Consider the bounded variable logics $L^k_{\infty\omega}$ (with k variable symbols), and $C^k_{\infty\omega}$ (with k variables in the presence of counting quantifiers $\exists^{\geq m}$ ). These fragments of infinitary logic $L_{\infty\omega}$ are well known to provide an adequate logical framework for some important issues in finite model theory. This paper deals with a translation that associates equivalence of structures in the k-variable fragments with bisimulation equivalence between derived structures. Apart from a uniform and intuitively appealing treatment of these equivalences, (...) this approach relates some interesting issues for the case of an arbitrary number of variables to the case of just three variables. Invertibility of the invariants for $\equiv_{C^3}$ , in particular, would imply a positive answer to the tempting conjecture that fixed-point logic with counting captures $\bigcup_k$ Ptime $\cap C^k_{\infty\omega}$. (shrink)

Given a regular cardinal κ such that κ<κ=κ (e.g., if the generalized continuum hypothesis holds), we develop a proof system for classical infinitary logic that includes heterogeneous quantification (i.e., infinite alternating sequences of quantifiers) within the language Lκ+,κ, where there are conjunctions and disjunctions of at most κ many formulas and quantification (including the heterogeneous one) is applied to less than κ many variables. This type of quantification is interpreted in Set using the usual second-order formulation in terms (...) of strategies for games, and the axioms are based on a stronger variant of the axiom of determinacy for game semantics. With respect to this axiom system we prove the soundness and completeness theorem with respect to a class of set-valued structures that we call well determined. We also investigate intuitionistic systems with heterogeneous quantifiers for Lκ+,κ,κ (when only conjunctions of less than κ many formulas are allowed), and prove analogously a completeness theorem with respect to well-determined structures in categories in general, in κ-Grothendieck topoi in particular, and, when κ<κ=κ, also in Kripke models. Finally, we consider an extension of our system in which heterogeneous quantification with bounded quantifiers is expressible, and extend our completeness results to that case. (shrink)

ABSTRACT Quine insisted that the satisfaction of an open modalised formula by an object depends on how that object is described. Kripke's ‘objectual’ interpretation of quantified modal logic, whereby variables are rigid, is commonly thought to avoid these Quinean worries. Yet there remain residual Quinean worries for epistemic modality. Theorists have recently been toying with assignment-shifting treatments of epistemic contexts. On such views an epistemic operator ends up binding all the variables in its scope. One might worry that this (...) yields the undesirable result that any attempt to ‘quantify in’ to an epistemic environment is blocked. If quantifying into the relevant constructions is vacuous, then such views would seem hopelessly misguided and empirically inadequate. But a famous alternative to Kripke's semantics, namely Lewis' counterpart semantics, also faces this worry since it also treats the boxes and diamonds as assignment-shifting devices. As I'll demonstrate, the mere fact that a variable is bound is no obstacle to binding it. This provides a helpful lesson for those modelling de re epistemic contexts with assignment sensitivity, and perhaps leads the way toward the proper treatment of binding in both metaphysical and epistemic contexts: Kripke for metaphysical modality, Lewis for epistemic modality. (shrink)

Let L ω ∞ω be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. It is shown that if p(n) ≪ n -1 satisfies certain simple conditions on its growth rate, then for every σ∈ L ω ∞ω , the probability that σ holds for (...) the random graph on n vertices converges. In fact, if $p(n) = n^{-\alpha}, \alpha > 1$ , then the probability is either smaller than 2 -n d for some $d > 0$ , or it is asymptotic to cn -d for some $c > 0, d \geq 0$ . Results on the difficulty of computing the asymptotic probability are given. (shrink)

We carry out a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory, such as least fixpoint logic LFP, partial fixpoint logic PFP, infinitary logic Lω∞ω with a finite number of variables, as well as the closures of these logics under implicit definitions. Moreover, we show that the (...) upper and lower bounds established here cannot be made substantially tighter, unless outstanding conjectures in complexity theory are resolved at the same time. (shrink)

