This chapter contains sections titled: Beliefs Access to truth Cogito ergo sum Mathematical certainty Classical logic C. I. Lewis' empiricism Access as a metaphor J. F. Fries and K. Popper Voluntarism and linearity One‐way justification Beginning in the middle Justification, contextual and comparative Justification in the empirical sciences Circularity versus linearity Democratic controls Interactionism.

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as (...) tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

This paper shows how to conservatively extend theories formulated in non-classical logics such as the Logic of Paradox, the Strong Kleene Logic and relevant logics with Skolem functions. Translations to and from the language extended by Skolem functions into the original one are presented and shown to preserve derivability. It is also shown that one may not always substitute s=f(t) and A(t, s) even though A determines the extension of a function and f is a Skolem function (...) for A. (shrink)

Semantic justifications of the classical rules of logical inference typically make use of a notion of bivalent truth, understood as a property guaranteed to attach to a sentence or its negation regardless of the prospects for speakers to determine it as so doing. For want of a convincing alternative account of classical logic, some philosophers suspicious of such recognition-transcending bivalence have seen no choice but to declare classical deduction unwarranted and settle for a weaker system; intuitionistic (...) logic in particular, buttressed by assertion-conditional semantics, is often considered to enjoy a degree of meaning-theoretical respectability unattainable by classical logic.The decision to forgo the classical inference rules is not always made lightly. Thus, Dummett: " In the resolution of the conflict between [the view that generally accepted classical modes of inference ought to be theoretically accommodated, and the demand that any such accommodation be achieved without recourse to bivalence] lies, as I see it, one of the most fundamental and intractable problems in the theory of meaning; indeed, in all philosophy. "The present article ventures to suggest that the conflict need not be irresoluble. Helping ourselves only to such conceptual resources as are standardly invoked by proponents of intuitionistic logic, we shall formulate a rudimentary meaning theory for a selection of logical constants, define a relation of valid inferability, and prove that the latter includes all of classical first-order logic. The result is essentially that reported in Sandqvist as Theorem 2.20.2. TheoryWe will be considering a standard-syntax first-order language with ⊃, ⊥ and ∀ as its only primitive logical constants. We assume countable supplies of individual constants, individual functors , and predicates , as well as a denumerably infinite set of individual variables. By a basic 1 formula we will mean one that …[Full Text of this Article]. (shrink)

We study here equiformity, the standard identity criterion for sentences. This notion was put forward by Lesniewski, mentioned by Tarski and defined explicitly by Presburger. At the practical level this criterion seems workable but if the notion of sentence is taken as a fundamental basis for logic and mathematics, it seems that this principle cannot be maintained without vicious circle. It seems also that equiformity has some semantical features ; maybe this is not so clear for individual (...) signs but sentences are often considered as meaningful combinations of signs. If meaning has to play a role, we are thus maybe in no better position than when dealing with identity criterion for propositions. In formal logic, one speaks rather about well-formed formulas, but closed formulas are called sentences because they are meaningful in the sense that they can be true or false. Formulas look better like mathematical objects than material inscriptions and equiformity does not seem to apply to them. Various congruencies can be considered as identities between formulas and in particular "to have the same logical form". One can say that the objects of study of logic are rather logical forms than sentences conceived as material inscriptions. (shrink)

We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are (...) class='Hi'>well-suited to provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant. (shrink)

Connectionist attention to variables has been too restricted in two ways. First, it has not exploited certain ways of doing without variables in the symbolic arena. One variable-avoidance method, that of logical combinators, is particularly well established there. Secondly, the attention has been largely restricted to variables in long-term rules embodied in connection weight patterns. However, short-lived bodies of information, such as sentence interpretations or inference products, may involve quantification. Therefore short-lived activation patterns may need to achieve the (...) effect of variables. The paper is mainly a theoretical analysis of some benefits and drawbacks of using logical combinators to avoid variables in short-lived connectionist encodings without loss of expressive power. The paper also includes a brief survey of some possible methods for avoiding variables other than by using combinators. (shrink)

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for (...) class='Hi'>classical modal systems by Stolpe, 557–575, 2003), but unfortunately that work used only models and not frames. (shrink)

