Starting from certain metalogical results, I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity (...) of its truths. Consequently, each philosophical approach to these truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate. (shrink)

W. V. Quine thinks logical truth can be defined in purely extensional terms, as follows: a logical truth is a true sentence that exemplifies a logical form all of whose instances are true. P. F. Strawson objects that one cannot say what it is for a particular use of a sentence to exemplify a logical form without appealing to intensional notions, and hence that Quine's efforts to define logical truth in purely extensional terms cannot succeed. (...) Quine's reply to this criticism is confused in ways that have not yet been noticed in the literature. This may seem to favour Strawson's side of the debate. In fact, however, a proper analysis of the difficulties that Quine's reply faces suggests a new way to clarify and defend the view that logical truth can be defined in purely extensional terms. (shrink)

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural (...) way. (shrink)

This paper introduces the logic _Q__L__E__T_ _F_, a quantified extension of the logic of evidence and truth _L__E__T_ _F_, together with a corresponding sound and complete first-order non-deterministic valuation semantics. _L__E__T_ _F_ is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (_FDE_) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘_A_ entails that _A_ behaves classically, ∙_A_ follows from _A_’s violating some classically valid inferences. The (...) semantics of _Q__L__E__T_ _F_ combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of _FDE_, _K3_, and _LP_, we show how these tools, which we call here the method of _anti-extensions + valuations_, can be naturally applied to a number of non-classical logics. (shrink)

This paper shows that the logic known as Information-friendly logic (IF-logic) introduced by Jaakko Hintikka and Gabriel Sandu defines its own truth-predicate. The result is interesting given that IF logic is a much stronger logic than ordinary first-order logic and has also a well behaved notion of negation which, on its first-order subfragment, behaves like classical, contradictory negation.

I consider the well-known criticism of Quine's characterization of first-order logical truth that it expands the class of logical truths beyond what is sanctioned by the model-theoretic account. Briefly, I argue that at best the criticism is shallow and can be answered with slight alterations in Quine's account. At worse the criticism is defective because, in part, it is based on a misrepresentation of Quine. This serves not only to clarify Quine's position, but also to (...) crystallize what is and what is not at issue in choosing the model-theoretic account of first-order logical truth over one in terms of substitutions. I conclude by highlighting the need for justifying the belief that the definition of first-order logical truth in terms of models is superior to its definition in terms of substitutions. (shrink)

First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in (...) the aforementioned proof systems when capacity limits are put on the cognitive resources. Finally, we investigate the correlation between a number of mathematical complexity measures defined on graphs and sentences and some psychological complexity measures that were recorded in the experiment. (shrink)

The Language of First-Order Logic is a complete introduction to first-order symbolic logic, consisting of a computer program and a text. The program, an aid to learning and using symbolic notation, allows one to construct symbolic sentences and possible worlds, and verify that a sentence is well formed. The truth or falsity of a sentence can be determined by playing a deductive game with the computer.

This paper begins the study of first-order functions, which are a generalization of truth-functions. The concepts of truth-table and systems of truth-functions, both introduced in propositional logic by Post, are also generalized and studied in the quantificational setting. The general facts about these concepts are given in the first five sections, and constitute a “general theory” of first-order functions. The central theme of this paper is the relation of definition among notions expressed by formulas of (...) first-order logic. We emphasize that logic is not concerned only with the consequence relation among notions expressed by formulas. It also attends to the relation of definition among notions, where a notion is defined from other notions. Sections 5 and 6 deal exclusively with the relation of definition among notions expressed by formulas of first-order logic. In these sections, we study the systems of first-order functions, which are the sets of first-order functions closed under definitions. Sections 7 and 8 are concerned with the relativization of first-order functions to a class of structures. The relativization to a class of structures is a fundamental operation which is used in order to relate the theory of first-order functions with set theory and first-order model theory, a subject which we have barely scratched the surface. The apparatus developed in this paper enables us to define what is a vehicle for the foundation of classical mathematics in set theory, and, in Sect. 8, we prove that first-order logic with one binary predicate variable is not a minimal vehicle for the foundation of classical mathematics in set theory. Sections 9 and 10 introduce further operations and ideals of first-order functions. Besides some results on the influence of the arguments of a first-order function, a result about definability is proved in Sect. 10.1. It is this theorem that provides necessary and sufficient conditions for a first-order function to be in a finitely generated ideal. In Sect. 11, this result is applied to the problem of predicate definability in classes of structures, the problem with which Beth’s theorem dealt in the case of elementary classes. (shrink)

