of the first importance, and by no less a figure than Alain Badiou: “It would be no exaggeration to say that Quentin Meillassoux has opened up a new path in the history of philosophy…” (Preface, vii). But rather than taking Badiou’s word for it, or rejecting his word, we might experiment by asking how Meillassoux could be radicalized or reversed. Are there interesting ways of doing this that might launch whole new schools of philosophy, unexpected or even condemned by Meillassoux himself? While no one can see the future, the present is poor when it is not riddled with virtual futures. The relation between philosophers and their predecessors and successors is always somewhat complicated, of course. But generally there is one central divergence at stake, which might be taken as the key to all the others. On this basis we could say that new thinkers primarily radicalize or primarily reverse the main ideas of their chief philosophical forerunner. There may be specific historical conditions and perhaps even personality traits connected with these two types, but this question can be left aside for now. More important for us is that radicalizers will generally be followed by reversers, and vice versa. Consider the textbook example of a reversal in the history of philosophy: Kant’s Copernican Revolution, which inverts the so-called dogmatic tradition that addresses the world itself, and makes the world revolve instead around the conditions by which it is known. While it is not completely impossible that Kant’s successors might have re-reversed this principle back into a new and stronger dogmatic realism, conditions were premature for such a move. Anyone doing this too early would likely have been an angry anti-Kantian reactionary rather than an original thinker in command of a genuinely new realist principle. The far more likely outcome is the one that actually happened: Kant’s reversal of his predecessors was viewed as incomplete, or as retaining lamentable bits of the traditional view, which despite his admirable breakthrough he was unable to shake off. This was the view of German Idealism, anyway. In similar fashion, Spinoza could also be viewed as a radicalizer of Descartes, who is equally accused of preserving various Scholastic dogmas in an otherwise radical project of philosophical reversal. The point is this: reversals in the history of thought tend to be followed soon thereafter by radicalizations of those reversals. The same may hold true in reverse: radicalizations might generally be followed by reversals, given that it is not always possible to be more radical than the radicals have already been. Consider the case of Husserl, who radicalizes Brentano’s early vagueness about what lies beyond immanent objectivity, and Twardowski’s assertion that there must be an external object lying outside the intentional content, by collapsing everything into the intentional sphere: there is no difference between the Berlin in my consciousness and the actual Berlin that is home to millions of people. It is difficult to see how one could be even more radical than Husserl’s idealist turn here. And thus the road is paved to Heidegger’s reversal of classical phenomenology, in which the key point is what lies deeper than any presence to consciousness: the Sein whose power and obscurity cannot be made exhaustively present, but only sends itself in historical epochs. In similar fashion we might also read Leibniz as a reverser of Spinoza’s radicalization, retrieving a strong sense of individual substance and a certain validity of what the Scholastics had said. Returning to Meillassoux, we might ask which kind of philosopher he is: a radicalizer or a reverser? At present, Meillassoux looks to me like a radicalizer (though for now his future remains shrouded in mist). He takes the correlationist tradition, which allows us to speak only of the relation between human and world, and tries to raise it into an even more extreme claim about the absolute contingency of everything. But whereas German Idealism did this by trying to collapse the distinction between thought and world entirely into the “thought” side, Meillassoux does it by trying to shift the non-absolute contingency of the thought-world correlate from epistemology to ontology. It is no longer a question of the inability of human knowledge to know what lies outside the correlate, but the inability of reality itself to be rooted in any definite laws. Furthermore, if we look at the various features of Meillassoux’s philosophy identified earlier tonight, all but one are already so radical that there is no obvious way to push them further. The one exception would be his claim that the world as a whole can change for no reason at any moment, coupled with the inconsistent claim that within a given world there are laws of nature that everything must follow. If gravitational attraction between all masses is a current law in our world, then for Meillassoux there can be no exceptions to this law for as long as it remains in force. A toppled vase will fall to the floor every time for sure,unless there is a cosmic change by which the