Implicational tonoid logics and their extensions with abstract Galois properties have been introduced by Yang and Dunn. They introduced matrix semantics for the implicational tonoid logics but did not do for the extensions. Here we provide such semantics for implicational partial gaggle logics as one sort of such extensions. To this end, first we discuss implicational partial gaggle logics in Hilbert-style. We next introduce one kind of matrix semantics based on Lindenbaum– Tarski (...) matrices for the logics and show that those logics are complete with respect to the matrix semantics. Finally, we further introduce a slightly different kind of matrix semantics based on reduced models for the logics and show that those logics are complete with respect to this matrix semantics. (shrink)

Implicational tonoid logics and their relational semantics have been introduced by Yang and Dunn. This paper extends this investigation to implicational partial Galois logics. For this, we first define some implicational partial gaggle logics as special kinds of implicational tonoid logics called “implicational partial Galois logics.” Next, we provide Routley–Meyer-style relational semantics for finitary those logics.

This paper combines two classes of generalized logics, one of which is the class of weakly implicative logics introduced by Cintula and the other of which is the class of gaggle logics introduced by Dunn. For this purpose we introduce implicational tonoid logics. More precisely, we first define implicational tonoid logics in general and examine their relation to weakly implicative logics. We then provide algebraic semantics for implicational tonoid logics. Finally, we consider relational semantics, called Routley–Meyer–style (...) semantics, for finitary those logics. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking (...) each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula phi a formula f(phi) such that for every formula psi, phi -> BOX psi is provable in L if and only if f(phi) -> psi is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

A major worry in self-deception research has been the implication that people can hold a belief that something is true and false at the same time: a logical as well as a psychological impossibility. However, if beliefs are held with imperfect confidence, voluntary self-deception in the sense of seeking evidence to reject an unpleasant belief becomes entirely plausible and demonstrably real.

Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with only one designated value are provided.

The aim of the paper is to outline a treatment of cotenability inspired by a perspective which had strong roots in ancient logic since Chrysippus and was partially recovered in the XX Century by E. Nelson and the exponents of so-called connexive logic. Consequential implication is a modal reinterpretation of connexive implication which permits a simple reconstruction of Aristotle's square of conditionals, in which proper place is given not only to ordinary cotenability between A and B, represented by (...) ¬, but to its “secondary” variant ¬ . After showing that logics of strict implication, of Stalnaker-Lewis conditionals and of relevant implication do not satisfy the intuitive properties required for cotenability , it is proved that such conditions are satisfied by consequential cotenability and by its secondary variant . The notions of strong cotenability and of bilateral cotenability are then introduced: the latter stands to standard cotenabilty as contingency stands to possibility, and in Section 9 it is shown to be a useful tool in the reconstruction of Boethius' theory of adjuncta. (shrink)

The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier.

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested (...) implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)