In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early (...) work of Hintikka’s, we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503. (shrink)

For first order languages with no individual constants, empty structures and truth values (for sentences) in them are defined. The first order theories of the empty structures and of all structures (the empty ones included) are axiomatized with modus ponens as the only rule of inference. Compactness is proved and decidability is discussed. Furthermore, some well known theorems of model theory are reconsidered under this new situation. Finally, a word is said on other approaches to the (...) whole problem. (shrink)

In an earlier defense of the view that the fundamental logical properties of logical truth and logical consequence obtain or fail to obtain only relative to contexts, I focused on a variation of Kaplan’s own modal logic of indexicals. In this paper, I state a semantics and sketch a system of proof for a first-order logic of demonstratives, and sketch proofs of soundness and completeness. (I omit details for readability.) That these results obtain for the (...) first-order logic of demonstratives shows that the significance of demonstratives for logic exceeds their behavior as rigid designators in counterfactual reasoning, or reasoning about alternative possibilities. Furthermore, the results in this paper help address one common objection to the view that logical truth and consequence obtain only relative to contexts. According to this objection, the view entails that logical consequence is not formal. (shrink)

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical (...) real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants. (shrink)

The typical rules for truth-trees for first-order logic without functions can fail to generate finite branches for formulas that have finite models–the rule set fails to have the finite tree property. In 1984 Boolos showed that a new rule set proposed by Burgess does have this property. In this paper we address a similar problem with the typical rule set for first-order logic with identity and functions, proposing a new rule set that does have the finite (...) tree property. (shrink)

In my response to Guillermo Rosado-Haddock I discuss the two main issues raised in his paper. The first is that by allowing Henkin’s general models as a legitimate model-theoretic interpretation of second-order logic, I undermine my defense of second-order logic against Quine’s views concerning the primacy of first-order logic. The second is that my treatment of logical truth and logical properties does not take into account various systems of logic and properties of systems (...) of logic such as the Löwenheim-Skolem property.Em minha réplica à Guillermo Rosado-Haddock discuto as duas questões centrais levantadas em seu artigo. A primeira é que ao permitir modelos gerais de Henkin como uma interpretação legítima da lógica de segunda ordem, desvirtuo minha defesa da lógica de segunda ordem contra a visão de Quine respeito à primazia da lógica de primeira ordem. A segunda é que meu tratamento da verdade lógica e das propriedades lógicas não leva em consideração diversos sistemas de lógica e propriedades de sistemas de lógica tais como a propriedade de Löwenheim-Skolem. (shrink)

-/- Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? (...) -/- The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧φ (if and) only if Σ⊢φ ∸2−n for all n < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ. -/- Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ, where φ T o is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable. (shrink)

Classical first-order logic \ is commonly used to study logical connections between statements, that is sentences that in every context have an associated truth-value. Inquisitive first-order logic \ is a conservative extension of \ which captures not only connections between statements, but also between questions. In this paper we prove the disjunction and existence properties for \ relative to inquisitive disjunction Open image in new window and inquisitive existential quantifier \. Moreover we extend these results (...) to several families of theories, among which the one in the language of \. To this end, we initiate a model-theoretic approach to the study of \. In particular, we develop a toolkit of basic constructions in order to transform and combine models of \. (shrink)