Western (deductive) logic originated in Greek antiquity. It found its first expression in those works of the great philosopher Aristotle (384–322 BC) which have come to be known as the Organon, i.e., ‘instrument’. Aristotle’s logic, also known as syllogistics, was unsystematically concerned with patterns of reasoning and argumentation. It remained in this rudimentary state relatively unchanged and unchallenged until the second half of the nineteenth century. At that time, logic underwent a period of unprecedented reform and modernization, (...) due in large part to the German mathematician Gottlob Frege (1848– 1925) It thus became more and more a mathematical endeavor of studying the structure and peculiarities of artificial, formal languages. In this new form, logic gave rise in the twentieth century to disciplines such as theoretical informatics and programming languages, and transformed our lives through computation, information processing, and the Internet. A formal language consists, in effect, of a particular alphabet and some precise rules of forming, and transforming, strings over this alphabet. There exist several types of formal languages analyzed in logic. Depending on their structure, they are called first-order languages, second-order languages, and so on. Today a logic is considered a theory of such a language and is, correspondingly, referred to as a first-order logic, a second-order logic, and so on. In this chapter, we shall outline a first-order logic as a paragon of deductive logic. Its full name is: “Classical, first-order predicate logic with identity”. What all these expressions mean exactly, will become clear below. The logic we shall study first is termed classical logic because the idea to create such an instrument, or ‘organon’, is rooted in Greek antiquity. Owing to its origin, it is based on three time-honored Aristotelian doctrines. For these and several other reasons that we shall discuss later in this chapter, it is also called a logic of the Aristotelian style, or an Aristotelian logic for short. (shrink)

The paper proposes the first unified account of deictic/sentence-external and sentence-internal readings of singular different . The empirical motivation for such an account is provided by a cross-linguistic survey and an analysis of the differences in distribution and interpretation between singular different , plural different and same (singular or plural) in English. The main proposal is that distributive quantification temporarily makes available two discourse referents within its nuclear scope, the values of which are required by sentence-internal uses of singular (...) different to be distinct, much as its deictic uses require the values of two discourse referents to be distinct. Thus, we take sentence-internal readings to be a form of ‘association with distributivity’ that is similar to association with focus. The contrast between singular different , plural different and same is explained in terms of several kinds of quantificational distributors that license their internal readings. The analysis is executed in a stack-based dynamic system couched in type logic, so we get compositionality in the usual Montagovian way. Quantificational subordination and dependent indefinites in various languages provide additional motivation for the account. Investigating the connections between items with sentence-internal readings and the quantificational licensors of these readings opens up a larger project of formally investigating (i) the typology of quantificational distributors and distributivity-dependent items and (ii) the fine-grained contexts of evaluation needed to capture this typological variation. (shrink)

Logics of significance have been proposed in an attempt to overcome the shortcomings of classical logic as a model of reasoning in the presence of nonsignificant (e.g., meaningless, ill-formed, unverifiable) sentences. Many-valued logicians have addressed this problem by introducing logics with infectious truth values. Cases in point are the weak Kleene logics B3 (paracomplete weak Kleene logic) and PWK (paraconsistent weak Kleene logic). Over time, it has become clear that the valid entailments of these significance (...) logics obey variable inclusion patterns that link them to other, usually better known, logics—such patterns, however, allow for disturbing exceptions. Logics of pure (left or right) variable inclusion have been introduced with an eye to removing these exceptions. In this paper, we consider the pure left variable inclusion companion of classical logic and give a complete description of its subclassical extensions. We also provide relative axiomatizations and characteristic (sets of) matrices for each one of these extensions, as well as syntactic descriptions (in terms of variable inclusion criteria) for the valid entailments of some of them, and determine in each case the algebra reducts of the Suszko-reduced matrix models. (shrink)

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim (...) of this paper is to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus. We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

Within Linguistics the semantic analysis of natural languages (English, Swahili, for example) has drawn extensively on semantical concepts first formulated and studied within classical logic, principally first order logic. Nowhere has this contribution been more substantive than in the domain of quantification and variable binding. As studies of these notions in natural language have developed they have taken on a life of their own, resulting in refinements and generalizations of the classical quantifiers as well (...) as the discovery of new types of quantification which exceed the expressive capacity of the classical quantifiers. We refer the reader to Keenan and Westerståhl (1997) for an overview of results in this area. Here, we focus on one property of quantification in natural language—its inherently sortal nature—which distinguishes it from quantification in classical logic. (shrink)

