In this paper a proposal by Henkin of a nominalistic interpretation for second and higher-order logic is developed in detail and analysed. It was proposed as a response to Quine’s claim that second and higher-order logic not only are committed to the existence of sets, but also are committed to the existence of more sets than can ever be referred to in the language. Henkin’s interpretation is rarely cited in the debate on semantics and ontological commitments for (...) these logics, though it has many interesting ideas that are worth exploring. The detailed development will show that it employs an early strategy of using substitutional quantification in order to reduce ontological commitments. It will be argued that the perspective adopted for the predicate variables renders it a natural extension of Quine’s nominalistic interpretation for first-order logic. However, we will argue that, with respect to Quine’s nominalistic program and his notion of ontological commitment, still holds and thus Henkin’s interpretation is not nominalistic. Nevertheless, it will be seen that is addressed successfully and this provides further insights on the so-called “Skolem Paradox”. Moreover, the interpretation is ontologically parsimonious and, in this respect, it arguably fares better than a recent proposal by Bob Hale. (shrink)

A simple Henkin-style completeness proof for Gödel 3-valued propositional logic G3 is provided. The idea is to endow G3 with an under-determined semantics of the type defined by Dunn. The key concept in u-semantics is that of “under-determined interpretation”. It is shown that consistent prime theories built upon G3 can be understood as u-interpretations. In order to prove this fact we follow Brady by defining G3 as an extension of Anderson and Belnap’s positive fragment of First Degree (...) Entailment Logic. (shrink)

Reviews of the papers mentioned in the title.

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with an abundant (...) or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

This paper focuses on the evolution of the notion of completeness in contemporary logic. We discuss the differences between the notions of completeness of a theory, the completeness of a calculus, and the completeness of a logic in the light of Gödel's and Tarski's crucial contributions.We place special emphasis on understanding the differences in how these concepts were used then and now, as well as on the role they play in logic. Nevertheless, we can still observe a certain ambiguity in (...) the use of the close notions of completeness of a calculus and completeness of a logic. We analyze the state of the art under which Gödel's proof of completeness was developed, particularly when dealing with the decision problem for first-order logic. We believe that Gödel had to face the following dilemma: either semantics is decidable, in which case the completeness of the logic is trivial or, completeness is a critical property but in this case it cannot be obtained as a corollary of a previous decidability result. As far as first-order logic is concerned, our thesis is that the contemporary understanding of completeness of a calculus was born as a generalization of the concept of completeness of a theory. The last part of this study is devoted to Henkin's work concerning the generalization of his completeness proof to any logic from his initial work in type theory. (shrink)

In this paper, we introduce a new approach to independent quantifiers, as originally introduced in Informational independence as a semantic phenomenon by Hintikka and Sandu [9] under the header of independence-friendly languages. Unlike other approaches, which rely heavily on compositional methods, we shall analyze independent quantifiers via equilibriums in strategic games. In this approach, coined equilibrium semantics, the value of an IF sentence on a particular structure is determined by the expected utility of the existential player in any of (...) the game’s equilibriums. This approach was suggested in Henkin quantifiers and complete problems by Blass and Gurevich [2] but has not been taken up before. We prove that each rational number can be realized by an IF sentence. We also give a lower and upper bound on the expressive power of IF logic under equilibrium semantics. (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)

The paper proposes a semantic value for the logical constants (connectives and quantifiers) within the framework of proof-theoretic semantics, basic meaning on the introduction rules of a meaning conferring natural deduction proof system. The semantic value is defined based on Fregecontributions” to sentential meanings as determined by the function-argument structure as induced by a type-logical grammar. In doing so, the paper proposes a novel proof-theoretic interpretation of the semantic types, traditionally interpreted in Henkin models. The compositionality of the (...) resulting attribution of semantic values is discussed. Elsewhere, the same method was used for defining proof-theoretic meaning of subsentential phrases in a fragment of natural language. Doing the same for (the simpler and clearer case of) logic sheds more light on the proposal. (shrink)