Classical first-order logic \ is commonly used to study logical connections between statements, that is sentences that in every context have an associated truth-value. Inquisitive first-order logic \ is a conservative extension of \ which captures not only connections between statements, but also between questions. In this paper we prove the disjunction and existence properties for \ relative to inquisitive disjunction Open image in new window and inquisitive existential quantifier \. Moreover we extend these results (...) to several families of theories, among which the one in the language of \. To this end, we initiate a model-theoretic approach to the study of \. In particular, we develop a toolkit of basic constructions in order to transform and combine models of \. (shrink)

In contrast to Hintikka’s enormously complex distributive normal forms of first- order logic, this paper shows how to generate minimized disjunctive normal forms of first-order logic. An effective algorithm for this purpose is outlined, and the benefits of using minimized disjunctive normal forms to explain the truth conditions of propo- sitions expressible within pure first-order logic are presented.

First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is (...) uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized. (shrink)

Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures to various natural classes of complete metric structures. With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?The primary purpose of this article is to show that a certain, interesting set of axioms does (...) indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is satisfiable if it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊨φ only if Σ⊢φ∸ 2-n for all n < ω. This approximated form of strong completeness asserts that if Σ⊨φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ.Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory recursive? Say that a complete theory T is decidable if for every sentence φ, the value φT is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φT∘ is uniformly recursive from φ, where φT∘is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive axiomatization then it is decidable. (shrink)

This paper investigates the “general” semantics for first-order logic introduced to Antonelli, 637–58, 2013): a sound and complete axiom system is given, and the satisfiability problem for the general semantics is reduced to the satisfiability of formulas in the Guarded Fragment of Andréka et al. :217–274, 1998), thereby showing the former decidable. A truth-tree method is presented in the Appendix.

The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the (...) sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs. (shrink)

DEALING INITIALLY WITH QC, THE STANDARD QUANTIFICATIONAL CALCULUS OF ORDER ONE, THE AUTHOR COMMENTS ON A SHORTCOMING, REPORTED IN 1956 BY MONTAGUE AND HENKIN, IN CHURCH'S ACCOUNT OF A PROOF FROM HYPOTHESES, AND SKETCHES THREE WAYS OF RIGHTING THINGS. THE THIRD, WHICH EXPLOITS A TRICK OF FITCH'S, IS THE SIMPLEST OF THE THREE. THE AUTHOR INVESTIGATES IT SOME, SUPPLYING FRESH PROOF OF UGT, THE UNIVERSAL GENERALIZATION THEOREM. THE PROOF HOLDS GOOD AS ONE PASSES FROM QC TO QC asterisk , (...) THE PRESUPPOSITION-FREE VARIANT OF QC. TURNING NEXT TO QC subscript = , THE STANDARD QUANTIFICATIONAL CALCULUS OF ORDER ONE WITH '=', AND TO THE PRESUPPOSITION-FREE VARIANT QC subscript = OF QC subscript = , THE AUTHOR NEXT ESTABLISHES THE LEMMAS NEEDED THERE TO PROVE UGT. THAT, GIVEN FITCH'S ACCOUNT OF A PROOF FROM HYPOTHESES, UGT HOLDS FOR QC subscript = WAS ARGUED IN LEBLANC'S "TRUTH-VALUE SEMANTICS", BUT THE PROOF WAS IN ERROR. (shrink)

Truth predicates are widely believed to be capable of serving a certain logical or quasi-logical function. There is little consensus, however, on the exact nature of this function. We offer a series of formal results in support of the thesis that disquotational truth is a device to simulate higher-order resources in a first-order setting. More specifically, we show that any theory formulated in a higher-order language can be naturally and conservatively interpreted in a (...) class='Hi'>first-order theory with a disquotational truth or truth-of predicate. In the first part of the paper we focus on the relation between truth and full impredicative sentential quantification. The second part is devoted to the relation between truth-of and full impredicative predicate quantification. (shrink)

Well-written undergraduate-level introduction begins with symbolic logic and set theory, followed by presentation of statement calculus and predicate calculus. First-order theories are discussed in some detail, with special emphasis on number theory. After a discussion of truth and models, the completeness theorem is proved. "...an excellent text."—Mathematical Reviews. Exercises. Bibliography.