First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L∞ωω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L∞ωω to obtain the logics L∞ωω, where Q (...) = {Qi: iε I} is a family of generalized quantifiers on finite structures. Using the logics L∞ωω, we can express polynomial-time properties that are not definable in L∞ωω, such as “there is an even number of x” and “there exists at least n/2 x” , without going to second-order logic. We show that equivalence of finite structures relative to L∞ωω can be characterized in terms of certain pebble games that are a variant of the Ehrenfeucht—Fraïssé games. We combine this game-theoretic characterization with sophisticated combinatorial tools from Ramsey theory, such as van der Waerden's Theorem and Folkman's Theorem, in order to investigate the scope and limits of generalized quantifiers in finite model theory. We obtain sharp lower bounds for expressibility in the logics L∞ωω and discover an intrinsic difference between adding finitely many simple unary generalized quantifiers to L∞ωω adding infinitely many. In particular, we show that if Qis a finite sequence of simple unary generalized quantifiers, then the equicardinality, or Härtig, quantifier is not definable in L∞ωω. We also show that the query “does the equivalence relation E have an even number of equivalence classes” is not definable in the extension L∞ωω of L∞ωω by the Härtig quantifier I and any finite sequence Q of simple unary generalized quantifiers. (shrink)

We establish a general hierarchy theorem for quantifier classes in the infinitary logic L∞ωωon finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure.This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This (...) proves a conjecture of Immerman.We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog. (shrink)

We provide a strongly complete infinitary proof system for hybrid logic. This proof system can be extended with countably many sequents. Thus, although these logics may be non-compact, strong completeness proofs are provided for infinitary hybrid versions of non-compact logics like ancestral logic and Segerberg’s modal logic with the bounded chain condition. This extends the completeness result for hybrid logics by Gargov, Passy, and Tinchev.

As a result of these restrictions, an implementation of L [subscript lambda] does not need to implement full higher-order unification. Instead, an extension to first-order unification that respects bound variable names and scopes is all that is required. Such unification problems are shown to be decidable and to possess most general unifiers when unifiers exist. A unification algorithm and logic programming interpreter are described and proved correct. Several examples of using L[subscript lambda] as a meta-programming language are presented.

(Though it is now known that many pronouns once lumped under ”bound variables” are in fact referential indefinites or other phenomena better accounted for in a DRT-like view of referents, there remain many true instances of sentenceinternally bound anaphora: this talk concerns only the latter.) Almost all versions of categorial grammar (CG) are differentiated from other syntactic theories in treating a multi-argument verb as an Ò-place predicate phrase (PrdP) that combines with a NP or other argument to yield a (Ò-1)-place (...) PrdP (which, if (n–1) 1, then combines with another argument to yield a ´Ò ¾µ-place PrdP, until a sentence (0-place PrdP) results) – a ”curried function” account of argument structure. (In CG, Ú ÔÜ ÒÔ · ÒÔ Üµ. A number of CG analyses of Ú Ô bound anaphora (Bach & Partee 1980, Chierchia 1988, Szabolcsi 1992, Jacobson 1991, Dowty 1993, Jacobson 1999), though otherwise diverse, have in common that they treat anaphoric binding as a process that affects (only) a predicate phrase (PrdP) — usually (finite or non-finite) Ú Ô, else ÚÔ ÒÔ — by binding an internal pronoun, i.e. ”binding at the VP level”; semantically, this is usually indicated ´Üµ . The result of this binding is that the next argument the.. (shrink)

Infinitary action logic can be naturally expanded by adding exponential and subexponential modalities from linear logic. In this article we shall develop infinitary action logic with a subexponential that allows multiplexing (instead of contraction). Both non-commutative and commutative versions of this logic will be considered, presented as infinitary sequent calculi. We shall prove cut admissibility for these calculi, and estimate the complexity of the corresponding derivability problems: in both cases it will turn out (...) to be between complete first-order arithmetic and the \(\omega ^\omega \) level of the hyperarithmetical hierarchy. Here the complexity upper bound is much lower than that for the system with a subexponential that allows contraction. The complexity lower bound in turn is much higher than that for infinitary action logic. (shrink)

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider (...) class='Hi'>infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time or polynomial space can be characterized as collections of all classes of finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth. (shrink)

