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

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies the axioms iff it embeds into a first order algebra. Importantly, our argument is modular and also works for, e.g., the positive existential algebras and the quantifier-free algebras. We also explain the relationship to theories, and (...) indicate how to add in function symbols. (shrink)

This paper deals with the interpretation and distribution of universal Free Choice (FC) items, such as English FC any or Spanish cualquiera. Crosslinguistically, universal FC items can be characterized as follows. First, they have a restricted distribution. Second, they express freedom of choice: the sentence You can take any card conveys the information that the addressee is free to pick whichever card she chooses. Under standard assumptions, the truth conditions of sentences like You can take any card (...) are taken to be captured by formalizations in which a universal quantifier ranging over individuals has wide scope over the possibility modal. The crucial observation that informs the account in this paper is that this type of formalization cannot capture the freedom of choice component. I will argue that in order to derive the right interaction between possibility modals and FC items, we need to add an exclusiveness condition to the standard wide scope paraphrases. The same proposal that guarantees freedom of choice will automatically account for the distribution restrictions of FC items. The formal implementation of this proposal is cast in the Hamblin semantics proposed in Kratzer and Shimoyama (Indeterminate phrases: The view from Japanese, 2002). (shrink)

A universally free logic is a system of quantification theory, with or without identity, whose theses remain logically true if the domain of quantification is empty and some of the singular terms present in the language do not denote existing objects. In the West, logics satisfying and ones satisfying were developed starting in the 1950s. But Stanisław Jaśkowski preceded all this work by some twenty years: his paper “On the Rules of Supposition in Formal Logic” of (...) 1934 can be regarded as containing, at least implicitly, the first universally free logic. The system proposed there is an inclusive logic as it is, and a straightforward extension of it is also free. (shrink)

Relativistic quantum mechanics has been formulated as a theory of the evolution ofevents in spacetime; the wave functions are square-integrable functions on the four-dimensional spacetime, parametrized by a universal invariant world time τ. The representation of states with spin is induced with a little group that is the subgroup of O(3, 1) leaving invariant a timelike vector nμ; a positive definite invariant scalar product, for which matrix elements of tensor operators are covariant, emerges from this construction. In a previous study (...) a second-order equation was introduced similar to the second-order Dirac equation, based on a quadratic function of two operators which are the self-adjoint parts, in this new scalar product, of γμpμ. It is shown in this paper that one of these operators, in fact the one from which the gyromagnetic moment is obtained, can be used to construct a first-order equation. The corresponding quantum theory is somewhat analogous to Dirac's spinor form; the Hamilton equations appear to describe dynamical degrees of freedom in a spacelike hyperplane orthogonal to nμ (in Dirac's theory the motion appears to be lightlike). It is shown that the integration over nμ required by unitarity results in timelike motion (as in the expectation value of Dirac's α). Explicit forms are obtained for the wave functions and currents for free motion. The general form of the theory is written for the (five-dimensional) pre-Maxwell fields required by gauge invariance. (shrink)

Online trading invariably involves dealings between strangers, so it is important for one party to be able to judge objectively the trustworthiness of the other. In such a setting, the decision to trust a user may sensibly be based on that user’s past behaviour. We introduce a specification language based on linear temporal logic for expressing a policy for categorising the behaviour patterns of a user depending on its transaction history. We also present an algorithm for checking whether the transaction (...) history obeys the stated policy. To be useful in a real setting, such a language should allow one to express realistic policies which may involve parameter quantification and quantitative or statistical patterns. We introduce several extensions of linear temporal logic to cater for such needs: a restricted form of universal and existential quantification; arbitrary computable functions and relations in the term language; and a “counting” quantifier for counting how many times a formula holds in the past. We then show that model checking a transaction history against a policy, which we call the history-based transaction monitoring problem, is PSPACE-complete in the size of the policy formula and the length of the history, assuming that the underlying interpreted func-. (shrink)

Van Lambalgen (1990) proposed a translation from a language containing a generalized quantifierQ into a first-order language enriched with a family of predicatesR i, for every arityi (or an infinitary predicateR) which takesQxg(x, y1,..., yn) to x(R(x, y1,..., y1) (x,y1,...,yn)) (y 1,...,yn are precisely the free variables ofQx). The logic ofQ (without ordinary quantifiers) corresponds therefore to the fragment of first-order logic which contains only specially restricted quantification. We prove that it is decidable using (...) the method of analytic tableaux. Related results were obtained by Andréka and Németi (1994) using the methods of algebraic logic. (shrink)