As far as logic is concerned, the conclusion of Michael Dummett's manifestability argument is that intuitionistic logic, as first developed by Heyting, satisfies the semantic requirements of antirealism. The argument may be roughly sketched as follows: since we cannot manifest a grasp of possibly justification-transcendent truth conditions, we must countenance conditions which are such that, at least in principle and by the very nature of the case, we are able to recognize that they are satisfied whenever they are. (...) Intuitionistic logic satisfies the semantic requirement that we should either eschew the notion of truth altogether and replace it by provability in principle, or constrain it by provability in principle (Dummett [1973] 1978). (Handout: 1-5). Some philosophers have argued that the traditional antirealist desideratum of decidability in principle is too weak, so that semantic antirealism properly construed must be committed to effective decidability. As such, it either leads to strict finitism (Wright [1982] 1993) [Handout: 6] or to a much stronger kind of logical revisionism than the one considered by intuitionists (whether or not they accept the manifestability argument): substructural logics, and in particular linear logics, rather than intuitionistic logic, satisfy the semantic requirements of strict antirealism (Dubucs and Marion 2004) [Handout: 8-9]. I shall develop two different kinds of replies. The first is concerned with the notion of meaning per se and looks to strict finitism directly, although not on the ground that it would provide a correct way of dealing with Soritic-type paradoxes (the original and primary focus of discussion in Wright [1982] 1993). The second is concerned with the justification of structural and logical rules in a natural deduction system à la Gentzen. It will deal in particular with the criticism of the structural rules of Weakening and Contraction [Handout: 10 and 11]. The first kind of reply, which Dummett has partially taken into consideration, is that if we jettison the effectively vs. in principle distinction, as applied to manifestability-type arguments, we end up with an unsatisfactory explanation of how the meaning of statements covering the practically unsurveyable or pro tempora undecided cases is fixed. The idea is that if we have a method which may be used over some small range, we have determined a way of applying the method everywhere in principle and that this is enough as far as fixing meaning is concerned. Decidability in principle is just what we need with respect to manifestation of grasp of meaning. In this perspective, antirealism shouldn't be strict and manifestability-type arguments need not be applied as far as the strict finitist would want to [Handout: 7]. It follows that there is no reason to think that practical feasibility should be built in assertibility conditions and proofs be construed dynamically as acts in the strict antirealist's acception of that term [Handout: 8]. I shall then look at one radical antirealist principle disqualifying structural rules, namely Token Preservation, arguing against Bonnay and Cozic's criticisms of Dubucs and Marion (Bonnay and Cozic 3 2007) that some conceptual support may be provided for Token Preservation, which doesn't rely on a causal misreading of the turnstile [Handout: 13, 14]. I shall then assess the merits and limits of radical antirealism and the logic of feasible proofs with respect to the original Dummettian argument in favour of semantic antirealism (provided it has indeed revisionist implications for logic), whether the radical antirealist merely stipulates what human feasibility amounts to, or dispenses with structural rules in order to argue in favour of a curb on the epistemic idealizations they unwarrantedly embed. It will be noted here that there is a great difference, conceptually speaking, between the rejection of classical logic via the curbing of the epistemic idealizations embedded in structural rules, and the rejection of classical logic via the criticism of the introduction and elimination rules which fix the meaning of the classical constants. The kind of logical revisionism envisaged by intuitionists from Heyting on is in many respects stronger than the one envisaged by advocates of linear logic, should they ground their arguments on an endorsement of strict antirealism. The crucial case of excluded middle is telling. Because of the splitting of the constants in linear logic (Handout: 12], two distinct laws of excluded middle may be formulated (Handout: 15), but the traditional arguments of Brouwer and Heyting against the classical law yield a stronger revision than the substructural revision with its admission of these two (very) weak versions of the law. (Project: a clearer conception is needed of how the introduction and elimination rules for the logical connectives in the intuitionistic calculus depend on the structural rules which the radical antirealist wishes to reject.). (shrink)

A minimal theorem in a logic L is an L-theorem which is not a non-trivial substitution instance of another L-theorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has been known to be partially positive and generally negative. It is shown here that a minimal implicational theorem A in intuitionistic logic has a unique -normal proof in NJ (...) whenever A is provable without non-prime contraction. The non-prime contraction rule in NJ is the implication introduction rule whose cancelled assumption differs from a propositional variable and appears more than once in the proof. Our result improves the known partial positive solutions to Komori's problem. Also, we present another simple example of a minimal implicational theorem in intuitionistic logic which does not have a unique -normal proof in NJ. (shrink)

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; that (...) it is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability. (shrink)