We describe an extension to our quantifier-free computational logic to provide the expressive power and convenience of bounded quantifiers and partial functions. By quantifier we mean a formal construct which introduces a bound or indicial variable whose scope is some subexpression of the quantifier expression. A familiar quantifier is the Σ operator which sums the values of an expression over some range of values on the bound variable. Our method is to represent expressions of the logic as objects (...) in the logic, to define an interpreter for such expressions as a function in the logic, and then define quantifiers as "mapping functions." The novelty of our approach lies in the formalization of the interpreter and its interaction with the underlying logic. Our method has several advantages over other formal systems that provide quantifiers and partial functions in a logical setting. The most important advantage is that proofs not involving quantification or partial recursive functions are not complicated by such notions as "capturing," "bottom," or "continuity." Naturally enough, our formalization of the partial functions is nonconstructive. The theorem prover for the logic has been modified to support these new features. We describe the modifications. The system has proved many theorems that could not previously be stated in our logic. Among them are. (shrink)

A back and forth condition on interpretations for those second-order languages without functional variables whose non-logical vocabulary is finite and excludes functional constants is presented. It is shown that this condition is necessary and sufficient for the interpretations to be equivalent in the language. When applied to second-order languages with an infinite non-logical vocabulary, excluding functional constants, the back and forth condition is sufficient but not necessary. It is shown that there is a class of infinitary second-order languages whose (...) non-logical vocabulary is infinite for which the back and forth condition is both necessary and sufficient. It is also shown that some applications of the back and forth construction for second-order languages can be extended to the infinitary second-order languages. (shrink)

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke–Feferman with its intended semantics. The method we apply is based on infinitary proof systems containing an ω-rule.

We are used to the idea that computers operate on numbers, yet another kind of data is equally important: the syntax of formal languages, with variables, binding, and alpha-equivalence. The original application of nominal techniques, and the one with greatest prominence in this paper, is to reasoning on formal syntax with variables and binding. Variables can be modelled in many ways: for instance as numbers (since we usually take countably many of them); as links (since they may `point' to a (...) binding site in the term, where they are bound); or as functions (since they often, though not always, represent `an unknown'). None of these models is perfect. In every case for the models above, problems arise when trying to use them as a basis for a fully formal mechanical treatment of formal language. The problems are practical—but their underlying cause may be mathematical. The issue is not whether formal syntax exists, since clearly it does, so much as what kind of mathematical structure it is. To illustrate this point by a parody, logical derivations can be modelled using a Gödel encoding (i.e., injected into the natural numbers). It would be false to conclude from this that proof-theory is a branch of number theory and can be understood in terms of, say, Peano's axioms. Similarly, as it turns out, it is false to conclude from the fact that variables can be encoded e.g., as numbers, that the theory of syntax-with-binding can be understood in terms of the theory of syntax-without-binding, plus the theory of numbers (or, taking this to a logical extreme, purely in terms of the theory of numbers). It cannot; something else is going on. What that something else is, has not yet been fully understood. In nominal techniques, variables are an instance of names, and names are data. We model names using urelemente with properties that, pleasingly enough, turn out to have been investigated by Fraenkel and Mostowski in the first half of the 20th century for a completely different purpose than modelling formal language. What makes this model really interesting is that it gives names distinctive properties which can be related to useful logic and programming principles for formal syntax. Since the initial publications, advances in the mathematics and presentation have been introduced piecemeal in the literature. This paper provides in a single accessible document an updated development of the foundations of nominal techniques. This gives the reader easy access to updated results and new proofs which they would otherwise have to search across two or more papers to find, and full proofs that in other publications may have been elided. We also include some new material not appearing elsewhere. (shrink)

We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an$\omega $-rule, and prove that the derivability problem in this calculus is$\Pi _1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a (...) Lambek grammar with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007). (shrink)

If the language is extended by new individual variables, in classical first order logic, then the deduction system obtained is a conservative extension of the original one. This fails to be true for the logics with infinitary predicates. But it is shown that restricting the commutativity of quantifiers and the equality axioms in the extended system and supposing the merry-go-round property in the original system, the foregoing extension is already conservative. It is shown that these restrictions are crucial (...) for an extension to be conservative. The origin of the results is algebraic logic. (shrink)

Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $, we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that (...) is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory, that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic. (shrink)

What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this (...) claim.) As a counterproposal, then, we define a large fragment of predicate logic characterized by its use of only bounded quantification. This so-called guarded fragment enjoys the above nice properties, including decidability, through an effectively bounded finite model property. (These are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like modal operators. This observation inspires a generalized semantics for first-order predicate logic with accessibility constraints on available assignments (cf. N´emeti 1986, 1992) which moves the earlier quantifier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candidates sharing various desirable features with basic modal logic – in particular, its decidability. (shrink)

The complexity of the satisfiability problems of various arrow logics and cylindric modal logics is determined. As is well known, relativising these logics makes them decidable. There are several parameters that can be set in such a relativisation. We focus on the following three: the number of variables involved, the similarity type and the kind of relativised models considered. The complexity analysis shows the importance and relevance of these parameters.

We consider infinitary logic with only two variable symbols, both with and without counting quantifiers, i.e. L2 L∞ω2 and C2 L∞ω2mεω. The main result is that finite relational structures admit canonization with respect to L2 and C2: there are polynomial time com putable functors mapping finite relational structures to unique representatives of their equivalence class with respect to indistinguishability in either of these logics. In fact we exhibit in verses to the natural invariants that characterize structures up to (...) L2- or C2-equivalence, respectively. As a corollary we obtain recursive presentations of the classes of all boolean queries that are closed with respect to L2- or C2-equivalence. (shrink)

We identify the computational complexity of the satisfiability problem for FO 2 , the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO 2 has the finite-model property, which means that if an FO 2 -sentence is satisfiable, then it has a (...) finite model. Moreover, Mortimer showed that every satisfiable FO 2 -sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO 2 -sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO 2 is NEXPTIME-complete. (shrink)

We consider a new fragment of first-order logic with two variables. This logic is defined over interval structures. It constitutes unary predicates, a binary predicate and a function symbol. Considering such a fragment of first-order logic is motivated by defining a general framework for event-based interval temporal logics. In this paper, we present a sound, complete and terminating decision procedure for this logic. We show that the logic is decidable, and provide a NEXPTIME complexity bound (...) for satisfiability. This result shows that even a simple decidable fragment of first-order logic has NEXPTIME complexity. (shrink)

We study first-order logic with two variables FO² and establish a small substructure property. Similar to the small model property for FO² we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO² under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO² has the finite model property and is complete for (...) non-deterministic exponential time, just as for plain FO² With two equivalence relations, FO² does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO² is undecidable. (shrink)

In this paper it is argued that Hintikka's game theoreticalsemantics for Independence Friendly logic does not formalize theintuitions about independent choices; it rather is aformalization of imperfect information. Furthermore it is shownthat the logic has several remarkable properties (e.g.,renaming of bound variables is not allowed). An alternativesemantics is proposed which formalizes intuitions aboutindependence.

We show that both the $n$-density and the bounded $n$-width of Kripke frames can be modally defined not only with natural and well-known Sahlqvist formulae containing a linear number of different propositional variables but also with formulae of polynomial length with a logarithmic number of different propositional variables and then we prove that this exponential decrease in the number of variables leads us outside the class of Sahlqvist formulae.

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p 1, . . . , p n , = (p 1, . . . , p n-1, p n ) intuitively states that the value of p n is determined by those of p 1, . . . , p n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is (...) complete for nondeterministic exponential time.In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ${\wedge, \square, \lozenge}$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness.We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction.In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster. (shrink)

Inspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε 0 -recursive function [] 0 : T → ω so that a reduces to b implies [a] $_0 > [b]_0$ . The construction of [] 0 is based on a careful analysis of the Howard-Schütte (...) treatment of Gödel's T and utilizes the collapsing function ψ: ε 0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [] 0 is also crucially based on ideas developed in the 1995 paper "A proof of strongly uniform termination for Gödel's T by the method of local predicativity" by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper. Indeed, for given n let T n be the subsystem of T in which the recursors have type level less than or equal to n+2. (By definition, case distinction functionals for every type are also contained in T n .) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the T n -derivation lengths in terms of ω n+2 -descent recursive functions. The derivation lengths of type one functionals from T n (hence also their computational complexities) are classified optimally in terms of $ -descent recursive functions. In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T 0 is primitive recursive, thus any type one functional a in T 0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T 1 in terms of multiple recursion. As proof-theoretic corollaries we reobtain the classification of the IΣ n+1 -provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO ( $\varepsilon_0) \vdash \Pi^0_2$ - Refl(PA) and PRA + PRWO ( $\omega_{n+2}) \vdash \Pi^0_2$ - Refl(I Σ n+1 ), hence PRA + PRWO( $\epsilon_0) \vdash$ Con(PA) and PRA + PRWO( $\omega_{n+2}) \vdash$ Con(IΣ n+1 ). For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity. (shrink)

Kaplan (1989a) insists that natural languages do not contain displacing devices that operate on character—such displacing devices are called monsters. This thesis has recently faced various empirical challenges (e.g., Schlenker 2003; Anand and Nevins 2004). In this note, the thesis is challenged on grounds of a more theoretical nature. It is argued that the standard compositional semantics of variable binding employs monstrous operations. As a dramatic first example, Kaplan’s formal language, the Logic of Demonstratives, is shown to contain monsters. (...) For similar reasons, the orthodox lambda-calculus-based semantics for variable binding is argued to be monstrous. This technical point promises to provide some far-reaching implications for our understanding of semantic theory and content. The theoretical upshot of the discussion is at least threefold: (i) the Kaplanian thesis that “directly referential” terms are not shiftable/bindable is unmotivated, (ii) since monsters operate on something distinct from the assertoric content of their operands, we must distinguish ingredient sense from assertoric content (cf. Dummett 1973; Evans 1979; Stanley 1997), and (iii) since the case of variable binding provides a paradigm of semantic shift that differs from the other types, it is plausible to think that indexicals—which are standardly treated by means of the assignment function—might undergo the same kind of shift. (shrink)

-/- 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 proposed using vbtos to formalize definite descriptions, set abstracts {x: F}, minimalization in recursive function theory, etc. However, they gave no sematics for vbtos. Hatcher gave a semantics but one that has flaws. 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 was later proved independently by the authors and by Newton da Costa. -/- The expression (vy:F) is called a variable bound term (vbt). In case F has only y free, (vy:F) has the syntactic propreties of an individual constant; and under a suitable interpretation of the language vy:F) denotes an individual. By a semantic analysis of vbtos we mean a proposal for amending the standard notions of (1) "an interpretation o f a first -order language" and (2) " the denotation of a term under an interpretation and an assignment", such that (1') an interpretation o f a first -order language associates a set-theoretic structure with each vbto and (2') under any interpretation and assignment each vb t denotes an individual. (shrink)

By a free logic is generally meant a variant of classical first-order logic in which constant terms may, under interpretation, fail to refer to individuals in the domain D over which the bound variables range, either because they do not refer at all or because they refer to individuals outside D. If D is identified with what is assumed by the given interpretation to exist, in accord with Quine’s dictum that “to be is to be the value of (...) a [bound] variable,” then a free variation on classical semantics does not require that all constant terms refer to existents, and in this sense such terms lack existential import. (shrink)

We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of (...) first order logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic. (shrink)

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives (...) cannot be so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games. (shrink)

Montague [7] translates English into a tensed intensional logic, an extension of the typed -calculus. We prove that each translation reduces to a formula without -applications, unique to within change of bound variable. The proof has two main steps. We first prove that translations of English phrases have the special property that arguments to functions are modally closed. We then show that formulas in which arguments are modally closed have a unique fully reduced -normal form. As a corollary, translations (...) of English phrases are contained in a simply defined proper subclass of the formulas of the intensional logic. (shrink)

Standard (classical) logic is not independent of set theory. Which formulas are valid in logic depends on which sets we assume to exist in our set-theoretical universe. Second-order logic is just set theory in disguise. The typically logical notions of validity and consequence are not well defined in second-order logic, at least as long as there are open issues in set theory. Such contentious issues in set theory as the axiom of choice, the continuum hypothesis or (...) the existence of inaccessible cardinals, can be equivalently transformed into question about the logical validity of pure sentences of second-order logic, where “pure” means that they only contain logical symbols and bound variables. Even standard first-order logic depends on the acceptance on infinite sets in our set-theoretical universe. Should we choose to admit only finite sets, the number of logically valid pure first-order formulas would increase dramatically and first-order logic would not be recursively enumerable any longer. (shrink)