We present a compositional semantics for first-order logic with imperfect information that is equivalent to Sevenster and Sandu’s equilibrium semantics (under which the truth value of a sentence in a finite model is equal to the minimax value of its semantic game). Our semantics is a generalization of an earlier semantics developed by the first author that was based on behavioral strategies, rather than mixed strategies.

This paper is an attempt to develop the many-valued first-order fuzzy logic. The set of its truth, values is supposed to be either a finite chain or the interval 0, 1 of reals. These are special cases of a residuated lattice L, , , , , 1, 0. It has been previously proved that the fuzzy propositional logic based on the same sets of truth values is semantically complete. In this paper the syntax and semantics of the (...) class='Hi'>first-order fuzzy logic is developed. Except for the basic connectives and quantifiers, its language may contain also additional n-ary connectives and quantifiers. Many propositions analogous to those in the classical logic are proved. The notion of the fuzzy theory in the first-order fuzzy logic is introduced and its canonical model is constructed. Finally, the extensions of Gödel's completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete. (shrink)

It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) (...) and of the authors' temporal logic of linear discrete time with gaps follows. (shrink)

According to a common definition, the argument of truthful ones is an argument in which the existence of Necessary Being is proved with no presumption of the existence of the possible being. Avicenna proposed different versions of this style of argument and the version in the book of Nejat is one of them. This paper is intended to examine the possibility of proving the logical validity of this version in first-order predicate logic and explain the principles which (...) the argument is based on. In this way, it becomes clear to what extent Avicenna's version is in accordance with the criteria he himself introduces for the argument of truthful ones. In this respect, while providing a clear exposition of the version in the book of Nejat, bugs on some of the statements and proposals to obviate them in terms of replacement arises. Then explaining the concepts and premises which the argument in the book of Nejat is besed on, it is formulated in first-order predicate logic with its full proof. Finally, the proposed model's accordance with the presented criteria for the argument of truthful ones will be shown. (shrink)

In this dissertation, we construct a consistent, complete quotational logic G$\sb1$. We first develop a semantics, and then show the undecidability of circular quotation and anaphorism . Next, a complete axiom system is presented, and completeness theorems are shown for G$\sb1$. We show that definable truth exists in G$\sb1$. ;Later, we replace equality in G$\sb1$ with an equivalence relation. An axiom system and completeness theorems are provided for this equality-free version of G$\sb1$, which is useful in program verification. ;Interpolation (...) and definability are discussed in intensional and extensional contexts. We show that G$\sb1$ contains the complete first order theory of arithmetic by implicit definitional extension. ;The last chapter deals with representability and fixed points. Derived induction schemata are produced, and G$\sb1$ is shown to be a recursion theory. We give an example of how G$\sb1$ may be used to solve domain equations of Scott. (shrink)

The aim of this paper is to introduce a formal system STW of self-referential truth, which extends the classical first-order theory of pure combinators with a truth predicate and certain approximation axioms. STW naturally embodies the mechanisms of general predicate application/abstraction on a par with function application/abstraction; in addition, it allows non-trivial constructions, inspired by generalized recursion theory. As a consequence, STW provides a smooth inner model for Myhill's systems with levels of implication.