In the first chapter I have introduced Carnapian intensional logic against the background of Frege's and Quine's puzzles. The main body of the dissertation consists of two parts. In the first part I discussed Carnapian modal logic and arithmetic with descriptions. In the second chapter, I have described three Carnapian theories, CCL, CFL, and CNL. All three theories have three things in common. First, they are formulated in languages containing description terms. Second, they contain a system of modal (...) logic. Third, they do not contain the unrestricted classical substitution principle, but they do contain the classical substitution principle restricted to non-modal formulas and the Carnapian substitution principle, which says that two terms can be substituted salva veritate if they are necessarily coreferential. There are two major differences between the three theories. First, CCL and CFL allow universal instantiation with description terms, whereas CNL does not. Moreover, the quantificational theory of the CCL is classical, whereas the quantificational theory of CFL is a free logic. Another difference is that CCL and CFL contain different description principles. Most importantly, the description principle of CCL ensures that even improper descriptions have a denotation, whereas the description principle of CFL does not guarantee this. CNL does not have a description principle. In the third chapter, I have studied collapse arguments for CCL, CFL, and CNL. A collapse argument is an argument for the following statement: if p is true, then it is necessarily true. A crucial role in the proofs of these collapse results was played by so-called self-predication principles, which say that under certain conditions the predicate that expresses the descriptive condition can be combined by the description term formed out of that predicate with the result being a true sentence. In this chapter I have discussed a collapse argument for the extension of CCL with a self-predication principle, I have given a collapse argument for a similarly extended CFL, and most importantly, I have given a collapse argument for the extension of CNL with a self- predication principle. Finally, I have argued that the relevant self-predication principles are unsound under a Carnapian interpretation. In the fourth chapter, I have studied the extension of Peano Arithmetic with a Carnapian modal logic C, which is a dummy letter standing for either CCL or CFL. One can prove that the principle of the necessity of identity is a theorem of CPA. This implies that one gets a collapse result for CPA. The standard principle of weak induction was crucial for the proof. One can also prove that, if one assumes a particular self-predication principle, and if one assumes the principle of strong induction or, equivalently, the least-number principle, then one gets a partial collapse of de re modal truths in de dicto modal truths. I have argued that, if the box operator is interpreted as a metaphysical necessity operator, then Platonists would not be inimical to the collapse result. But if CPA is extended with a physical theory, then there is a threat that physical truths become physical necessiti es. It was shown that, under a Carnapian interpretation, the standard principle of weak induction is unsound, and that it can be replaced by a Carnapian principle of weak induction that is sound. The problem of logical and mathematical omniscience prevents ordinary Carnapian intensional logic from being taken seriously as a logic adequate for describing the principles of demonstrability. Yet many of the proof-theoretic results of the first part carry over to the part on Carnapian epistemic arithmetic with descriptions, since proof-theoretic results are independent of the informal reading of the operators. In the fifth chapter, I looked at extensions of arithmetic with a modal logic in which the box operator is interpreted as a demonstrability operator. A first extension in that sense is Shapiro s Epistemic Arithmetic. Shapiro himself offered the problem of mathematical omniscience as a reason why it is difficult to find a model theory for EA. Horsten attempted to provide a model theory via the detour of Modal-Epistemic Arithmetic. The attention of the reader was drawn to an incoherence in the model theory of. Two alternative solutions were presented and, after a short discussion of the problem of de re demonstrability one of those alternatives was chosen. The discussion of the problem of de re demonstrability made it clear that it would be interesting to study the epistemic properties of notation systems. Horsten himself provided a framework for this, viz. Carnapian Epistemic Arithmetic, and he started a systematic study of the epistemic properties of notation systems within that framework. However, he did not provide non-trivial but adequate models. To make a start with solving the problem of finding good models for CEA, I introduced Carnapian Modal-Epistemic Arithmetic In constructing CMEA I incorporated the lesson about the principle of weak induction learnt in the fourth chapter. In the sixth chapter, I gave a critical assessment of an argument concerning the limits of de re demonstrability about the natural numbers. The conclusion of the Description Argument is that it is undemonstrable that there is a natural number that has a certain property but of which it is undemonstrable that it has that property. A crucial step in the Description Argument involved a self-predication principle. Making good use of one of the results obtained in the third chapter, I proved a collapse result for the background theory against which the Description Argument was formulated. I concluded that either the either the Description Argument is sound but its conclusion is trivial, o r the Description Argument is unsound, or it is a cheapshot. As an appendix I included an article co-authored by prof. dr. Leon Horsten and me. The topic of the article is indirectly related to some other topics investigated in my dissertation. Also, it backs up one of the addition al theses I might be asked to publicly defend during my doctoral exam. T he topic of the appendix is the set of the so-called paradoxes of strict implication. Jonathan Lowe has argued that a particular variation on C.I. Lewis notion of strict implication avoids the paradoxes of strict implication. Pace Lowe, it is argued that Lowe s notion of implication does not achieve this aim. Moreover, a general argument is offered to the effect that no other variation on Lewis notion of constantly strict implication describes the logical behaviour of natural language conditional s in a satisfactory way. (shrink)

Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the way that every quantum (...) B-algebra admits a natural embedding into a logical quantale, the enveloping quantale. Partially defined products of algebras related to effect algebras are handled efficiently in this way. The unit group of the enveloping quantale of a quantum B-algebra X is shown to be always contained in X, which gives a functorial subgroup X× of X. Similar subfunctors are obtained for the non-commutative extensions of BCK-algebras and effect algebras. The results of Galatos, Jónsson, and Tsinakis on the splitting of generalized BL-algebras into a semidirect product of a partially ordered group operating on an integral residuated poset are extended to a characterization of twisted semidirect products of a po-group by a quantum B-algebra. (shrink)