A sentence of first-order logic is satisfiable if it is true in some structure, and finitely satisfiable if it is true in some finite structure. The question arises as to whether there exists an algorithm for determining whether a given formula of first-order logic is satisfiable, or indeed finitely satisfiable. This question was answered negatively in 1936 by Church and Turing (for satisfiability) and in 1950 by Trakhtenbrot (for finite satisfiability).In contrast, the satisfiability and finite satisfiability problems are (...) algorithmically solvable for restricted subsets---or, as we say, fragments---of first-order logic, a fact which is today of considerable interest in Computer Science. This book provides an up-to-date survey of the principal axes of research, charting the limits of decision in first-order logic and exploring the trade-off between expressive power and complexity of reasoning. Divided into three parts, the book considers for which fragments of first-order logic there is an effective method for determining satisfiability or finite satisfiability. Furthermore, if these problems are decidable for some fragment, what is their computational complexity? Part I focusses on fragments defined by restricting the set of available formulas. Topics covered include the Aristotelian syllogistic and its relatives, the two-variable fragment, the guarded fragment, the quantifier-prefix fragments and the fluted fragment. Part II investigates logics with counting quantifiers. Starting with De Morgan's numerical generalization of the Aristotelian syllogistic, we proceed to the two-variable fragment with counting quantifiers and its guarded subfragment, explaining the applications of the latter to the problem of query answering in structured data. Part III concerns logics characterized by semantic constraints, limiting the available interpretations of certain predicates. Taking propositional modal logic and graded modal logic as our cue, we return to the satisfiability problem for two-variable first-order logic and its relatives, but this time with certain distinguished binary predicates constrained to be interpreted as equivalence relations or transitive relations. The work finishes, slightly breaching the bounds of first-order logic proper, with a chapter on logics interpreted over trees. (shrink)

Combinatory logic (Curry and Feys 1958) is a “variable-free” alternative to the lambda calculus. The two have the same expressive power but build their expressions differently. “Variable-free” semantics is, more precisely, “free of variable binding”: it has no operation like abstraction that turns a free variable into a bound one; it uses combinators—operations on functions—instead. For the general linguistic motivation of this approach, see the works of Steedman, Szabolcsi, and Jacobson, among others. The standard view in linguistics is that (...) reflexive and personal pronouns are free variables that get bound by an antecedent through some coindexing mechanism. In variable free semantics the same task is performed by some combinator that identifies two arguments of the function it operates on (a duplicator). This combinator may be built into the lexical semantics of the pronoun, into that of the antecedent, or it may be a free-floating operation applicable to predicates or larger chunks of texts, i.e. a typeshifter. This note is concerned with the case of cross-sentential anaphora. It adopts Hepple’s and Jacobson’s interpretation of pronouns as identity maps and asks how this can be extended to the cross-sentential case, assuming the dynamic semantic view of anaphora. It first outlines the possibility of interpreting indefinites that antecede non-ccommanded pronouns as existential quantifiers enriched with a duplicator. Then it argues that it is preferable to use the duplicator as a type-shifter that applies “on the fly”. The proposal has consequences for two central ingredients of the classical dynamic semantic treatment: it does away with abstraction over assignments and with treating indefinites as inherently existentially quantified. However, cross-sentential anaphora remains a matter of binding, and the idea of propositions as context change potentials is retained. (shrink)