Fractional semantics provides a multi-valued interpretation of a variety of logics, governed by purely proof-theoretic principles. This approach employs a method of systematic decomposition of formulas through a well-disciplined sequent calculus, assigning a fractional value that measures the “quantity of identity” (intuitively, “quantity of truth”) within a sequent. A key consequence of this framework is the breakdown of the traditional symmetry between truth and contradiction. In this paper, we explore the ramifications of this novel perspective on classical (...) logic. Specifically, we (i) introduce an alternative _paraconsistent_ consequence relation, and (ii) show how the gradual character of contradictions induces a corresponding characterization of tautologies, thereby obtaining a full-fledged informational refinement of classical logic. (shrink)

This anthology consists of forty-nine of Geach’s previously published papers on logic. He opens his Preface by writing: "I bring together here almost all my English articles that I have previously published and have not already collected or cannibalized in other books." It contains his first published paper "Designation and Truth" from Analysis 1947-48 as well as his elegant sketch of a decidable entailment system in the 1970 Philosophical Review. For the most part he has made only stylistic (...) changes and added some notes. So, he claims: "all articles appear without substantial change." However, he admits that he divided his "Quantification Theory and the Problem of Identifying Objects of Reference" from the now classic 1963 volume of Acta Philosophica Fennica. He divided it because he wanted to separate his strong advocacy of a substitution interpretation of the quantifiers from some views on quotation marks, about which he is less than certain. The papers are not in mere chronological order. They are organized by topic within chapters entitled: Historical Essays, Traditional Logic, Theory of Reference and Syntax, Intentionality, Quotation and Semantics, Set Theory, Identity Theory, Assertion, Imperatives and Practical Reasoning, and Logic in Metaphysics and Theology. The chapter titles are a fair guide to content. But note the following. Two lucid papers defending von Wright’s notion of entailment are in the Intentionality chapter. The essay "Ascriptivism" in the Assertion chapter is relevant to discussions of free will because he argues: "to ascribe an act to agent is a causal description of the act" and he urges a search for a non-Humean analysis of causality. The essays in the last chapter are: "Nominalism," "Some problems about time," and "God’s relation to the world.". (shrink)

The paper proposes a novel account to the problem of exceptional scope (ES) of (in)definites, e.g. the widest and intermediate scope readings of the sentence Every student of mine read every poem that a famous Romanian poet wrote before World War II. We propose that ES readings are available when the sentence is interpreted as anaphoric to quantificational domains and quantificational dependencies introduced in the previous discourse. For example, the two every quantifiers and the indefinite elaborate on the sets of (...) individuals and the correlations between them introduced by a previous sentence like Every student chose a poet and read every poem written by him (for the intermediate scope reading) or a sentence like Every student chose a poet - the same poet - and read every poem written by him (for the widest scope reading). Our account, formulated within a compositional dynamic system couched in classical type logic, relies on two independently motivated assumptions: (a) the discourse context stores not only (sets of) individuals, but also quantificational dependencies between them, and (b) quantifier domains are always contextually restricted. Under this analysis, (in)definites are unambiguous and we do not resort to movement or special storage mechanisms, nor do weposit special choice-functional variables. (shrink)

This article presents a new semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an (...) informational refinement of traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions. (shrink)

This paper gives a semantical account for the (i)ordinary propositional calculus, enriched with quantifiers binding variables standing for sentences, and with an identity-function with sentences as arguments; (ii)the ordinary theory of quantification applied to the special quantifiers; and (iii)ordinary laws of identity applied to the special function. The account includes some thoughts of Roman Suszko as well as some thoughts of Wittgenstein's Tractatus.

Epistemic contradictions are now a well-known and often discussed phenomenon among those who study epistemic modals. These contradictions are expressed by sentences like ‘It is raining and it might not be raining’ whose oddness to the common ear demands an explanation. However, it has turned out to be a rather controversial enterprise to provide such an explanation in a sufficiently precise and general manner. According to pragmatic explanations, epistemic contradictions are semantically consistent but pragmatically defective. According to semantic (...) explanations, one should regard epistemic contradictions as plain semantic inconsistencies. Endorsing such a semantic solution is, however, tantamount to rejecting classical logic, which predicts the consistency of epistemic contradictions. After arguing that all existing solutions to this problem face as yet unmet challenges, I will present a new solution that adequately overcomes these challenges. I will propose to view epistemic contradictions as semantically consistent but epistemically defective sentences. The main thesis emerging will be that we need neither abandon the inference rules of classical logic nor the classical truth-conditional approach to semantics to deal adequately with epistemic contradictions. (shrink)