How does causation in the physical world relate to implication in logic? This article presents implication as fundamentally a relation of inclusion between propositions. Given this, it is argued that an event cannot “causally imply” another, also given the laws of nature. Then, by applying the notion of inclusion to physical objects, a relation “within the possibilities of” is developed, which generates a partial order on sets of entities and is independent of time. Based on this, it is (...) shown that changes of physical objects in time (at any rate, a great many of them) imply, and thus counterfactually depend on, what we call “causes” – an asymmetric dependence which is robust despite the perspectival nature of the concept of “cause”. (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)

This book is both a history of philosophy of logic told from the Kantian viewpoint and a reconstruction of Kant’s theory of logic from a historical perspective. Kant’s theory represents a turning point in a history of philosophical debates over the following questions. (1) Is logic a science, instrument, standard of assessment, or mixture of these? (2) If logic is a science, what is the subject matter that differentiates it from other sciences, particularly metaphysics? (3) If (...) logic is a necessary instrument to all philosophical inquiries, how is it so entitled? (4) If logic is both a science and an instrument, how are these two roles related? Kant’s official answer to these questions centers on three distinctions: general versus particular logic; pure versus applied logic; pure general logic versus transcendental logic. The true meaning and significance of each distinction becomes clear, I argue, only if we consider two factors. First, Kant was mindful of various historical views on how logic relates to other branches of philosophy (viz. metaphysics and physics) and to the workings of common human understanding. Second, he first coined ‘transcendental logic’ while struggling to secure metaphysics as a proper “science,” and this conceptual innovation would in turn have profound implications for his mature theory of logic. Against this backdrop, I reassess the place of Kant’s theory in the history of philosophy of logic and highlight certain issues that are still debated today, such as normativity of logic and the challenges posed by logical pluralism. In Chapter 1, “Kant and a Philosophical History of Logic – Methodological Reflections,” I discuss certain exegetical challenges posed by Kant’s logic corpus and argue for a “history of philosophical problems” method by which to reconstruct a Kantian theory of logic. In Chapter 2, “The Nature and Place of Logic – A History of Controversies,” I construct a partial history of philosophy of logic that revolves around the (supposedly) scientific status of logic on the one hand and its value or utility on the other. In Chapter 3, “The Making of a Scientific Logic from Bacon to Wolff,” I examine how four representative early modern philosophers – Francis Bacon, John Locke, G.W. Leibniz, and Christian Wolff – approached the four questions mentioned above. In Chapter 4, “Kant on the Way to His Own Philosophy of Logic,” I consider how Kant, in the decade between around mid-1760s and mid-1770s, navigated between existing accounts of logic until he finally found his own voice. I highlight the breakthroughs that would mark his critical departures from previous views and pave the way for the final articulation of his own view. In Chapter 5, “Logic and the Demands of Kantian ‘Science’,” I zero in on Kant’s official theory of logic in his monumental publication, Critique of Pure Reason. I foreground both what makes the theory original and what leaves it vulnerable to criticisms from post-Kantian thinkers. (shrink)

Three semantic relations are analyzed: the relation of implication of a question by a question and a set of declarative sentences, the relation of implication of a question by a question, and the relation of strong implication of a question by a question and a set of declarative sentences. The connections between these concepts and the concepts of relative soundness, partial answerhood and presupposition are examined. The principal results are theorems about, to speak generally, epistemic reducibility of well-posed questions (...) to sequences of implied yes-no questions. (shrink)