The year 2012 has witnessed worldwide celebrations of Alan Turing’s 100th birthday. A great number of conferences and workshops were organized by logicians, computer scientists and researchers in AI, showing the continued flourishing of computer science, and the fruitful interfaces between logic and computer science. Logic is no longer just the concept that Frege had about one hundred years ago, let alone that of Aristotle twenty centuries before. One of the prominent features of contemporary logic is its (...) interdisciplinary character, connecting across mathematics, philosophy, modern computer science, and even the cognitive and social sciences. -/- China is no exception. This special issue explores interfaces of logic with AI and other disciplines, and it attempts to give a bird’s eye view of the developing landscape of logic and AI in China. Our contributors come from computer science, AI, philosophy and linguistics, united by their interest in using formal logical tools and notions to deal with substantial problems in their fields. We are aware that AI is a vast area, and that the limitations of space in a volume like this may provide a somewhat incomplete or biased view. Still we hope that the six papers selected here illustrate the concerns and curiosity driving Chinese researchers, on topics shared with our colleagues around the world. -/- One of the active research areas in AI is planning under uncertainty, with incomplete knowledge, where a well-established logical framework is the action language of Son and Baral (Son and Baral 2001). This language can express the notions of sensing actions and making conditional plans. However, its current semantics is not yet satisfactory, since according to the semantics an agent would not have knowledge of what happens after a plan is executed, which can be a serious problem in a safety-critical environment. In their paper Proof Systems for Planning under Cautious Semantics (Shen and Zhao 2013), Yuping Shen and Xishun Zhao address this problem and propose a cautious and weakly cautious semantics for the action language. This new semantics enables the agent to know exactly what happens after executing an action. Hoare style proof systems are then proposed and proved to be sound and complete. What is more, the proposed new system reduces the reasoning complexity of the action language. -/- In Preferential Semantics for Plausible Subsumption in Possibility Theory (Qi and Zhang 2013), Guilin Qi and Zhizheng Zhang investigate another important issue in AI: how to handle exceptions in a knowledge-based system. This has become a hot topic in recent years, especially in ontological design, where description logics are widely used. The question then arises whether we can get a nonmonotonic extension of description logic. To answer this, the authors first propose three preferential semantics for plausible subsumption in the framework of possibility theory (Dubois and Prade 1985) and study their properties and their relationships. Then they introduce preferential subsumptions to a description logic-based knowledge system, and define an entailment relation of plausible subsumption. Moreover, they study reductions of the new semantics to standard semantics for an expressive description logic. -/- Counterfactual reasoning has been explored extensively by philosophers, cf. Lewis (1973), Stalnaker (1968). But in the last two decades, causal models have been proposed for modeling counterfactual reasoning by researchers in AI. In A Lewisian Logic of Causal Counterfactuals (Zhang 2013), Jiji Zhang studies classes of causal models and provides formal characterizations. In particular, a new logic system is proposed and proved complete for the class of ‘Lewisian models’ satisfying all the principles of David Lewis’s logic of counterfactuals. Moreover, Zhang compares his new framework with those of Lewis and Stalnaker, and also establishes formal connections with more recent approaches by Galles and Pearl, and by Halpern. -/- Consistency is a fundamental notion in logic, but it also plays a key role in many techniques for solving constraint satisfaction problems (Mackworth 1977). In Variable-Centered Consistency in Model RB (Li et al. 2013), Liang Li, Tian Liu and Ke Xu start with a discussion of i-consistency in ‘Model RB’ and ‘Model RD’, two models of random constraint satisfaction problems with growing domain size, then compare it with the notion of t-consistency and strong t-consistency in well-known earlier work by Freuder. After that, the authors define a new kind of variable-centered consistency, and show an upper bound on a suitably chosen parameter in Model RB and Model RD, such that up to this bound, all typical instances of Model RB are variable-centered consistent. -/- Language understanding is a cognitive process that human beings carry out every day. Metaphor is a pervasive phenomenon in natural languages, and its cognitive role calls for interpretation. There is a growing body of literature on cognitive mechanisms underlying metaphor. The paper An Ontology-Based Approach to Metaphor Cognitive Computation (Huang et al. 2013) by Xiaoxi Huang, Huaxin Huang, Beishui Liao, and Cihua Xu proposes a theoretical framework for metaphor understanding based on recent results from cognitive science (Shutova 2010). They then introduce ontology as a knowledge representation method, and present a new ontological model to formalize metaphor-based knowledge. An algorithmic description is provided using a quantitative measure of integrated degree, and its effectiveness is demonstrated. -/- Puzzles and their solution mechanisms are a recurrent theme in many fields. Dynamic epistemic logic (van Ditmarsch et al. 2007); (van Benthem 2011) is a recent framework that can handle information dynamics in knowledge puzzles. In Reasoning About Agent Types and the Hardest Logic Puzzle Ever (Liu and Wang 2013), Fenrong Liu and Yanjing Wang propose a new dynamic-epistemic logic that can specify different types of agents and analyse agents’ knowledge, communication and reasoning. To demonstrate its power, Smullyan’s well-known Knights and Knaves puzzles are formalized, as well as the Hardest Logic Puzzle Ever (HLPE) of Boolos (Boolos 1996). Moreover, a spectrum of new puzzles is found involving knowledge-based agent types, and their solution complexity is shown to be harder than that for existing logic puzzles. It is also shown that a version of HLPE in which the agents do not know the others’ types does not have a solution at all. -/- Finally, we would like to thank the editor of Minds and Machines, Gregory Wheeler, for his efficient assistance and kind support. We also thank Ties Nijssen at Springer for his encouragement to start this project at the first place. And of course, our appreciation goes to our authors, as well as all the colleagues who have helped us review the papers in this collection. (shrink)