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional part (...) of ‘strict finitistic logic’ from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model. (shrink)

This paper develops a semantics for a fragment of English that is based on the idea of `impossible possible worlds'. This idea has earlier been formulated by authors such as Montague, Cresswell, Hintikka, and Rantala, but the present set-up shows how it can be formalized in a completely unproblematic logic---the ordinary classical theory of types. The theory is put to use in an account of propositional attitudes that is `hyperfine-grained', i.e. that does not suffer from the well-known (...) problems involved with replacing expressions by logical equivalents. (shrink)

Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, propositions, (...) and identity, as well as on the metaphysics of modality. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan (...) Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural (...) semantics, a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered. (shrink)

Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ \ with the intended meaning that “the probability of truthfulness of \ belongs to the interval [a, b]”. This method makes it possible to define a system (...) of derivations based on ’axioms’ of the form \, obtained as a result of empirical research, and then infer conclusions of the form \. We discuss the consistency, define the models, and prove the soundness and completeness for the defined probabilized sequent calculus. (shrink)

It is a well-known fact that the Russell’s antinomy arises within distributive set theory whereas it does not do so within collective set theory. n this paper, I shall propose what I shall call a “collective” understanding of a sentence as opposed to the standard, truth-functional approach which I shall term a “distributive" approach. Similar to the case with sets, the liar antinomy appears when the liar sentence is treated distributively. If, however, the sentence is understood collectively, then the (...) liar antimony does not appear. (shrink)

Predicate logic has proved a very useful tool for the expression of theories of natural language semantics. Hurford's suggestion that predicate–argument structures mirror certain properties of the human sensorimotor architecture can be seen as an explanation of why this is so. Although I support this view, I think that the correspondences that Hurford draws between linguistic and sensorimotor structures not only involve natural language semantics, but include some elements of natural language syntax as well.

Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as (...) a necessary property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing. (shrink)

Semantics connected to some information based metaphor are well-known in logic literature: a paradigmatic example is Kripke semantic for Intuitionistic Logic. In this paper we start from the concrete problem of providing suitable logic-algebraic models for the calculus of attribute dependencies in Formal Contexts with information gaps and we obtain an intuitive model based on the notion of passage of information showing that Kleene algebras, semi-simple Nelson algebras, three-valued ukasiewicz algebras and Post algebras of order three (...) are, in a sense, naturally and directly connected to partially defined information systems. In this way wecan provide for these logic-algebraic structures a raison dêetre different from the original motivations concerning, for instance, computability theory. (shrink)

Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades (...) it has been argued that we have good reason to admit among our primitive logical notions also the plural quantifiers ∀xx and ∃xx. More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as ‘pure logic’; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood. In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic. For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic.” Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects. (shrink)

This paper investigates the meaning of restricted quantification when the embedded conditional is taken as the conditional of some first-order connexive logics. The study is carried out by checking the suitability of RQ for defining a connexive class theory, in analogy to the definition of Boolean class theory by using RQ in classical logic. Negative results are obtained for Wansing’s first-order connexive logic QC and one variant of Priest’s first-order connexive logic QP. A positive result (...) is obtained for another variant of QP. (shrink)

A first-order language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language. Subatomic natural deduction systems are defined for this naturalistic first-order language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula property (...) with respect to atomic and identity formulae as a special case. The systems admit a proof-theoretic semantics which does not only apply to logically compound but also to atomic and identity formulae—as well as to their components. The potential of the defined systems for a meticulous first-order analysis of natural inferences whose validity crucially depends on expressions of some of the aforementioned categories is demonstrated. (shrink)

Presents a set-theoretic version of the analysis of "there be" as predicating instantiation of a property, a property-theoretic version of which was developed in McNally 1992. This paper provides a solution to the criticism that McNally 1992's analysis could not account for sentences in which postverbal nominal contains a monotone decreasing or nonmonotonic determiner.

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of ‘increasingly classical’ logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by (...) taking a somehow different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician. (shrink)

We describe a relational framework that uniformly supports formalization and automated reasoning in varied propositional modal logics. The proof system we propose is a relational variant of the classical Rasiowa-Sikorski proof system. We introduce a compact graph-based representation of formulae and proofs supporting an efficient implementation of the basic inference engine, as well as of a number of refinements. Completeness and soundness results are shown and a Prolog implementation is described.