laws of nature as a whole have altered. (This is reminiscent of the late medieval distinction between the absolute and ordained power of God, according to which God has the power to set or change the laws of nature, but not to contravene those laws locally once they are set.) On this point, to radicalize Meillassoux would simply be to say: there are no laws of nature even in the local sense. Everything that happens, even in the world here and now, is purely contingent and not governed by even a trace of law. And while this would be a more consistent development of Meillassoux’s thoughts on contingency, it is difficult to see how it could lead to a new philosophy. Instead, the admiring successors of Meillassoux are more likely to reverse one of his already sufficiently radical points. At least four candidates come to mind: *First, we have seen that Meillassoux thinks correlationism is challenged by a time before or after consciousness, but not by a space lying outside it. Perhaps this could be reversed into saying that spatial exteriority is the really crucial point. The arguments on this point are perhaps the least convincing in After Finitude (and do not even occur in the original French edition), and therefore it might be a candidate for the “blind spot” of which no philosopher is ever free. *Second, Meillassoux uses Cantor to claim that the contingency of laws of nature would not entail that they are unstable. A successor of Meillassoux might claim that it does make them unstable, and celebrate this fact. This person would then have to explain why common sense seems to encounter a relatively stable world despite its truly rampant instability. Whereas Meillassoux’s problem is to show how stability might exist despite contingency, this successor’s problem would be slightly different: to show why actual, full-blown instability might have the appearance of stability. *Third, Meillassoux claims that the primary qualities of things are those that can be mathematized. He might be reversed by a successor who says the opposite: the mathematizable qualities are the secondary ones, and the primary ones are those that elude symbolic formulation. While this is a perfectly valid possible objection to Meillassoux, it is one that is made in advance by some of his predecessors and is still made by some of his peers, making it less interesting for futurology than some of his other points. *Fourth and finally, whereas Meillassoux claims that God does not exist but might exist in the future, a successor might argue even more bizarrely that God has always existed but might vanish in the future. Let’s arbitrarily select the first of these possibilities, and imagine briefly where it might lead, if pursued in the future by admiring detractors. Meillassoux comes from the circle of Badiou, and some of Badiou’s most ardent admirers are found in Latin America. So, let’s imagine that towards mid-century some ingenious reversers of Meillassoux emerge in that portion of the Spanish-speaking world. Just for fun, let’s call them Castro and Chávez. And in order to avoid any confusion with the present-day politicians of those names, we will stipulate that Meillassoux’s great successors are both women. The philosopher Castro (we will suppose she comes from Peru) reverses Meillassoux’s argument that the ancestral or diachronic are what most threaten the human-world correlate. Instead, she claims that the diachronic does not threaten the correlate at all, and that we must instead look at space as what ruins the correlate and demands a strange new realism. What would such a philosophy look like? In order to determine this, we might ask what price Meillassoux pays for doing it the opposite way. As I see it, he pays in two separate ways. One is that laws of nature for him are contingent over time . The laws of nature apply to the universe as a whole at any given moment, and would be changed globally if they are ever changed at all. The second price he pays is that Meillassoux has no mereology , or theory of parts and wholes. Everything for him is on the level of the given, or immanent in experience, with the sole proviso that the laws governing this immanence might change without notice at any given moment. In reversing Meillassoux, Castro makes the following claims in the preface to her stunning debut book of 2045, The Cosmos and its Neighborhoods , rapidly translated from Spanish into all the languages of the world: Despite his brilliant analysis of the contingency of laws of nature over time, Meillassoux gets two important assumptions wrong. First, he allows for only one set of contingent laws to govern nature as a whole. Second, he allows laws to govern only the world that is immanent in experience, and thereby fails to explore the contingency among part-whole relations. In this book I will argue, first, that the laws of nature vary in any given instant between one region of the universe and the next; and second, that the world is made up of layers of parts and wholes that are also contingent with respect to one another. Those are the words of Castro. This may sound like a hopeless