The paper briefly surveys the sentential proof-theoretic semantics for fragment of English. Then, appealing to a version of Frege’s context-principle (specified to fit type-logical grammar), a method is presented for deriving proof-theoretic meanings for sub-sentential phrases, down to lexical units (words). The sentential meaning is decomposed according to the function-argument structure as determined by the type-logical grammar. In doing so, the paper presents a novel proof-theoretic interpretation of simple type, replacing Montague’s model-theoretic type interpretation (in arbitrary Henkin models). (...) The domains of derivations are collections of derivations in the associated “dedicated” natural-deduction proof-system, and functions therein (with no appeal to models, truth-values and elements of a domain). The compositionality of the semantics is analyzed. (shrink)

In [1] we have characterized four types vagueness related to negation, and constructed the corresponding propositional calculi adequate to formalize each type of vagueness. The calculi obtained were named V0; V1; V2 and C1 . The relations among these calculi and the classical propositional calculus C0 can be represented in the following diagram, where the arrows indicate that a system is a proper subsystem of the other V0 V1 C0 V2 C1 6 1 PP PP PP PiP 1 PP PP (...) PP PiP In this paper we present a two-valued semantics for each of these sys- tems. The semantics used here is the Henkin-style semantics which has proven fruitful in treating other paraconsistent logics. (shrink)

We show that every modal logic (with arbitrary many modalities of arbitrary arity) can be seen as a multi-dimensional modal logic in the sense of Venema. This result shows that we can give every modal logic a uniform "concrete" semantics, as advocated by Henkin et al. This can also be obtained using the unravelling method described by de Rijke. The advantage of our construction is that the obtained class of frames is easily seen to be elementary and that (...) the worlds have a more uniform character. (shrink)

A semantical proof of Craig's interpolation theorem for the intuitionistic predicate logic and some intermediate prepositional logics will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.

ABSTRACT The two main directions pursued in the present paper are the following. The first direction was started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than (...) 2. To extend the characterization beyond multi-modal logics, we look at arbitrary algebraizable logics. We will introduce an algebraic property equivalent with the Craig interpolation property in algebraizable logics, and prove that the superamalgamation property implies the Craig interpolation property. The problem of extending the characterization result to non-normal non-unary modal logics also will be discussed. In the second direction pursued herein: for non-normal modal logic with one unary modality Lemmon [Lem 66] gave a possible worlds semantics. Here we give a more general possible worlds semantics for not necessarily normal multi-modal logics with arbitrarily many not necessarily unary modalities. Strongly related to the above is the theorem, proved, e.g., in Jóns son-Tarski [JT 52] and Henkin-Monk-Tarski [HMT 71], that every normal Boolean algebra with operators can be represented as a subalgebra of the complex algebra of some relational structure. We extend this result to not necessarily normal BAO's as follows. We define partial relational structures and show that every not necessarily normal BAO is embeddable into the complex algebra of a partial relational structure. This gives a possible worlds semantics for not necessarily normal multi-modal logics. (shrink)

The main part of the proof of Kripke's completeness theorem for intuitionistic logic is Henkin's construction. We introduce a new Kripke-type semantics with semilattice structures for intuitionistic logic. The completeness theorem for this semantics can he proved without Henkin's construction.

In this article, a qualitative notion of subjective plausibility and its revision based on a preorder relation are implemented in higher-order logic. This notion of plausibility is used for modeling pragmatic aspects of communication on top of traditional two-dimensional semantic representations.

We consider an example of a sentence which according to Hintikka's claim essentially requires for its logical form a Henkin quantifier. We show that if Hintikka is right then recognizing the truth value of the sentence in finite models is an NP-complete problem. We discuss also possible conclusions from this observation.

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)

In the literature over the Ramsey-sentence approach to structural realism, there is often debate over whether structural realists can legitimately restrict the range of the second-order quantifiers, in order to avoid the Newman problem. In this paper, I argue that even if they are allowed to, it won’t help: even if the Ramsey sentence is interpreted using such restricted quantifiers, it is still an implausible candidate to capture a theory’s structural content. To do so, I use the following observation: if (...) a Ramsey sentence did encode a theory’s structural content, then two theories would be structurally equivalent just in case they have logically equivalent Ramsey sentences. I then argue that this criterion for structural equivalence is implausible, even where frame or Henkin semantics are used. (shrink)