This thesis consists in six articles and a comprehensive summary. • The pourpose of the summary is to introduce the AGM theory of belief change and to exemplify the diversity and significance of the research that has been inspired by the AGM article in the last 25 years. The research areas associated with AGM was divided in three parts: criticisms, where we discussed some of the more common criticisms of AGM. Extensions where the most common extensions and variations of AGM (...) are presented and applications where we provided an overview of applications and connections with other areas of research. • Article I elaborates on the connection between partial meet contractions [AGM85] and kernel contractions [Han94a] in belief change theory. Also both functions are equivalent in belief sets, there are notequivalent in belief bases. A way to define incision functions from selection functions and vice versa is presented. It is explained under which conditions there are exact correspondences between selection and incision functions so that the same contraction operations can be obtained by using either of them. • Article II proposes an axiomatic characterization for ensconcement-based contraction functions, belief base functions proposed byWilliams and relates this function with other kinds of base contraction functions. • Article III adapts the Fermé and Hansson model of Shielded Contraction [FH01] as well as Hansson et all Credibility-Limited Revision [HFCF01] for belief bases, to join two of the many variations of the AGM model [AGM85], i.e. those in which knowledge is represented through belief bases instead of logic theories, and those in which the object of the epistemic change does not get the priority over the existing information as it is the case in the AGM model. • Article IV introduces revision by comparison a refined method for changing beliefs by specifying constraints on the relative plausibility of propositions. Like the earlier belief revision models, the method proposed is a qualitative one, in the sense that no numbers are needed in order to specify the posterior plausibility of the new information. The method uses reference beliefs in order to determine the degree of entrenchment of the newly accepted piece of information. Two kinds of semantics for this idea are proposed and a logical characterization of the new model is given. • Article V focuses on the extension of AGM that allows change for a belief base by a set of sentences instead of a single sentence. In [FH94], Fuhrmann and Hansson presented an axiomatic for Multiple Contraction and a construction based on the AGM Partial Meet Contraction. This essay proposes for their model another way to construct functions: Multiple Kernel Contraction, that is a modification of Kernel Contraction,proposed by Hansson [Han94a] to construct classical AGM contractions and belief base contractions. • Article VI relates AGM model with the DFT model proposed by Carlos Alchourrón [Alc93]. Alchourrón devoted his last years to the analysis of the notion of defeasible conditionalization. His definition of the defeasible conditional is given in terms of strict implication operator and a modal operator f which is interpreted as a revision function at the language level. This essay points out that this underlying revision function is more general than AGM revision. In addition, a complete characterization of that more general kind of revision that permits to unify models of revision given by other authors is given. (shrink)

The article introduces extended models for the propositional dynamic logic PDL. In extended models, valuation assigns to every state a set of atomic formulas and a PDL program. The program is informally construed as an action preferred by a contextually fixed agent. PDL is then extended by introducing a conditional connective expressing partial correctness claims. The main contribution of the article is the observation that the partial correctness conditional is in fact a substructural implication. It is shown (...) that a variant of the minimal distributive substructural logic DB is a fragment of PDLS, an extension of PDL that contains the partial correctness conditional and fusion. It is also shown that the main results can be extended to a specific class of positive substructural logics. (shrink)

For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is known. (...) In this paper, we provide partial answers to the above open problems and among other results, we establish the following:1.For every integer \, if \ is true for all integers i with \, then \ is true for all integers i with \.2.If \ are any integers such that for some prime p we have \ and \, then in \: \ and \.3.For \, \\\ implies \, and \ implies neither \ nor \ in \.4.For every integer \, \ implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in \ ). In particular, \ strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.5.The Chain-AntiChain Principle implies neither \ nor \ in \, for every integer \. (shrink)

For Heidegger, Hegel understands being, ‘the highest actuality’, as the categories which pervade and thereby form all objects and events. Since, Heidegger argues, the categories are, in Hegel, present-at-hand, Hegel conceives of being as presence-at-hand. This is a problem, for Heidegger, because it entails the full transparency and knowability of being, whereas, in his view, being is partially hidden and unknowable. I consider the objection to this Heideggerian critique of Hegel that Hegelian logic understands being not only as the (...) list of categories but also as their derivation and movement from pure being to the absolute idea, which (derivation and movement) establish being not only as presence but also as implication. Since being-as-implication is (a) not presence-at-hand and (b) necessary to being, it cannot be said that Hegel's account of being turns it into full transparency and knowability. Heidegger's critique should, therefore, be rejected. I argue that this objection is unsuccessful because there is strong evidence in the Logic that Hegel ‘subordinates’ being-as-implication to presence-at-hand. Implication's way of being is, in Hegel, only a collapse into presence-at-hand and hence ‘merely a modification of presence’. Consequently, Heidegger's critique of Hegel should not be rejected based on the objection. I conclude the article with a remark on the relation between language and being-as-implication. I argue that Hegel's account of being-as-implication in language disrespects the autonomy of being-as-implication therein and that Haas's argument for such an autonomy based on the phenomenon of the syntactic ellipsis of ‘is’ fails to undermine Hegel's account. (shrink)