We investigate how the sentence choice semantics for propositional superposition logic developed in Tzouvaras could be extended so as to successfully apply to first-order superposition logic. There are two options for such an extension. The apparently more natural one is the formula choice semantics based on choice functions for pairs of arbitrary formulas of the basis language. It is proved however that the universal instantiation scheme of first-order logic, $\varphi \rightarrow \varphi $, is false, as a (...) scheme of tautologies, with respect to FCS. This causes the total failure of FCS as a candidate semantics. Then we turn to the other option, which is a variant of SCS, since it uses again choice functions for pairs of sentences only. This semantics however presupposes that the applicability of the connective | is restricted to quantifier-free sentences, and thus the class of well-formed formulas and sentences of the language is restricted too. Granted these syntactic restrictions, the usual axiomatizations of FOLS turn out to be sound and conditionally complete with respect to this second semantics, just like the corresponding systems of PLS. (shrink)

We define process algebras with a generalised operation ∑ for choice. For every infinite cardinal κ, we prove that the algebra of transition trees with branching degree <κ is free in the class of process algebras in which ∑ is defined for all subsets with a cardinality <κ. We explain how the expressions of a fragment of the specification language μCRL may be used to denote elements of our process algebras. In particular, we explain how choice quantifiers may be (...) used to denote infinite sums. We show that choice quantifiers can simulate both the existential and the universal quantifiers of first-order logic, while the input prefix mechanism can only simulate the universal quantifier. (shrink)

A well established technique toward developing the proof theory of a Hilbert-style modal logic is to introduce a Gentzen-style equivalent (a Gentzenisation), then develop the proof theory of the latter, and finally transfer the metatheoretical results to the original logic (e.g., [1, 6, 8, 18, 10, 12]). In the first-order modal case, on one hand we know that the Gentzenisation of the straightforward first-order extension of GL, the logic QGL, admits no cut elimination (if the rule (...) is included as primitive; or, if not included, then the rule is not admissible [1]). On the other hand the (cut-free) Gentzenisations of the first-order modal logics M3 and ML3 of [10, 12] do have cut as an admissible rule. The syntactic cut admissibility proof given in [18] for the Gentzenisation of the propositional provability logic GL is extremely complex, and it was the basis of the proofs of cut admissibility of the Gentzenisations of M3 and ML3, where the presence of quantifiers and quantifier rules added to the complexity and length of the proof. A recent proof of cut admissibility in a cut-free Gentzenisation of GL is given in [5] and is quite short and easy to read. We adapt it here to revisit the proofs for the cases of M3 and ML3, resulting to similarly short and easy to read proofs, only slightly complicated by the presence of quantification and its relevant rules. (shrink)

We refine the constructions of Ferrante-Rackoff and Solovay on iterated definitions in first-order logic and their expressibility with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets and can be eliminated. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtained from the iterated definitions. In the quantifier-free case and in the case of purely existential or universal quantifiers, we show (...) that Ω quantifiers are necessary and sufficient. The last lower bounds are obtained with the aid of the Yao-Håstad switching lemma. (shrink)

We define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities ∑ i a i · x i ≥ b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CP- inequalities, we prove (...) a monotone interpolation theorem obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Theorem 5.1]. LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantification is allowed only over the former sort. We shall give a construction of a theory LK(M) for any discretely ordered module M (e.g., LK(Z) extends LK(CP)). The interpolation theorem generalizes to these theories obtained from discretely ordered Z-modules. We shall also discuss a connection to quantifier elimination for such theories. We formulate a communication complexity problem whose (suitable) solution would allow to improve the monotone interpolation theorem and the lower bound for R(CP). (shrink)

We give a syntactic translation from first-order intuitionistic predicate logic into second-order intuitionistic propositional logic IPC2. The translation covers the full set of logical connectives ∧, ∨, →, ⊥, ∀, and ∃, extending our previous work, which studied the significantly simpler case of the universal-implicational fragment of predicate logic. As corollaries of our approach, we obtain simple proofs of nondefinability of ∃ from the propositional connectives and nondefinability of ∀ from ∃ in the second-order intuitionistic propositional (...) logic. We also show that the ∀-free fragment of IPC2 is undecidable. (shrink)

Henry Leonard and Karel Lambert first introduced so-called presupposition-free (or just simply: free) logics in the 1950’s in order to provide a logical framework allowing for non-denoting singular terms (be they descriptions or constants) such as “the largest prime” or “Pegasus” (see Leonard [1956] and Lambert [1960]). Of course, ever since Russell’s paradigmatic treatment of definite descriptions (Russell [1905]), philosophers have had a way to deal with such terms. A sentence such as “the..

We prove that each ∀1 free amalgamation class K over a finite relational language L admits a countable generic structure M isometrically embedding all countable structuresin K relative to a fixed metric. We expand L by infinitely many binary predicates expressingdistance, and prove that the resulting expansion of K has a model companion axiomatizedby the first-order theory of M. The model companion is non-finitely axiomatizable, evenover a strong form of the axiom scheme of infinity.