free-for-all of chaos, yet the book somehow succeeds in drawing some compelling deductions about how laws must vary from one place or level in the world to the next. Trapped in the limited horizon of 2011, and not yet inspired by the heavily balkanized political and technological situation of 2050 that somehow lends additional credence to Castro’s vision, we can only vaguely grasp what such a philosophy might look like. After this reversal of Meillassoux by Castro, the usual pattern leads us to expect a radicalization by Chávez, a young Argentine student of Castro. How could the already strange theories of Castro be radicalized? Perhaps as follows, in a disturbing new book entitled The Implosion of the Neighborhoods , which argues as follows: Castro was right to shift the Meillassouxian framework of contingency from time to space. However, in this respect she retained a surprisingly traditional opposition between the two. In this book I will show that time and space collapse into one another. This may sound too much like the discredited four-dimensional block universe of twentieth century physics and philosophy. However, the four-dimensional universe is a model biased in favor of space, merely adding an extra dimension to the commonsense spatial continuum while stipulating that the serial passage of time is an illusion. In this book I will argue instead for a one-dimensional space-time modeled after our experience of time, in which there is no simultaneous co-existence at all between different parts of the universe, or ‘neighborhoods’ as my esteemed teacher Castro has called them. Instead, the various portions of the universe link to one another by succession rather than by coexistence. Buenos Aires, New York, and Amsterdam do not exist simultaneously in the same landscape, but one after the other in the mind of some observer, and this observer can only be an observer much larger than any human. Against Meillassoux’s notion of a virtual God that does not exist now but might exist in the future, I will argue for an actual God that surveys the universe in sequence, thereby generating the illusion of spatial diversity and even the illusion of individual minds located within that diversity. Once this divine observer dies, the universe as a whole must perish. Again, these ideas are so bizarre that we of 2011 can barely comprehend them, just as Aristotle would have had a difficult time grasping the theories of Descartes. We could then perhaps imagine a further reversal of this theory, emanating from the intellectually resurgent Philippines of the twenty-second century. The Filipino School might argue that the universe is already dead, given the collapse of its spatial richness into the serial observations of a flimsy and mortal God. The virtual universe does not yet exist, but might exist in fully spatial form in the future, and this would require the death of God and the resulting liberation of God’s succession of images as independent, spatially situated realities. With a bit of sharpening, we might be able to make all of these imaginary thinkers more intuitively clear. Along with the history of philosophy, there might arise a new discipline generating imaginary futures for philosophy. The richness of Meillassoux’s system comes not from the fact that he is plausibly right about so many things, but because his philosophy offers such a treasury of bold statements ripe for being radicalized or reversed. He is a rich target for many still-unborn intellectual heirs, and this is what gives him the chance to be an important figure. NOTES 1. Quentin Meillassoux, After Finitude . Trans. R Brassier. (London: Continuum, 2008.) Page 5. The word “correlationism” does not appear in his doctoral thesis. As Meillassoux informed me in an email of February 8, 2011, he first coined this term in 2003 or 2004, while editing for publication a lecture he had given at the École normale supérieure on a day devoted to the theme of “Philosophy and Mathematics,” an event including Alain Badiou as one of the participants. 2. Graham Harman, Quentin Meillassoux: Philosophy in the Making . (Edinburgh: Edinburgh University Press, forthcoming 2011). 3. In an email of December 6, 2010, Meillassoux clarifies that in After Finitude he only deduces the impossibility of a “universal contradiction,” not of a determinate contradiction. In the same email he suggests that he can also prove the latter, though the proof is somewhat lengthier than the one found in After Finitude . 4. After Finitude (18-26), in the passage falling between the two sets of triple asterisks. These pages were sent by Meillassoux to translator Ray Brassier (in French) during the translation process, and do not appear in the original French version of the book. 5. Graham Harman, “Quentin Meillassoux: A New French Philosopher.” Philosophy Today 51.1 (2007): Pages 104-117. The passage where I raise the question of space can be found in the first column of page 107. 6. Vernes is first cited on p. 95 of After Finitude . See Jean-René Vernes, Critique de la raison aléatoire, ou Descartes contra Kant . (Paris: Aubier, 1982).  . (shrink)

Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, (...) showing that any arithmetical theorem proved in analytic number theory is a theorem in Peano's arithmetic. In doing so, the author applies Gentzen's cut elimination theorem. Although the results of Part One may be regarded as straightforward consequences of the spectral theorem in function analysis, the use of Boolean- valued models makes explicit and precise analogies used by analysts to lift results from ordinary analysis to operators on a Hilbert space. Essentially expository in nature, Part Two yields a general method for showing that analytic proofs of theorems in number theory can be replaced by elementary proofs. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion (...) follows from the premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on: admissible or permissible inference rules the derivability of the admissible inference rules the structural completeness of logics the bases for admissible and valid inference rules. There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered. The book is basically (...) self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included. The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook. (shrink)

This paper studies a propositional logic which is obtained by interpreting implication as formal provability. It is also the logic of finite irreflexive Kripke Models.A Kripke Model completeness theorem is given and several completeness theorems for interpretations into Provability Logic and Peano Arithmetic.

For each truth-functionally complete set of connectives, we construct a sound and complete natural deduction system containing no axioms and the smallest possible number of inference rules, namely one.

The paper discusses a four-valued propositional logic FOUR≤, similar to Belnap's logic, which can be used to describe incomplete or inconsistent knowledge. In addition to the two classical logical values tt, ff, FOUR≤ features also two nonclassical values: ⊥, representing incomplete information, and ⊤, representing inconsistency. The nonclassical values are incomparable, and together with the classical ones they form a diamond-shaped lattice L4 known from Belnap's logic, which underlies the semantics of FOUR≤. The set of (...) connectives contains those of Belnap's logic, together with an additional operator representing the lattice order, which decisively increases the expressive power of the logic. In the paper we develop a complete decompositional deduction system in Rasiowa-Sikorski style for FOUR≤. (shrink)

George Boole published his account on hypotheticals in his pamphlet The Mathematical Analysis of Logic in 1847. Hypothetical deductions were not as developed as categorical ones by Boole’s time. It was still common practice to reduce hypotheticals to categoricals. Boole innovated by proposing an algebraic method to derive (equations expressing) the conclusions of hypotheticals. He had developed a calculus of classes for categoricals in his first pamphlet chapters and seemingly intended extending it to hypotheticals. Nonetheless, propositions can be only (...) true or false. In contradistinction with his approach for categorical propositions, here it was not possible to consider things or classes of things but only the cases and conjunctures of circumstances in which the propositions are true. Some criticisms have been raised concerning Boole’s view of logic as the solution of algebraic equations. In order to circumvent such criticisms, we construct a deductive system based on Boole's algebraic treatment and apply it to deductions of those hypotheticals analyzed by Boole. We found that Barbara translated in our notation is useful to eliminate symbols in such deductions. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to hypotheticals in The Mathematical Analysis of Logic. (shrink)

The Logic of Causation: Definition, Induction and Deduction of Deterministic Causality is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part (...) of a larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004). The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts). Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians. The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction. Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses. This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation. Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – but also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated. In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions. This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed. (shrink)

According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order (...) matrix A is Fregean if, for any subset X of A, the set of all pairs a,b such that X{a} and X{b} have the same closure is a congruence relation on A. Hence a deductive system is Fregean if interderivability is compositional. The logics intermediate between the classical and intuitionistic propositional calculi are the paradigms for Fregean logics. Normal modal logics are non-Fregean while quasi-normal modal logics are generally Fregean.The main results of the paper: Fregean deductive systems that either have the deduction theorem, or are protoalgebraic and have conjunction, are completely characterized. They are essentially the intermediate logics, possibly with additional connectives. All the full matrix models of a protoalgebraic Fregean deductive system are Fregean, and, conversely, the deductive system determined by any class of Fregean second-order matrices is Fregean. The latter result is used to construct an example of a protoalgebraic Fregean deductive system that is not strongly algebraizable. (shrink)

Extant semantic theories for languages containing vague expressions violate intuition by delivering the same verdict on two principles of classical propositional logic: the law of noncontradiction and the law of excluded middle. Supervaluational treatments render both valid; many-Valued treatments, Neither. The core of this paper presents a natural deduction system, Sound and complete with respect to a 'mixed' semantics which validates the law of noncontradiction but not the law of excluded middle.

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the (...) standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this. (shrink)

A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as a multi-graph (m-graph) where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is a multi-graph (m-graph) where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is (...) an m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. (shrink)

Solovay proved the arithmetical completeness theorem for the system GL of propositional modal logic of provability. Montagna proved that this completeness does not hold for a natural extension QGL of GL to the predicate modal logic. Let Th(QGL) be the set of all theorems of QGL, Fr(QGL) be the set of all formulas valid in all transitive and conversely well-founded Kripke frames, and let PL(T) be the set of all predicate modal formulas provable in Tfor any arithmetical (...) interpretation. Montagna’s results are described as Th(QGL) ⊊ (Fr(QGL), PL(PA) ⊈ Fr(QGL), and Th(QGL) ⊊ PL(PA). In this paper, we prove the following three theorems: (1) Fr(QGL) ⊈ PL(T) for any Σ1-sound recursively enumerable extension T of I Σ1, (2) PL(T) ⊈ Fr(QGL) for any recursively enumerable S1755020312000275_inline1 A -theory T extending I Σ1, and (3) Th(QGL) ⊊ Fr(QGL) ∩ PL(T) for any recursively enumerable S1755020312000275_inline2 A -theory T extending I Σ2. To prove these theorems, we use iterated consistency assertions and nonstandard models of arithmetic, and we improve Artemov’s lemma which is used to prove Vardanyan’s theorem on the Π0 2-completeness of PL(T). (shrink)