I explain what haecceities are and the role they play in our thought and talk about specific individual things, whether those things are concrete or abstract. Everything that can be referred to by using a singular term has a haecceity. I distinguish between singular terms and general terms, on the one hand, and subject terms and predicate terms, on the other. I distinguish three types of sentence: singular predications; general predications; and singular quantifications. I show how singular predications can be (...) eliminated without loss of content, provided we have singular quantifications as well as general predications. This should not be confused with Quine’s “elimination of singular terms.” I say a few words about propositions. (shrink)

Although Harry Woolf’s great collective volume Quantification mostly overlooked biology, Thomas Kuhn’s chapter there on the role of quantitative measurement within the physical sciences maps quite well onto the forms of reasoning that actually persuade us as biologists 50 years later. Kuhn distinguished between two contexts, that of producing quantitative anomalies and that of resolving them. The implied form of reasoning is actually C. S. Peirce’s abduction or inference to the best explanation: “The surprising fact C is observed; (...) but if A were true, C would be a matter of course; hence there is reason to suspect that A is true.” This article reviews abduction and the Kuhnian dichotomy in a range of classic examples where quantitative reasoning has ended arguments in the natural sciences. Included are John Snow’s discovery of the cause of cholera, Jean Perrin’s proof that atoms exist, the discovery of the double helix, the Alvarezes’ explanation of the extraterrestrial origin of the Cretaceous-Tertiary extinction, and current examples in passive smoking, ulcers, and the anthropogenicity of global warming. Modern biology is a quantitative science to the extent that we operate by “strong inference,” the insistence that our data are surprising on everybody else’s hypotheses but follow as a matter of course from our own, and that we demand numerical consilience whenever we infer across levels of analysis or across disciplines. (shrink)

This is the 3rd edition. Although a number of new technological applications require classical deductive computation with non-classical logics, many key technologies still do well—or exclusively, for that matter—with classical logic. In this first volume, we elaborate on classical deductive computing with classical logic. The objective of the main text is to provide the reader with a thorough elaboration on both classical computing – a.k.a. formal languages and automata theory – and (...) classical deduction with the classical first-order predicate calculus with a view to computational implementations, namely in automated theorem proving and logic programming. The present third edition improves on the previous ones by providing an altogether more algorithmic approach: There is now a wholly new section on algorithms and there are in total fourteen clearly isolated algorithms designed in pseudo-code. Other improvements are, for instance, an emphasis on functions in Chapter 1 and more exercises with Turing machines. (shrink)

Identification of distinct items is a basic technique in mathematics. However, identification suffers from a certain weakness of resolve in that it is (classically) accompanied by dropping the original disidentification, which causes a loss of information about the theory which sources the identity. This article proposes an alternative, namely keeping the disidentification along with the identification. This produces an inconsistent theory which is generally an extension of the source theory. The concept of a Dunn–Meyer extension is defined to study (...) these properties. It is seen that this technique is sensitive to the choice of background logic, particularly RM3 as opposed to closed-set logic. By employing the Routley functor, a best-choice logic is found for this construction. (shrink)

The use of the primitive notion of mereological fusion (also known as composition and sum) has been considered by various philosophers and logicians, including Aristotle, G. Leibniz, S. Leśniewski, K. Fine, J. Ketland, T. Schindler, and S. Kleishmid. The problem of finding an axiomatization of Classical Mereology with primitive fusion, instead of the primitive notion of being a part, is quite old and was formally considered by C. Lejewski. Lejewski somehow axiomatized classical mereology using primitive fusion (1962, and (...) also later in 1984). However, his mereology is formulated in a very rich nonclassical language of Leśniewski’s ontology with the essential use of quantification over function symbols, including function symbols “part of” and “sum of.” Lejewski’s approach is not expressible in modern mereologies, and we do not use it in any way. We prove that classical mereology is axiomatizable using primitive mereological fusion within the framework of two-sorted logic. The theory presented here is the first contemporary axiomatization of classical mereology with primitive mereological fusion. (shrink)

Mares and Goldblatt, 163–187, 2006) provided an alternative frame semantics for two quantified extensions of the relevant logic R. In this paper, I show how to extend the Mares-Goldblatt frames to accommodate identity. Simpler frames are provided for two zero-order logics en route to the full logic in order to clarify what is needed for identity and substitution, as opposed to quantification. I close with a comparison of this work with the Fine-Mares models for relevant (...) logics with identity and a discussion of constant and variable domains. (shrink)