In the final chapter of Speech Matters, Seana Shiffrin argues that institutions have especially stringent duties to protect speech freedoms. In this article, I develop a few lines of criticism. First, I question whether Shiffrin’s framework of justified suspended contexts is appropriate for institutional settings. Second, I challenge the presumption that the knowledge-gathering function performed by police is necessarily compromised by insincere practices. Third, I criticize Shiffrin’s characterization of the university as involving a complete repudiation of enforced consensus, and (...) I express doubts about the close connection between education and democratic legitimation that Shiffrin endorses. Finally, I raise a problem with the book’s overall argument: even if one agrees that speech freedoms are necessary for moral development, they also may be threatening to moral development. The upshot is that the protection of speech should be modulated in order to account for the potential conflicts between sincerity and other valuable ends, rather than being oriented above all to sincerity. (shrink)

Standard first-order logic plus quantifiers of all finite orders ("SFOLω") faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOLω models. The typed structure of SFOLω models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of (...) meaning is removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOLω type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOLω models. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction (...) rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

The result of combining classical quantificational logic with modal logic proves necessitism – the claim that necessarily everything is necessarily identical to something. This problem is reflected in the purely quantificational theory by theorems such as ∃x t=x; it is a theorem, for example, that something is identical to Timothy Williamson. The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted that (...) in order to specify the truth conditions of certain sentences involving constants or variables that don’t denote, one has to apparently quantify over things that are not identical to anything. In this paper I defend a contingentist, non-Meinongian metaphysics within a positive free logic. I argue that although certain names and free variables do not actually refer to anything, in each case there might have been something they actually refer to, allowing one to interpret the contingentist claims without quantifying over mere possibilia. (shrink)

We examine second order intuitionistic propositional logic, IPC². Let $F_\exists $ be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in $F_\exists $ that is, for φ € $F_\exists $ φ is a classical tautology if and only if ¬¬φ is a tautology of IPC². We show that for each sentence φ € $F_\exists $ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. (...) As a corollary we obtain a semantic argument that the quantifier V is not definable in IPC² from ⊥, V, ^, →. (shrink)

Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type (...) theory is assumed. However, the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell’s notion of range of significance. (shrink)

In this paper we prove that $\forall \textrm{FO}$, the universal fragment of first-order logic, is superfluous in $\varSigma _2^p$ and $\varPi _2^p$. As an example, we show that this yields a syntactic proof of the $\varSigma _2^p$-completeness of value-cost satisfiability. The superfluity method is interesting since it gives a way to prove completeness of problems involving numerical data such as lengths, weights and costs and it also adds to the programme started by Immerman and Medina about the syntactic (...) approach in the study of completeness. (shrink)

The still unsettled decision problem for the restricted purely universal formulae 0-formulae) of the first order set-theoretic language based over =, ∈ is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the 0-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the (...) hereditarily finite sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set-satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set-satisfiability problem is also solvable . Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the set-satisfiability problem for the full class of Δ0-formulae is shown to be semidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the Δ0-formulae. (shrink)

George Boolos's employment of plurals to give an ontologically innocent interpretation of monadic higher‐order quantification continues and extends a minority tradition in thinking about quantification and ontological commitment. An especially prominent member of that tradition is Stanislaw Leśniewski, and shall first draw attention to this work and its relation to that of Boolos. Secondly I shall stand up briefly for plurals as logically respectable expressions, while noting their limitations in offering ontologically deflationary accounts of higher‐order (...) quantification. Thirdly I shall focus on the key idea of ontological commitment and investigate its connection with the idea of truth‐making. Fourthly I shall consider how different interpretations of quantification may sideline Boolos's work, but finally I shall largely support his analysis of quantification involving nominal expressions, while arguing, in the spirit of Arthur Prior, that non‐nominal quantification is non‐committing. (shrink)

This paper investigates a problem related to quantifiers which has some analogies to that of propositional completeness I give a definition of quantifier in many-valued logics generalizing the cases which already occur in first order many- valued logics. Though other definitions are possible, this particular one, which I call distribution quantifiers, generalizes the classical quantifiers in a very natural way, and occurs in finite numbers in every m-valued logic. We then call the problem of quantificationa2 completeness in m-valued (...) logic the problem of characterizing which are the quantifiers in a given language which can generate all other quantifiers in this language, using the connectives, as is the case, for example, of the universal and exis- tential quantifiers in classical logic, using negation. We are interested, in particular, in those many-valued quantifiers which closely mimic the behavior of existential an universal quantifiers in generating all other quantifiers using negation: these I call perfect quantifiers, as defined below. The main result of this paper is the characterization of all perfect quantifiers in 3-valued logics, which are complete if the logic is functionally complete. As a byproduct, we obtain the same result for the classical logic, which we include mainly for motivation. (shrink)