In [1] J.-Y. Bèziau formulated a logic called Z. Bèziau’s idea was generalized independently in [6] and [7]. A family of logics to which Z belongs is denoted in [7] by K. In particular; it has been shown in [6] and [7] that there is a correspondence between normal modal logics and logics from the class K. Similar; but only partial results has been obtained also for regular logics. In a logic N has been investigated in the (...) language with negation; implication; conjunction and disjunction by axioms of positive intuitionistic logic; the right-to-left part of the second de Morgan law; and the rules of modus ponens and contraposition. From the semantical point of view the negation used by Došen is the modal operator of impossibility. It is known this operator is a characteristic of the modal interpretation of intuitionistic negation. In the present paper we consider an extension of N denoted by N+. We will prove that every extension of N+ that is closed under the same rules as N+; corresponds to a regular logic being an extension of the regular deontic logic D21. The proved correspondence allows to obtain from soundnesscompleteness result for any given regular logic containing D2, similar adequacy theorem for the respective extension of the logic N+. (shrink)

In reverse mathematics, it is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on how to weaken the assumption while preserving the conclusion (other than reducing all the way to the tautology of assuming the conclusion). A main cause of this phenomenon is the proof of a $\Pi^1_2$ sentence from the theory $\mathbf{\Pi^{\textbf{1}}_{\textbf{1}}-CA_{\textbf{0}}}$. Using methods based on the functional interpretation, we introduce a family of weakenings of $\mathbf{\Pi^{\textbf{1}}_{\textbf{1}}-CA_{\textbf{0}}}$ (...) and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs. (shrink)

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs (...) are MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hájek's BL-algebras, and the basic algebras of Chajda, Halaš, and Kühr. (shrink)

Putnam and Finkelstein have proposed the abandonment of distributivity in the logic of quantum theory. This change results from defining the connectives, not truth-functionally, but in terms of a certain empirical ordering of propositions. Putnam has argued that the use of this ordering ("implication") to govern proofs resolves certain paradoxes. But his resolutions are faulty; and in any case, the paradoxes may be resolved with no changes in logic. There is therefore no reason to regard the partially ordered (...) set of propositions as a logic--i.e. as embodying a criterion for soundness of proofs. Its role in quantum theory ought to be understood in an entirely different way. (shrink)

In the Remark to the final paragraph of the Chapter on “existence” (Dasein) in the Logic of the Encyclopedia of the Philosophical Sciences in Basic Outline (1830) Hegel states that the “ideality of the finite is the chief proposition of philosophy” and that “every true philosophy is for that reason idealism” (Enz § 95A). In turn, at the end of the Chapter on “existence” in the Science of Logic (1832) Hegel claims, further, that “every philosophy is essentially idealism (...) or at least has idealism for its principle, and the question then is only how far this principle is carried out” (GW 21.142). Along this line, Hegel conceives of absolute idealism not only as the result of the history of philosophy, but also as the philosophical system that reveals the essence of the philosophical systems before idealism, that is to say, as the philosophy that reveals, by developing it and formulating it adequately, what the precedent philosophies, mostly unknowingly, tried to develop and formulate, namely a general theory about reality based on the principle of the unity of being and thought. According to Hegel, every particular philosophy throughout history exposed in a successive, partial and complementary way the process of identification of being and thought; inasmuch as the system of absolute idealism assumes the latently idealist theses present in former philosophies, it makes those theses explicit and exposes the process of the identification of being and thought as its own internal development. Thus, absolute idealism is for Hegel the philosophy that exposes and shows what philosophy is actually about. The purpose of this essay is to render explicit the metaphilosophical implications of Hegel´s conception of absolute idealism as the “true philosophy” (wahre Philosophie, wahrhafte Philosophie). (shrink)

In this paper we show that several classes of partially ordered structures having paraorthomodular reducts, or whose sections may be regarded as paraorthomodular posets, admit a quite natural notion of implication, that admits a suitable notion of adjointness. Within this framework, we propose a smooth generalization of celebrated Greechie’s theorems on amalgams of finite Boolean algebras to the realm of Kleene lattices.