We introduce quantificational modal operators as dynamic modalities with (extensions of) Henkin quantifiers as indices. The adoption of matrices of indices (with action identifiers, variables and/or quantified variables as entries) gives an expressive formalism which is here motivated with examples from the area of multi-agent systems. We study the formal properties of the resulting logic which, formally speaking, does not satisfy the normality condition. However, the logic admits a semantics in terms of (an extension of) Kripke structures. As (...) a consequence, standard techniques for normal modal logic become available. We apply these to prove completeness and decidability, and to extend some standard frame results to this logic. (shrink)

The problem of identity across possible worlds is raised with the following questions: What does it mean to say of something in one possible world that it is identical with something in another possible world? How are we to decide whether an individual in one possible world is identical with an individual in another?1 Because the questions concern meaning and verification, it appears to be the case that they presuppose that there are one or more sentences whose meaning (...) and method of verification are in need of explanation. However, although it has been generally presumed to be easy, it turns out to be very difficult to produce a sentence that enables us to formulate the problem and that is not merely supposed to say, but really does say, of an individual in one world and of an individual in another world that the first individual is identical with the second. (shrink)

National identities are to a great extent based on common mythical stories (re)produced by literature and arts; in the long run, the core texts of literature themselves start to function as cultural myths. Performing classical works theatre relates them to the changing social context and thus actualises their meaning. Theatrical representations of national characters and mythical stories participate in reinforcing or redefining national identity. In independent Estonia of the 1990s–2000s the need for reconsidering national values and myths that (...) served to consolidate society in the Soviet period, has become evident. The article focuses on theatrical productions in the turn of the century, which are based on active rewriting of well-known Estonian classics (August Kitzberg, Oskar Luts, and the national epic Kalevipoeg). The article tries to answer two questions: how ingredients of national identity (for instance, the relation to the Other) are displayed and (de)constructed by adapting or rewriting of above-mentioned classics; how textual strategies aimed at semantic transformations are motivated and shaped by the principle of self-reflexive theatrical play. (shrink)

If we wish to formulate an axiomatic truth-theory interpreting a modal language and treat the symbol of necessity as a sentential operator and not as a quantifier over possible worlds, there arise various problems. These are due partly to the fact that words could have meant something other than what they actually mean and partly to certain principles of modal metaphysics. One of those principles is existentialism about propositions: a proposition that is expressed in a sentence containing a non-empty name (...) could not exist if the referent of the name did not exist. The paper explains how the problems arise. It also explains how we can avoid them using substitutional quantification in the metalanguage. The truth-theory we can construct in that way is compositional and homophonic, and its theorems interpreting the various sentences of the modal language are derived in a straightforward way. The paper develops one such theory for a propositional modal language and one for a first-order modal language. The first-order case involves a number of intricacies that are discussed. (shrink)

This chapter considers Geach's claims solely as pertaining to the philosophy of language and philosophical logic, though much of the interest of the concept of relative identity concerns its applicability to other areas: the metaphysical controversy about personal identity and the debate in philosophical theology on the doctrine of the Trinity. It describes Geach's views under six headings: the non‐existence of absolute identity; the sortal relativity of identity; the derelativization thesis; the counting thesis; the thesis (...) of the irreducibility of restricted quantification; and the 'name for an A'/'name of an A' distinction. The sortal relativity thesis depends for its significance on the distinction between sortal terms and non‐sortal terms. An important component of Geach's position is his thesis that for any sortal term 'A' there is a distinction between restricted quantification over As and unrestricted quantification over things that are As. (shrink)

In the present paper we give the first proof-theoretical example of an embedding of classical logic into a quantum-like logic. This is performed in the framework of basic logic, where a proof-theoretical approach to quantum logic is convenient. We consider basic orthologic, that corresponds to a sequential formulation of paraconsistent quantum logic, and which is given by basic orthologic added with weakening and contraction, in a language with Girard's negation. In the paper we first (...) consider a convenient cut-free calculus for classical logic, in the same language; then, in a language enriched with a new kind of literals, we introduce basic orthologic with a primitive modality, where classical logic is embeddable. Similarly to Girard's negation, our modality is not a connective but conceivable as an operator defined inductively on the set of formulae. This allows us to obtain a calculus which enjoys cut-elimination. (shrink)