The structure of strategies for semantical games is studied by means of a new formalism developed for the purpose. Rigorous definitions of strategy, winning strategy, truth, and falsity are presented. Non-contradiction and bivalence are demonstrated for the truth-definition. The problem of the justification of deduction is examined from this perspective. The rules of a natural deduction system are justified: they are seen to guarantee existence of a winning strategy for the defender in the semantical game for the conclusion, given winning (...) strategies for that player in the games for the premises. Finally, it is shown how semantical games and the truth-definition can be given for languages lacking individual constants. *** DIRECT SUPPORT *** AZ902009 00003. (shrink)

This chapter contains sections titled: Introduction Intensions Models About Quantification Truth in Models Equality Rigidity De Re/De Dicto Partial Designation Designation and Existence Definite Descriptions What Next?

This paper responds to criticism of the Kripkean account of logical truth in first-order modal logic. The criticism, largely ignored in the literature, claims that when the box and diamond are interpreted as the logical modality operators, the Kripkean account is extensionally incorrect because it fails to reflect the fact that all sentences stating truths about what is logically possible are themselves logically necessary. I defend the Kripkean account by arguing that some true sentences about (...) logical possibility are not logically necessary. (shrink)

In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a socalled master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show (...) that if attention is restricted to bidirectional frames, then the expressive power of the master modality is exactly what is needed to translate the bounded fragment of first-order logic into hybrid logic in a truth preserving way. We believe that this throws new light on Arthur Prior's fourth grade tense logic. (shrink)

This somewhat unusual introductory logic text has been clearly designed to bring the student into contact with the mathematical aspects and problems of logical systems as quickly and naturally as possible, at the expense of "fundamental" discussions of logical theory, language and philosophy. In the introductory chapter, the student is introduced to elementary logical technique via Gentzen-type rules of inference, given the requisite set-theoretical background, given a preliminary orientation with respect to the concept of an axiomatic theory, (...) and then shown the full first-order language of the basic predicate calculus. The following chapter develops an axiomatic version of first-order logic with identity, and exhibits proofs of the deduction theorem, completeness theorem, and various applications of equivalence and replacement theorems and other materials to formal number theory. Finally, in what is undoubtably the most valuable part of the book, a good treatment of first-order mathematical theories is given in the third chapter. This includes an extensive discussion of Tarski's semantical theory of truth and a detailed version of Henkin's completeness theorem. Sections on decidability and Gödel's theorem complete this fine introduction to basic mathematical logic. There are numerous exercises throughout the book.—H. P. K. (shrink)

All first-order Gödel logics G_V with globalization operator based on truth value sets V C [0,1] where 0 and 1 lie in the perfect kernel of V are axiomatized by Ciabattoni’s hypersequent calculus HGIF.

In this short paper I am concerned with basically two especially important issues in Oswaldo Chateaubriand’s Logical Forms II; namely, the dispute between first- and higher-order logic and his conception of logical truth and related notions, like logical property, logical state of affairs and logical falsehood. The first issue was also present in the first volume of the book, but the last is privative of the second volume. The extraordinary significance of (...) both issues for philosophy is emphasized and, though there is a basic agreement with Chateaubriand’s views, some critical remarks are interspersed.Neste pequeno artigo considero basicamente duas questões particularmente importantes em Logical Forms II de Oswaldo Chateaubriand; a saber, a disputa entre a lógica de primeira e de segunda ordem e sua concepção de verdade lógica e noções relacionadas, como as de propriedade lógica, estado de coisas lógico e falsidade lógica. A primeira questão também estava presente no primeiro volume do livro, mas a última apenas aparece no segundo volume. Enfatizo o significado extraordinário de ambas as questões para a filosofia e, embora haja uma concordância básica com as visões de Chateaubriand, algumas observações críticas são inseridas. (shrink)

Sedlár and Vigiani [18] have developed an approach to propositional epistemic logics wherein (i) an agent’s beliefs are closed under relevant implication and (ii) the agent is located in a classical possible world (i.e., the non-modal fragment is classical). Here I construct first-order extensions of these logics using the non-Tarskian interpretation of the quantifiers introduced by Mares and Goldblatt [12], and later extended to quantified modal relevant logics by Ferenz [6]. Modular soundness and completeness are proved for constant (...) domain semantics, using non-general frames with Mares–Goldblatt truth conditions. I further detail the relation between the demand that classical possible worlds have Tarskian truth conditions and incompleteness results in quantified relevant logics. (shrink)

This completely self-contained study, widely considered the best book in the field, is intended to serve both as an introduction to quantification theory and as ...