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem and Omitting Types Theorem . We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order logic , logic (...) of order-sorted algebras , preorder algebras , as well as their infinitary variants \ , \ , \ . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent model theory, which consists of borrowing the Omitting Types Property from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to partial algebras and higher-order logic with Henkin semantics , and from the institution of \ to \ and \ . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail. (shrink)

In 1958 Arnould Bayart, 1911-1998, produced a semantics for first and second-order S5 modal logic, and in 1959 a completeness proof for first-order S5, and what he calls a 'quasi-completeness' proof for second-order S5. The 1959 paper is the first completeness proof for modal predicate logic based on the Henkin construction of maximal consistent sets, and indeed may be the easier application of the Henkin method even to propositional modal logic. The semantics is in terms of (...) possible worlds, which, Bayart notes, can be anything at all. The present paper provides an English translation of both these papers together with an historical introduction and a logical commentary. (shrink)

The semantics of e-models for tense logics with binary operators for `until' and `since' (US-logics) was introduced by Bellissima and Bucalo in 1995. In this paper we show the adequacy of these semantics by proving a general Henkin-style completeness theorem. Moreover, we show that for these semantics there holds a Stone-like duality theorem with the algebraic structures that naturally arise from US-logics.

Hartry Field has recently examined the question whether our logical and mathematical concepts are referentially indeterminate. In his view, (1) certain logical notions, such as second-order quantification, are indeterminate, but (2) important mathematical notions, such as the notion of finiteness, are not (they are determinate). In this paper, I assess Field's analysis, and argue that claims (1) and (2) turn out to be inconsistent. After all, given that the notion of finiteness can only be adequately characterized in pure secondorder logic, (...) if Field is right in claiming that second-order quantification is indeterminate (see (1)), it follows that finiteness is also indeterminate (contrary to (2)). After arguing that Field is committed to these claims, I provide a diagnosis of why this inconsistency emerged, and I suggest an alternative, consistent picture of the relationship between logical and mathematical indeterminacy. (shrink)

The transition semantics presented in Rumberg (J Log Lang Inf 25(1):77–108, 2016a) constitutes a fine-grained framework for modeling the interrelation of modality and time in branching time structures. In that framework, sentences of the transition language L_t are evaluated on transition structures at pairs consisting of a moment and a set of transitions. In this paper, we provide a class of first-order definable Kripke structures that preserves L_t-validity w.r.t. transition structures. As a consequence, for a certain fragment of L_t, (...) validity w.r.t. transition structures turns out to be axiomatizable. The result is then extended to the entire language L_t by means of a quite natural ‘Henkin move’, i.e. by relaxing the notion of validity to bundled structures. (shrink)

The semantical structures called T x W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, $[Unrepresented Character]_{o}$ , which consists of the usual propositional language augmented with the Priorean operators P and F and with a possibility operator ◇. However, these structures are also suitable for interpreting an extended language, $[Unrepresented Character]_{so}$ , containing a further possibility operator $\lozenge^{s}$ which expresses synchronism among possibly incompatible histories and which can thus be thought of as a cross-history 'simultaneity' operator. (...) In the present paper we provide an infinite set of axioms in $[Unrepresented Character]_{so}$ , which is shown to be strongly complete for T x W-validity. Von Kutschera (1997) contains a finite axiomatization of T x W-validity which however makes use of the Gabbay Irreflexivity Rule (Gabbay, 1981). In order to avoid using this rule, the proof presented here develops a new technique to deal with reflexive maximal consistent sets in Henkin-style constructions. (shrink)

Chapin reviewed this 1972 ZEITSCHRIFT paper that proves the completeness theorem for the logic of variable-binding-term operators created by Corcoran and his student John Herring in the 1971 LOGIQUE ET ANALYSE paper in which the theorem was conjectured. This leveraging proof extends completeness of ordinary first-order logic to the extension with vbtos. Newton da Costa independently proved the same theorem about the same time using a Henkin-type proof. This 1972 paper builds on the 1971 “Notes on a Semantic Analysis (...) of Variable Binding Term Operators” (Co-author John Herring), Logique et Analyse 55, 646–57. MR0307874 (46 #6989). A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. Kalish-Montague 1964 proposed using vbtos to formalize definite descriptions “the x: x+x=2”, set abstracts {x: F}, minimization in recursive function theory “the least x: x+x>2”, etc. However, they gave no semantics for vbtos. Hatcher 1968 gave a semantics but one that has flaws described in the 1971 paper and admitted by Hatcher. In 1971 we give a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture, proved in this paper with Hatcher’s help, was proved independently about the same time by Newton da Costa. (shrink)