Although modern modal logic came about largely after Peirce’s death, he anticipated some of its key aspects, including strict implication and possible worlds semantics. He developed the Gamma part of Existential Graphs with broken cuts signifying possible falsity, but later identified the need for a Delta part without ever spelling out exactly what he had in mind. An entry in his personal Logic Notebook is a plausible candidate, with heavy lines representing possible states of things where propositions denoted (...) by attached letters would be true, rather than individual subjects to which predicates denoted by attached names are attributed as in the Beta part. New transformation rules implement various commonly employed formal systems of modal logic, which are readily interpreted by defining a possible world as one in which all the relevant laws for the actual world are facts, each world being partially but accurately and adequately described by a closed and consistent model set of propositions. In accordance with pragmaticism, the relevant laws for the actual world are represented as strict implications with real possibilities as their antecedents and conditional necessities as their consequents, corresponding to material implications in every possible world. (shrink)

The aim of this work is to investigate the problem of Logical Omniscience in epistemic logic by means of truthmaker semantics. We will present a semantic framework based on $$\varvec{W}$$ W -models extended with a partial function, which selects the body of knowledge of the agents, namely the set of verifiers of the agent’s total knowledge. The semantic clause for knowledge follows the intuition that an agent knows some information $$\varvec{\phi }$$ ϕ, when the propositional content that $$\varvec{\phi (...) }$$ ϕ is contained in her total knowledge. We will argue that this idea mirrors the philosophical conception of immanent closure by Yablo (2014), giving to our proposal a strong philosophical motivation. We will discuss the philosophical implications of the semantics and we will introduce its axiomatization. (shrink)

We investigate a theory of Frege structures extended by the Myhill-Flagg hierarchy of implications. We study its relation to a property theory with an approximation operator and we give a proof theoretical analysis of the basic system involved. MSC: 03F35, 03D60.

With a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e., a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory JC in a classical propositional (...) calculus such that the abacus logic determined by C is isomorphic to the poset of JC. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of JC. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting "hidden variables" it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the "classical observables" associated with B and the "quantum observables" associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the "uncertainty" of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable. (shrink)

In this thesis I investigate the implications, for one's account of mathematics, of holding an anti-realist view. The primary aim is to appraise the scope of revision imposed by anti-realism on classical inferential practice in mathematics. That appraisal has consequences both for our understanding of the nature of mathematics and for our attitude towards anti-realism itself. If an anti-realist position seems inevitably to be absurdly revisionary then we have grounds for suspecting the coherence of arguments canvassed in favour of anti-realism. (...) I attempt to defend the anti-realist position by arguing, i) that it is not internally incoherent for anti-realism to be a potentially revisionary position, and ii) that an anti-realist position can, plausibly, be seen to result in a stable intuitionistic position with regard to the logic it condones. The use of impredicative methods in classical mathematics is a site of traditional intuitionistic attacks. I undertake an examination of what the anti-realist attitude towards such methods should be. This question is of interest both because such methods are deeply implicated in classical mathematical theory of analysis and because intuitionistic semantic theories make use of impredicative methods. I attempt to construct the outlines of a set theory which is anti-realistically acceptable but which, although having no antecedent repugnance for impredicative methods as such, appears to be too weak to offer an anti-realistic vindication of impredicative methods in general. I attempt to exonerate intuitionistic semantic theories in their use of impredicative methods by showing that a partial order relying on the nature of our grasp of the intuitionistic meaning stipulations for the logical constants precludes a possible circularity. (shrink)

This book is a significant addition to studies of Parmenides and the foundation of Greek philosophy, with interesting implications for subsequent Western metaphysics. Within carefully drawn limits, Austin conducts a rigorous analysis of Parmenides' poem that is both creative and forceful. The resultant insights into Parmenidean logic, ontology and method cannot easily be discounted. Austin claims that Parmenides uses a consciously systematic and exhaustive method to describe being. Thus, he argues, all the arguments and distinctions of the "Truth" section--and (...) to some extent those of "Opinion"--are necessary for a complete description of being. Here Parmenides employs the principles of noncontradiction and excluded middle, and displays an understanding-albeit a partial one--of the functions of negation, double negation, copula and predicate. The most general challenge to Austin's thesis, obviously, is the charge of anachronism. Austin recognizes and embraces this challenge. His response is contained in the thorough arguments of the first five chapters. (shrink)

We consider finite multisets over some set of urelements equipped only with additive union [uplus ] and show that the {[otimes], -0}-Horn fragment of Intuitionistic Linear Logic has a sound and complete interpretation in them by interpreting [otimes] as [uplus ]. The linear implication is interpreted by ordered pairs of multisets expressing replacement. The operator ! is also defined in an asymptotic way. Soundness, completeness and partial completeness results are proved for the {×, -0, !}-Horn fragment as well.