This chapter offers an opinionated introduction to higher-order formal languages with an eye towards their applications in metaphysics. A simply relationally typed higher-order language is introduced in four stages: starting with first-order logic, adding first-order predicate abstraction, generalizing to higher-order predicate abstraction, and finally adding higher-order quantification. It is argued that both β-conversion and Universal Instantiation are valid on the intended interpretation of this language. Given these two principles, it is then (...) shown how we can use pure higher-order logic to ask, and begin to answer, metaphysical questions with non-trivial implications. In particular, while we must reject the popular idea that structural differences between sentences correspond to parallel distinctions in the logical structure of extra-linguistic reality, it may still be possible to give a purely logical characterization of objectual aboutness and related notions. (shrink)

Andrew Bacon, John Hawthorne, and Gabriel Uzquiano have recently argued that free logics—logics that reject or restrict Universal Instantiation—are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior. These logics encompass ramified and contextualist approaches to paradoxes, and broadly speaking, there are two kinds of criticism they face. First, they fail to address every version of the Priorean paradoxes. Second, the theoretical considerations behind the logics make absolutely general statements about all propositions, (...) properties of propositions, etc., but because this sort of intensional quantification is always restricted in the logics, they cannot even express those considerations. I present a novel sort of free logic, which rejects the standard Universal Instantiation but validates a restricted form of the rule, and which addresses both kinds of criticism by allowing some propositions to be unrestricted in their quantification. (shrink)

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

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due (...) in large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Pictorial free perception reports are sequences in comics or film of one unit that depicts an agent who is looking, and a following unit that depicts what they see. This paper proposes an analysis in possible worlds semantics and event semantics of such sequences. Free perception sequences are implicitly anaphoric, since the interpretation of the second unit refers to the agent depicted in the first. They are argued to be possibly non-extensional, because they can depict hallucination or (...) mis-perception. The semantics proposed here employs an account of anaphora using discourse referents, a formalized possible worlds semantics for pictorial narratives, and, to model the epistemic consequences of perceptual events, the event alternative construction from dynamic epistemic logic. In intensional examples, the second unit depicting what is seen is analyzed as embedded. It is argued that a semantics for embedding where the attitudinal state of the depicted agent is required to entail the semantic content of the picture attributes too much information to the agent. This is addressed with a model of normal looking, and a semantics for the embedding construction that uses existential quantification over alternatives, rather than universal quantification. (shrink)

The paper presents an epistemic logic with quantification over agents of knowledge and with a syntactical distinction between de re and de dicto occurrences of terms. Knowledge de dicto is characterized as ‘knowledge that’, and knowlegde de re as ‘knowledge of’. Transition semantics turns out to be an adequate tool to account for the distinctions introduced.

Reference [12] introduced a novel formula to formula translation tool that enables syntactic metatheoretical investigations of first-order modallogics, bypassing a need to convert them first into Gentzen style logics in order torely on cut elimination and the subformula property. In fact, the formulator tool,as was already demonstrated in loc. cit., is applicable even to the metatheoreticalstudy of logics such as QGL, where cut elimination is unavailable. This paper applies the formulator approach to show the independence of (...) the axiom schema ☐A → ☐∀ A of the logics M3and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics. (shrink)

We show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first‐order theory with an infinite model extending the theory of R‐modules and (where ⩽pp stands for “pure submodule”). Assume has the joint embedding and amalgamation properties. If or, then has a universal model of cardinality λ. As a special case, we get a recent result of Shelah [28, 1.2] concerning the existence of universal reduced torsion‐free (...) abelian groups with respect to pure embeddings.We begin the study of limit models for classes of R‐modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure‐injective and we characterize limit models with chains of countable cofinality. This can be used to answer [18, Question 4.25].As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both. (shrink)

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

Let be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the (...) point of view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number. Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures. (shrink)

For various reasons several authors have enriched classical first order syntax by adding a predicate abstraction operator. “Conservatives” have done so without disturbing the syntax of the formal quantifiers but “revisionists” have argued that predicate abstraction motivates the universal quantifier’s re-classification from an expression that combines with a variable to yield a sentence from a sentence, to an expression that combines with a one-place predicate to yield a sentence. My main aim is to advance the cause of predicate (...) abstraction while cautioning against revisionism. In so doing, however, I shall pursue a secondary aim by conveying mixed blessings to those who hold the view that in the logical sense of “existence” some existing object is such as to exist contingently. Advocates of this view must concede Williamson’s recent contention that the domain of unrestricted objectual quantification could not have been narrower than it is actually, but predicate abstraction affords them some hope of accommodating this concession. (shrink)