somewhat like Henkin's nonstandard interpretation of higher-order logics, while the right semantics [or logical modalities is an analogue to the standard of type theory in Henkin's sense. interpretation Another possibility would be to follow W.V. Quine's advice to give up logi­ cal modalities as being beyond repair. Or we could also try to develop a logic of conceptual possibility, restricting the range of our "possible worlds" to those compatible with the transcendental presuppositions of our own conceptual sys­ (...) tem. This looks in fact like one of the most interesting possible theories I have dreamt of developing but undoubtedly never will. Its kinship with Kant's way of thinking should be obvious. Besides putting the entire enterprise of possible-worlds semantics into a perspective, we can also see that the actual history of possible-worlds seman­ tics is more complicated than it might first appear to be. For the standard in­ terpretation of modal logics has reared its beautiful head repeatedly in the writings of Stig Kanger, Richard Montague the pre-Montague-semantics theorist, and Nino Cocchiarella. (shrink)

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take amodelto be one of these structures, α, together with some function or relation which associates with every formulaAa subset ofα. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and (...) Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take apropositionof α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers theprimaryinterpretation associated with the semantics. (shrink)

This much-needed book provides a thorough account of temporal logic, one of the most important areas of logic in computer science today. The book begins with a solid introduction to semantical and axiomatic approaches to temporal logic. It goes on to cover predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositional quantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results. With its inclusion of cutting-edge results and unifying methodologies, this book is an indispensable (...) reference for both the pure logician and the theoretical computer scientist. (shrink)

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, (...) it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. (shrink)

The history of building automated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of αβη-equality into (...) account. As a consequence, the higher-order unification problem is undecidable and sets of solutions need not even always have most general elements that represent them. Thus the mentioned calculi for higher-order logic have take special measures to circumvent the problems posed by the theoretical complexity of higher-order unification. In this paper, we will exemplify the methods and proof- and model-theoretic tools needed for extending first-order automated theorem proving to higherorder logic. For the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles of higher-order logic. We will characterize the deductive power of our calculus HT (which is roughly equivalent to these calculi) by the semantics of functional Σ-models. To arrive at a calculus that is complete with respect to Henkin models, we build on ideas from [6] and augment HT with tableau construction rules that use the extensionality principles in a goal-oriented way.. (shrink)

Regarding strictly monadic second‐order logic (SMSOL), which is the fragment of monadic second‐order logic in which all predicate constants are unary and there are no function symbols, we show that a standard deductive system with full comprehension is sound and complete with respect to standard semantics. This result is achieved by showing that in the case of SMSOL, the truth value of any formula in a faithful identity‐standard Henkin structure is preserved when the structure is “standardized”; that is, (...) the predicate domain is expanded into the set of all unary relations. In addition, we obtain a simpler proof of the decidability of SMSOL. (shrink)

In our previous paper [5], we have studied Kripke-type semantics for propositional logics without the contraction rule. In this paper, we will extend our argument to predicate logics without the structure rules. Similarly to the propositional case, we can not carry out Henkin's construction in the predicate case. Besides, there exists a difficulty that the rules of inference () and () are not always valid in our semantics. So, we have to introduce a notion of normal models.

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics (...) for plural logic that does not resort to sets or set-like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic. (shrink)

Church's simple theory of types is a system of higher-order logic in which functions are assumed to be total. We present in this paper a version of Church's system called PF in which functions may be partial. The semantics of PF, which is based on Henkin's general-models semantics, allows terms to be nondenoting but requires formulas to always denote a standard truth value. We prove that PF is complete with respect to its semantics. The reasoning mechanism (...) in PF for partial functions corresponds closely to mathematical practice, and the formulation of PF adheres tightly to the framework of Church's system. (shrink)