Which intermediate propositional logics can prove their own completeness? I call a logic reflexive if a second-order metatheory of arithmetic created from the logic is sufficient to prove the completeness of the original logic. Given the collection of intermediate propositional logics, I prove that the reflexive logics are exactly those that are at least as strong as testability logic, that is, intuitionistic logic plus the scheme $\neg φ ∨ \neg\neg φ. I show that (...) this result holds regardless of whether Tarskian or Kripke semantics is used in the definition of completeness. I also show that the operation of creating a second-order metatheory is injective, thereby insuring that I am actually considering each logic independently. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules (...) for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.

This book on mathematics, logic, and experience by the doyen of the Vienna Circle is an almost completely revised reprint of the volume originally published in 1957. The book is organized into four chapters, the first of which treats "mathematics and logic" and is one page in length. The other three chapters, "Logic and Experience," "Mathematics and Experience," and "Conventionalism and Its Refutation," share equally the remaining pages of the book. The validity of the principle of (...) class='Hi'>logic, Kraft argues, is the indispensable condition for order and knowledge. Consequently, logic cannot be an abstraction from the natural laws and cannot contain the formalized laws of reality. Rather logic has to be--and this is in accordance with Port-Royal and Wundt--a normative science. Kraft reminds us that only propositions which occur in predicate logic or class logic may have a truth value. Kraft therefore characterizes the principle of identity and the principle of contradiction as rules which are prelogical in nature, since they are the basis of order and serve as the norms of correct thinking. The validity of these prelogical rules rests on the fact that they constitute order and must consequently be accepted as necessary. The conception of logic as normative science puts the author in opposition to the view held by the Vienna Circle, namely, that the application of logic does not contain any problem. In Kraft's position, on the contrary, the necessity arises of showing the conditions under which logic is applicable. In respect of the applicability of logic, the nature of the reality serving as material in this application makes a difference; for the logical relation between the universal and the particular finds its fulfillment and its determination with regard to the content provided by reality. The condition which logic requires for its application is that the "manifold" be capable of being ordered and that it thereby make universality possible. In a similar way Kraft attempts to work out the conditions for the applicability of arithmetic and geometry, which like logic are independent of experience in respect to their validity. According to the author, arithmetic can be applied insofar as operative relations between numbers correspond to classes of sets. Geometry is applicable insofar as the coordination of certain empirical phenomena with geometrical elements and relations is possible. The applicability of geometry consequently presupposes the validity of certain natural laws, thereby denying the thesis of conventionalism that these can be arbitrarily chosen. Kraft argues convincingly in the third part of his book for his own thesis by showing the shortcomings of Dingler's procedure of exhaustion.--H. H. (shrink)

I address a type of circularity threat that arises for the view that we employ general basic logical principles in deductive reasoning. This type of threat has been used to argue that whatever knowing such principles is, it cannot be a fully cognitive or propositional state, otherwise deductive reasoning would not be possible. I look at two versions of the circularity threat and answer them in a way that both challenges the view that we need to (...) apply general logical principles in deductive reasoning and defuses the threat to a cognitivist account of knowing basic logical principles. (shrink)

Hybrid belief aggregation addresses aggregation of individual probabilistic beliefs into collective binary beliefs. In line with the development of judgment aggregation theory, our research delves into the identification of precise agenda conditions associated with some key impossibility theorems in the context of hybrid belief aggregation. We determine the necessary and sufficient level of logical interconnection between the propositions in an agenda for some key impossibilities to arise. Specifically, we prove three characterization theorems about hybrid belief aggregation: (i) Precisely the (...) path-connected and pair-negatable agendas lead to the ‘oligarchy result’—only oligarchic rules satisfy universal domain, proposition-wise independence, respect for unanimity, and deductive closure of collective beliefs. (ii) Precisely the negation-connected agendas lead to the ‘triviality result’—only unanimity rules satisfy those conditions as well as anonymity. (iii) Precisely the blocked agendas lead to the ‘non-existence result’—no rules satisfy those conditions as well as completeness and consistency of collective beliefs. Furthermore, we compare these novel findings with existing agenda-theoretic characterization theorems in the domains of judgment aggregation and belief binarization. (shrink)

This is an exposition of Lambek's strengthening and generalization of the deduction theorem in categories related to intuitionistic propositional logic. Essential notions of category theory are introduced so as to yield a simple reformulation of Lambek's Functional Completeness Theorem, from which its main consequences can be readily drawn. The connections of the theorem with combinatory logic, and with modal and substructural logics, are briefly considered at the end.