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

Over the last few decades, the idea that education should function to promote economic progress has played a major role in shaping educational policy. So far, however, philosophers of education have shown relatively little interest in analysing this notion and its implications. The present article critically examines, from a philosophical perspective, the link between education and the currently prevailing understanding of economic progress, which is grounded in human capital theory. A number of familiar philosophical objections to the idea that economic (...) progress is a worthy educational aim are raised, but it is held that many of these objections stem from a partial understanding of economic theory. The article then moves on to investigate the foundations of human capital theory's conception of economic progress. It is argued that there are inherent tensions between the philosophical foundations of the economist's conception of progress and the very essence of education. The article concludes by suggesting that in order to resolve these tensions we should consider grounding the economic theory that guides educational policy-making in an intersubjective or objective conception of welfare. (shrink)

A constructive proof is given which shows that every nonrecursive r.e. many-one degree is represented by the family of decision problems for partial implicational propositional calculi whose well-formed formulas contain at most two distinct variable symbols.

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and overcome the (...) failure of interpolation for the implication-free fragment. (shrink)

Havstad and Smith (2019) argue that Lakatos’ “methodology of scientific research programs” (MSRP) is a promising philosophical framework for explaining the perceived empirical success of the hypothesis that birds are maniraptoran theropod dinosaurs, and the perceived empirical failures or stagnation of alternatives to that hypothesis. These conclusions are rejected: Havstad and Smith’s account of the alternative “research programs” inadequately characterizes criticism of the hypothesis that birds are maniraptoran theropods and they neither offer sufficient modifications to MSRP to correct its known (...) difficulties in deriving logically or empirically satisfactory criteria for the assessment and preferential selection of “research programs” from historiographical data, nor proposals to mitigate its tendency to promote confirmation bias and dogmatism. Independent flight loss, an important problem in systematic ornithology with implications for the origin of birds, provides a supplementary demonstration of how the application of MSRP in the present context would tend systematically to mislead investigations of the evolutionary history of birds by promoting an uncritical perspective. Given these difficulties, MSRP is an unacceptable philosophical framework for evaluating alternative hypotheses for the origin of birds. (shrink)

A generalized inclusion frame consists of a set of points W and an assignment of a binary relation $R\sb{w}$ on W to each point w in W. Generalized inclusion frames whose $R\sb{w}$ are partial orders are called comparison frames. Conditional logics of various comparative notions, for example, Lewis's V-logic of comparative possibility and utilitarian accounts of conditional obligation, model the dyadic modal operator $>$ on comparison frames according to the following truth condition: $\alpha > \beta$ "holds at w" (...) iff every point in the truth set of $\alpha$ bears $R\sb{w}$ to some point where $\beta$ holds. ;In this essay I provide a relational frame theory which embraces both accessibility semantics and g.i. semantics as special cases. This goal is achieved via a philosophically significant generalization of universal strict implication which does not assume accessibility as a primitive. Within this very general setting, I provide the first axiomatization of the dyadic modal logic corresponding to the class of all g.i. frames. Various correspondences between dyadic logics and first order definable subclasses of the class of g.i. frames are established. Finally, some general model constructions are developed which allow uniform completeness proofs for important sublogics of Lewis' V. (shrink)

Answer-set programming (ASP) has emerged as a declarative programming paradigm where problems are encoded as logic programs, such that the so-called answer sets of theses programs represent the solutions of the encoded problem. The efficiency of the latest ASP solvers reached a state that makes them applicable for problems of practical importance. Consequently, problems from many different areas, including diagnosis, data integration, and graph theory, have been successfully tackled via ASP. In this work, we present such ASP-encodings for problems (...) associated to abstract argumentation frameworks (AFs) and generalisations thereof. Our encodings are formulated as fixed queries, such that the input is the only part depending on the actual AF to process. We illustrate the functioning of this approach, which is underlying a new argumentation system called ASPARTIX in detail and show its adequacy in terms of computational complexity. (shrink)

This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL ${}_{\scriptsize\mbox{e}}$ -algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, we (...) provide some philosophical upshots like, e.g., a discussion on the relevance of the above operation in relationship with G. Polya’s logic of plausible inference, and some characterization results on weak and strong connexivity. (shrink)

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and (...) several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. (shrink)