Alexius Meinong's Gegenstandstheorie is subject to a formal semantic paradox. The theory of defective objects originally developed by Meinong in response to Ernst Mally's paradox about self-referential thought is rejected as a general solution to paradox in the object theory. The intentionality thesis is also refuted by the counter-example of the unapprehended mountain. It is argued that despite these difficulties, an object theory is required in order to make intuitively correct sense of ontological commitment. ;A version of Meinong's theory is (...) formalized. The logic has non-standard truth value semantics and an objectual interpretation of quantifiers for a domain of existent and non-existent objects. Theories of intentional, referential, and extensional identity are offered, together with two theories of definite description, lambda abstraction, eight systems of alethic modal logic, and formal metatheory. The logic avoids the incompleteness and undecidability limitations of Godel and Church, and all logical, semantic, and set theoretical paradoxes, without type theory or Tarskian hierarchy of metalanguages. It is naively complete and mechanically decidable. ;Wittgenstein's private language argument is refuted as an obstacle to defining private mental objects in the logic. The argument is shown to be valid, but unsound. The object theory logic may therefore provide a mixed or impure private language, and its domain may contain private mental objects as well as ordinary public and non-existent objects. This means that it constitutes an object theory logic of intention, and can be used to represent the logical structure of intentional and specifically phenomenological philosophical theories that require the intelligible designation of private mental objects. ;Appendices include Henkin-type consistency, completeness, and compactness proofs ; criticism of Kazimierz Twardowski's arguments for the distinction between content and object in phenomenological psychology; translation of and commentary on Mally's essay, "Uber die Unabhangigkeit der Gegenstande vom Denken"; a discussion of the distinction between nuclear and extra-nuclear properties, and the eliminability of the modal moment; and a philosophical application of the logic to a solution of the paradox of analysis. (shrink)

Modal logic is the study of modalities - expressions that qualify assertions about the truth of statements - like the ordinary language phrases necessarily, possibly, it is known/believed/ought to be, etc., and computationally or mathematically motivated expressions like provably, at the next state, or after the computation terminates. The study of modalities dates from antiquity, but has been most actively pursued in the last three decades, since the introduction of the methods of Kripke semantics, and now impacts on a (...) wide range of disciplines, including the philosophy of language and linguistics ('possible words' semantics for natural language), constructive mathematics (intuitionistic logic), theoretical computer science (dynamic logic, temporal and other logics for concurrency), and category theory (sheaf semantics). This volume collects together a number of the author's papers on modal logic, beginning with his work on the duality between algebraic and set-theoretic modals, and including two new articles, one on infinitary rules of inference, and the other about recent results on the relationship between modal logic and first-order logic. Another paper on the 'Henkin method' in completeness proofs has been substantially extended to give new applications. Additional articles are concerned with quantum logic, provability logic, the temporal logic of relativistic spacetime, modalities in topos theory, and the logic of programs. (shrink)

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically (...) to a wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic. (shrink)

Theorem 1 states a negative result about the classical semantics j= ! of program schemes. Theorem 2 investigates the reason for this. We conclude that Theorem 2 justies the Henkin-type semantics j= for which the opposite of the present Theorem 1 was proved in [1]{[3] and also in a dierent form in part III of [5]. The strongest positive result on j= is Corollary 6 in [3].

This article is an extended promenade strolling along the winding roads of identity, equality, nameability and completeness, looking for places where they converge. We have distinguished between identity and equality; the first is a binary relation between objects while the second is a symbolic relation between terms. Owing to the central role the notion of identity plays in logic, you can be interested either in how to define it using other logical concepts or in the opposite scheme. In the first (...) case, one investigates what kind of logic is required. In the second case, one is interested in the definition of the other logical concepts in terms of the identity relation, using also abstraction. The present paper investigates whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic a reliable definition of identity is possible. However, the definition needs the standard semantics and we know that with this semantics completeness is lost. We have also studied the relationship of equality with comprehension and extensionality and pointed out the relevant role played by these two axioms in Henkin’s completeness method. We finish our paper with a section devoted to general semantics, where the role played by the nameable hierarchy of types is the key in Henkin’s completeness method. (shrink)

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame (...) class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope. (shrink)