Husserl’s first work formulated what proved to be an algorithmically complete arithmetic, lending mathematical clarity to Kronecker’s reduction of analysis to finite calculations with integers. Husserl’s critique of his nominalism led him to seek a philosophical justification of successful applications of symbolic arithmetic to nature, providing insight into the “wonderful affinity” between our mathematical thoughts and things without invoking a pre-established harmony. For this, Husserl develops a purely descriptive phenomenology for which he found inspiration in Mach’s proposal of a (...) “universal physical phenomenology.” To account for applications to any domain, Husserl envisages a theory of all possible deductive systems, which he develops extensively in his Göttingen lectures wherein he engages with Hilbert’s work on deductive systems for geometry, real arithmetic, and physics. This leads Husserl to formulate claims of decidability and proofs of completeness for various arithmetics that result from his analysis of Kronecker’s general arithmetic. Careful attention to these proofs seem to show that Husserl was not oblivious to problems that underlie our incompleteness theorems, namely that of showing that some inversions of his algorithmic arithmetic are undefined. His growing preoccupation with the issue of a pre-established harmony between mathematical thought and reality motivate his pursuit of a “supramathematics” of all possible complete theory forms to demystify such harmony, by having such a form on hand for describing any empirical domain. He soon decides that a transcendental idealism of nature will reveal the wonderful affinity of thoughts and things comprising such harmony, to be a wonderful “parallelism of objective unities and constituted manifolds of consciousness.” But the paradoxes of logic and set theory cloud the clarity of mathematics, which Weyl would restore with Brouwer’s intuitionism and Hilbert with his metamathematics. Husserl informed Weyl that his student Becker had formulated a phenomenological foundation not only for Weyl’s generalization of relativity theory but also for the Brouwer-Weyl continuum. But Weyl eventually rejected much of Becker’s work, especially when it became clear that his phenomenological intuitionism could not account for the success of Hilbert’s transfinite mathematics in quantum physics. Becker responded to this “crisis of phenomenological method” with his mantic phenomenology celebrating the magic of mathematical mysticism, which Husserl finally rejects in favor of a pluralistic phenomenology of mathematics and nature. (shrink)

This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by * ≡ ∃Q. In full topological models * is not generally definable but over Cantor-space and the reals it can be classically shown that *↔ ⅂⅂P; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order (...) predicate logic. Over [0, 1], the operator * is undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness. Parameters of the required kind can be obtained as so-called projections of lawless sequences. (shrink)

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite sets and series and quantum (...) information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as the set of (...) numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching! (shrink)

In this paper we investigate fuzzy propositional and first order logics which are complete or strongly complete with respect to a single chain, and we relate this properties with the existence of a universal chain for the logic.

A logical space is a pair of a non-empty set A and a subset of . Since is identified with {0, 1} A and {0, 1} is a typical lattice, a pair of a non-empty set A and a subset of for a certain lattice is also called a -valued functional logical space. A deduction system on A is a pair (R, D) of a subset D of A and a relation R between A* and A. In terms (...) of these simplest concepts, a general framework for studying the logical completeness is constructed. (shrink)

In [5], Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract “core” of logics in general, where logical syntax and semantics are “two sides of the same coin”. The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal (...) logic. However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics. (shrink)

We develop a predicate logical extension of a subintuitionistic propositional logic. Therefore a Hilbert type calculus and a Kripke type model are given. The propositional logic is formulated to axiomatize the idea of strategic weakening of Kripke''s semantic for intuitionistic logic: dropping the semantical condition of heredity or persistence leads to a nonmonotonic model. On the syntactic side this leads to a certain restriction imposed on the deduction theorem. By means of a Henkin argument (...) strong completeness is proved making use of predicate logical principles, which are only classically acceptable. (shrink)

Understanding propositional logic -- Deductive reasoning in propositional logic -- Understanding first-order logic -- Deductive reasoning in first-order logic -- Applications : mathematical proofs and automated reasoning -- Answers and solutions to selected exercises.