We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, φ, built up from propositional variables (p,q,r,...) and falsity $(\perp)$ using conjunction $(\wedge)$ , disjunction (∨) and implication (→). Write $\vdash\phi$ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula φ there exists a formula Apφ (effectively computable from φ), containing only variables not equal to p which occur in φ, and such that for (...) all formulas ψ not involving $p, \vdash \psi \rightarrow A_p\phi$ if and only if $\vdash \psi \rightarrow \phi$ . Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions. An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra. (shrink)

Reformulation strategies for solving Fitch’s paradox of knowability date back to Edgington. Their core assumption is that the formula \, from which the paradox originates, does not correctly express the intended meaning of the verification thesis, which should concern possible knowledge of actual truths, and therefore the contradiction does not represent a logical refutation of verificationism. Supporters of these solutions claim that can be reformulated in a way that blocks the derivation of the paradox. Unfortunately, these reformulation proposals (...) come with other problems, on both the logical and the philosophical side. We claim that in order to make the reformulation idea consistent and adequate one should analyze the paradox from the point of view of a quantified modal language. An approach in this line was proposed by, among others, Kvanvig but was not fully developed in its technical details. Here we approach the paradox by means of a first order hybrid modal logic, a tool that strikes us as more adequate to express transworld reference and the rigidification needed to consistently express this idea. The outcome of our analysis is ambivalent. Given a first order formula we are able to express the fact that it is knowable in a way which is both consistent and adequate. However, one must give up the possibility of formulating as a substitution free schema of the kind \. We propose that one may instead formulate by means of a recursive translation of the initial formula, being aware that many alternative translations are possible. (shrink)

This paper is about the concept of semantically closed languages. Roughly speaking, those are languages which can name their own sentences and apply to them semantic predicates, such as the truth or satisfaction predicates. Hence, they are “self-referential languages,” in the sense that they are capable of producing sentences about themselves or other sentences in the same language. In section one, we introduce the concept informally; in section two, we provide the formal definition of first-order semantically closed languages, (...) which is the Tarskian definition with some technical modifications. Then, we construct a semantic for this kind of language, and prove that the language is indeed semantically closed (according to Definition 1). Finally, we discuss whether the logic underlying the construction is classical, and future goals of this research. (shrink)

In the article we study the existence predicate \(\varepsilon\) in the context of semantics for first-order modal logic. For a formula \(\varphi\) we define \(\varphi^{\varepsilon}\) - the so called existence relativization. We point to a gap in the work of Fitting and Mendelsohn concerning the relationship between the truth of \(\varphi\) and \(\varphi^{\varepsilon}\) in classes of varying- and constant-domain models. We introduce operations on models which allow us to fill the gap and provide a more general perspective on (...) the issue. As a result we obtain a series of theorems describing the logical connection between the notion of truth of a formula with the existence predicate in constant-domain models and the notion of truth of a formula without the existence predicate in varying-domain models. (shrink)

The aim of this paper is to show what sorts of logics are required by externalist and internalist accounts of the meanings of natural kind nouns. These logics give us a new perspective from which to evaluate the respective positions in the externalist-internalist debate about the meanings of such nouns. The two main claims of the paper are the following: first, that adequate logics for internalism and externalism about natural kind nouns are second-order logics; second, that an internalist (...) second-order logic is a free logic—a second order logic free of existential commitments for natural kind nouns, while an externalist second-order logic is not free of existential commitments for natural kind nouns—it is existentially committed. (shrink)