The question motivating my investigation is: Are the basic philosophical principles underlying the "core" system of contemporary logic exhausted by the standard version? In particular, is the accepted narrow construal of the notion "logical term" justified? ;As a point of comparison I refer to systems of 1st-order logic with generalized quantifiers developed by mathematicians and linguists . Based on an analysis of the Tarskian conception of the role of logic I show that the standard division of terms into logical and (...) extra-logical is partly arbitrary. I argue that the semantic principles of Tarskian logic allow any higher order mathematical predicate or relation to function as a logical term in a 1st-order system, provided it is introduced in the right way into the syntactic-semantic apparatus. ;Formally, I propose a new, "constructive", semantic characterization of logical terms. A logical term is defined by a function which, given a model, "shows" how to construct relations, sets, or n-tuples of individuals which satisfy it. I discuss the linguistic applicability of the extended logic and bring numerous examples. ;One chapter is devoted to branching quantification--a non-standard logico-linguistic construction obtained by affixing a non-linear, partially-ordered quantifier-prefix to a well-formed formula. Although standard branching quantifiers were given a complete account by Henkin and Walkoe, it is not altogether clear what the meaning of branching generalized quantifiers is. I propose a new analysis of the generalized branching prefix which extends the existent, partial, definitions due to Barwise, and I develop a generalization leading to a "family" of branching structures. ;In conclusion I discuss the philosophical impact of the generalized conception of logic on such issues as the logicist thesis, the interaction between mathematics and logic, ontological commitment via logic, metaphysics and logical semantics. (shrink)

Fibring is defined as a mechanism for combining logics with a first-order base, at both the semantic and deductive levels. A completeness theorem is established for a wide class of such logics, using a variation of the Henkin method that takes advantage of the presence of equality and inequality in the logic. As a corollary, completeness is shown to be preserved when fibring logics in that class. A modal first-order logic is obtained as a fibring where neither the Barcan (...) formula nor its converse hold. (shrink)

Fix 2 < n < ω. Let CA n denote the class of cylindric algebras of dimension n, and let RCA n denote the variety of representable CA n ’s. Let L n denote first-order logic restricted to the first n variables. Roughly, CA n, an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of L n, namely, its proof theory, while RCA n algebraically and geometrically represents the Tarskian semantics of L n. Unlike (...) Boolean algebras having a Stone representation theorem, RCA n ⊊ CA n. Using combinatorial game theory, we show that the existence of certain finite relation algebras RA, which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for L n. Using special cases of such finite RA ’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on RCA n and we re-prove Hirsch and Hodkinson’s result that the class of completely representable CA n ’s is not first-order definable. We show that if T is an L n countable theory that admits elimination of quantifiers, λ is a cardinal < 2 ℵ 0, and F = 〈 Γ i : i < λ 〉 is a family of complete nonprincipal types, then F can be omitted in an ordinary countable model of T. (shrink)

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

The criterion of naturalness represents David Lewis’s attempt to answer some of the sceptical arguments in (meta-) semantics by comparing the naturalness of meaning candidates. Recently, the criterion has been challenged by a new sceptical argument. Williams argues that the criterion cannot rule out the candidates which are not permuted versions of an intended interpretation. He presents such a candidate – the arithmetical interpretation (a specific instantiation of The criterion of naturalness represents David Lewis’s attempt to answer some of (...) the sceptical arguments in (meta-) semantics by comparing the naturalness of meaning candidates. Recently, the criterion has been challenged by a new sceptical argument. Williams argues that the criterion cannot rule out the candidates which are not permuted versions of an intended interpretation. He presents such a candidate – the arithmetical interpretation (a specific instantiation of Henkin’s model), and he argues that it opens up the possibility of Pythagorean worlds, i.e. the worlds similar to ours in which the arithmetical interpretation is the best candidate for a semantic theory. The aim of this paper is a) to reconsider the general conditions for the applicability of Lewis’s criterion of naturalness and b) to show that Williams’s new sceptical challenge is based on a problematic assumption that the arithmetical interpretation is independent of fundamental properties and relations. As I show, if the criterion of naturalness is applied properly, it can respond even to the new sceptical challenge. (shrink)

The [lambda]-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of [lambda]-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is [lambda]-definable. These results subsume earlier ones using cartesian closed categories, as (...) well as those employing so-called Henkin and Kripke [lambda]-models. (shrink)

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid to which properties of theories result in the presence of which rules of inference, and to restrictions on the sets of formulas to which the rules may be employed, (...) restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck. (shrink)