We obtain a nonasymptotic lower time bound for deciding sentences of bounded second-order arithmetic with respect to a form of the random access machine with stored programs. More precisely, let P be an arbitrary program for the model under consideration which recognized true formulas with a given range of parameters. Let p be the length of P and let N be an arbitrary natural number. We show how to construct a formula G with one free variable with length not more (...) than 400 symbols and a value f of x such that the time required by P to decide the truth of G is at least N+1 steps. Furthermore, the G constructed does not depend on P and the length of f is less than p+400. (shrink)

Using only uncontentious principles from the logic of ground I construct an infinitely descending chain of ground without a lower bound. I then compare the construction to the constructions due to Dixon and Rabin and Rabern.

Traditionally, pronouns are treated as ambiguous between bound and demonstrative uses. Bound uses are non-referential and function as bound variables, and demonstrative uses are referential and take as a semantic value their referent, an object picked out jointly by linguistic meaning and a further cue—an accompanying demonstration, an appropriate and adequately transparent speaker’s intention, or both. In this paper, we challenge tradition and argue that both demonstrative and bound pronouns are dependent on, and co-vary with, antecedent expressions. Moreover, the semantic (...) value of a pronoun is never determined, even partly, by extra-linguistic cues; it is fixed, invariably and unambiguously, by features of its context of use governed entirely by linguistic rules. We exploit the mechanisms of Centering and Coherence theories to develop a precise and general meta-semantics for pronouns, according to which the semantic value of a pronoun is determined by what is at the center of attention in a coherent discourse. Since the notions of attention and coherence are, we argue, governed by linguistic rules, we can give a uniform analysis of pronoun resolution that covers bound, demonstrative, and even discourse bound readings. Just as the semantic value of the first-person pronoun ‘I’ is conventionally set by a particular feature of its context of use—namely, the speaker—so too, we will argue, the semantic values of other pronouns, including ‘he’, are conventionally set by particular features of the context of use. (shrink)

Local hidden-variable model of singlet-state correlations discussed in Czachor (Acta Phys Polon A 139:70, 2021a) is shown to be a particular case of an infinite hierarchy of local hidden-variable models based on an infinite hierarchy of calculi. Violation of Bell-type inequalities can be interpreted as a ‘confusion of languages’ problem, a result of mixing different but neighboring levels of the hierarchy. Mixing of non-neighboring levels results in violations beyond the Tsirelson bounds.

If to be is to be the value of a bound variable, then the acknowledgment and denial of the existence of chairs amounts to a serious disagreement about the range of a quantifier. However, by resorting to the intrinsic hierarchical structure of hi-world semantics, we find that the varying of domains from worlds to worlds can actually be accommodated within a unified framework. With the introduction of a universal domain D of hi-individuals and an existence predicate E that serves as (...) a realization operator, a new semantics for quantified modal logic is proposed. It allows individual variables to range over individuals in different levels of a hi-world, and is indifferent to the ontological debate about the existence of chairs. (shrink)