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

We propose an original use of techniques from random graph theory to find a Monadic ∑ 1 1 sentence without an asymptotic probability. Our result implies that the 0-1 law fails for the logics ∑ 1 1 and ∑ 1 1 . Therefore we complete the classification of first-order prefix classes with or without equality, according to the existence of the 0-1 law for the corresponding ∑ 1 1 fragment. In addition, our counterexample can be viewed as a single (...) explanation of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known. (shrink)

The provability problem in intuitionistic propositional second-order logic with existential quantifier and implication (∃,→) is proved to be undecidable in presence of free type variables (constants). This contrasts with the result that inutitionistic propositional second-order logic with existential quantifier, conjunction and negation is decidable.

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in (...) u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt on properties which has the same relationship with the Circuit Value Problem as reach has with the Directed Reachability Problem. We use alt to show that Πn⊈ FO, Σn⊈ FO, and Δn+1⊈ FOB, solving some open problems raised in [Mat98]. (shrink)

We introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin’s style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that (...) can be computed by a non-deterministic Turing machine with random access to the input in time $O^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin’s theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ is too weak as to define a total order of the domain. Nevertheless, $\textrm{SO}^{\textit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\textrm{SO}^{\textit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail. (shrink)

We start with a simple proof of Leivant's normal form theorem for ∑11 formulas over finite successor structures. Then we use that normal form to prove the following:1. over all finite structures, every ∑21 formula is equivalent to a ∑21 formula whose first-order part is a Boolean combination of existential formulas, and2. over finite successor structures, the Kolaitis-Thakur hierarchy of minimization problems collapses completely and the Kolaitis-Thakur hierarchy of maximization problems collapses partially.The normal form theorem for ∑21 fails (...) if ∑21 is replaced with ∑11 or if infinite structures are allowed. (shrink)

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's (...) attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms. (shrink)

Sellars [1960, ‘Grammar and existence: A preface to ontology’] argues that the truth of a second-order sentence does not incur commitment to there being any sort of abstract entity. This paper begins by exploring the arguments that Sellars offers for the above claim. It then develops those arguments by pointing out places where Sellars has been unclear or ought to have said more. In particular, Sellars's arguments rely on there being a means by which language users could come (...) to understand sentences of a second-order language wherein the truth of second-order existential sentences do not require there to be abstract entities. In addition to this, as Sellars [1979, Naturalism and Ontology] notes, a formal account of quantification is required that does not make use of the apparatus of sequences. Both such a translation of and a formal account of quantification are provided in this article. (shrink)

In 2001, Le Bars proved that there exists an existential monadic second-order sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence.

We show that the 0-1 law does not hold for the class Σ 1 1 (∀∃∀ without =) by finding a sentence in this class which almost surely expresses parity. We also show that every recursive real in the unit interval is the asymptotic probability of a sentence in this class. This expands a result by Lidia Tendera, who in 1994 proved that every rational number in the unit interval is the asymptotic probability of a sentence in the class Σ (...) 1 1 ∀∃∀ with equality. (shrink)

In this article we investigate the family of independence-friendly (IF) logics in the equality-free setting, concentrating on questions related to expressive power. Various natural equality-free fragments of logics in this family translate into existential second-order logic with prenex quantification of function symbols only and with the first-order parts of formulae equality-free. We study this fragment of existential second-order logic. Our principal technical result is that over finite models with a vocabulary consisting (...) of unary relation symbols only, this fragment of second-order logic is weaker in expressive power than first-order logic (with equality). Results about the fragment could turn out useful for example in the study of independence-friendly modal logics. In addition to proving results of a technical nature, we address issues related to a perspective from which IF logic is regarded as a specification framework for games, and also discuss the general significance of understanding fragments of second-order logic in investigations related to non-classical logics. (shrink)

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

Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G*0 and (...) G0, in which cuts are restricted to quantifier-free formulas, and prove that the Σqj-witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G*0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb0-CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G*0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID. (shrink)

Second-order logic is the extension of first-order logic obtaining by introducing quantification of predicate and function variables.

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

In this paper, I shall provide a defence of second-order logic in the context of its use in the philosophy of mathematics. This shall be done by considering three problems that have been recently posed against this logic: (1) According to Resnik [1988], by adopting second-order quantifiers, we become ontologically committed to classes. (2) As opposed to what is claimed by defenders of second-order logic (such as Shapiro [1985]), the existence of (...) non-standard models of first-order theories does not establish the inadequacy of first—order axiomatisations (Melia [1995]). (3) In contrast with Shapiro’s suggestion (in his [1985]), second-order logic does not help us to establish referential access to mathematical objects (Azzouni [1994]). As I shall argue, each of these problems can be neatly solved by the second-order theorist. As a result, a case for second-order logic can be made. The first two problems will beconsidered rather briefly in the next section. The rest of the paper is dedicate to a discussion of the third. (shrink)

We show that length initial submodels of S12 can be extended to a model of weak second order arithmetic. As a corollary we show that the theory of length induction for polynomially bounded second order existential formulae cannot define the function division.

Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure (...) second-order logic with second-order identity is equivalent to a member of a circumscribed class of formulas. As a corollary, pure second-order logic with second-order identity is compact, its notion of logical truth is decidable, and it satisfies a pure second-order analogue of model completeness. We end by mentioning an extension to n th-order pure logics. (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that (...) class='Hi'>second-order logic is actually a familiar part of our traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

Is second-order logic logic? Famously Quine argued second-order logic wasn't logic but his arguments have been the subject of influential criticisms. In the early sections of this paper, I develop a deeper perspective upon Quine's philosophy of logic by exploring his positive conception of what logic is for and hence what logic is. Seen from this perspective, I argue that many of the criticisms of his case against second- (...) class='Hi'>order logic miss their mark. Then, in the later sections, I go beyond Quine to develop a novel case that quantification into polyadic predicate position, understood as requiring quantifiers to range over relations, isn't intelligible. (shrink)

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness (...) Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, _positive logics_, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. (shrink)

We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order (...) set theory and second-order logic are not radically different: the latter is a major fragment of the former. (shrink)

We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to (...) be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory. (shrink)

The logic of paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we (...) canvass several of these, concluding that it will be extremely difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. (shrink)

Charles Peirce developed Existential Graphs as a diagrammatic syntax for representing and reasoning about propositions, with three parts: Alpha for propositional logic, Beta for first-order predicate logic, and Gamma for aspects of modal logic, second-order logic, and metalanguage. He made several adjustments between 1909 and 1911 that merit further consideration: using heavy lines to denote possible states of things in which attached propositions would be true, drawing a red line just inside the (...) edge of a page and writing postulates in the resulting margin, shading oddly enclosed areas instead of drawing thin lines as their boundaries, and using multiple sheets of paper to represent different subjects within the overall universe of discourse. These modifications can be combined to constitute a plausible candidate for Delta, a fourth part that Peirce intended to add but never spelled out. It implements modern formal systems of modal propositional logic in accordance with a version of possible worlds semantics in which the laws for the actual state of things are facts in every possible state of things. (shrink)

When we reason about strategic games, implicitly we need to reason about arbitrary strategy profiles and how players can improve from each profile. This structure is exponential in the number of players. Hence it is natural to look for subclasses of succinct games for which we can reason directly by interpreting formulas on the (succinct) game description rather than on the associated improvement structure. Priority separable games are one of such subclasses: payoffs are specified for pairwise interactions, and from these, (...) payoffs are computed for strategy profiles. We show that equilibria in such games can be described in Monadic Least Fixed Point Logic (MLFP). We then extend the description to games over arbitrarily many players, but using the monadic least fixed point extension of existential second order logic. (shrink)

In this report I motivate and develop a type-free logic with predicate quantifiers within the general ontological framework of properties, relations, and propositions. In Part I, I present the major ideas of the system informally and discuss its philosophical significance, especially with regard to Russell's paradox. In Part II, I prove the soundness, consistency, and completeness of the logic.

We consider the notion of everyday language. We claim that everyday language is semantically bounded by the properties expressible in the existential fragment of second–order logic. Two arguments for this thesis are formulated. Firstly, we show that so–called Barwise's test of negation normality works properly only when assuming our main thesis. Secondly, we discuss the argument from practical computability for finite universes. Everyday language sentences are directly or indirectly verifiable. We show that in both cases they (...) are bounded by second–order existential properties. Moreover, there are known examples of everyday language sentences which are the most difficult in this class (NPTIME–complete). (shrink)

We characterize the expressive power of extensions of Dependence Logic and Independence Logic by monotone generalized quanti ers in terms of quanti er extensions of existential second-order logic.

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (...) (Putnam, J Symbol Logic 45:464–482, 1980 ). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988 ), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970 ), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non - isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995 ). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985 )], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994 ). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs. (shrink)

Robert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is (...) not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete. (shrink)

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

In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether second-order logic can count as logic in some proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the (...) case, it might be argued that it does not have the "epistemic innocence" which would be needed for, e.g., foundational programmes in mathematics for which second-order logic is sometimes used. I suggest a Deductivist conception of logic, that characterises logical consequence by means of inference rules, and argue that on this conception second-order logic should count as logic in the proper sense. (shrink)

Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from (...) a logic, but rather must be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments. (shrink)

The aim of this thesis is to examine the debate between Quine and Boolos over the logical status of higher-order logic-with Quine taking the position that higher-logic is more properly understood as set theory and Boolos arguing in opposition that higher-order logic is of a genuinely logical character. My purpose here then will be to stay as neutral as possible over the question of whether or not higher-order logic counts as logic and (...) to instead focus on the exposition of the debate itself as exemplified in the work of Quine and Boolos. Chapter I will be a detailed consideration of Quine's conception of logic and its place within the wider context of his philosophy. Only once this backdrop is in place will I then examine his views on higher-order logic. In Chapter II, I turn to Boolos's response to Quine-his attempt to examine the extent to which we may want to count higher-order logic as logic and the extent to which we may want to count it as set theory. With each point Boolos raises, I attempt to give what I think would have been Quine's reply. Finally, in Chapter III, I consider Boolos's attempt to show that monadic second-order logic should be understood as pure logic as it does not commit us to the existence of classes, as we may take the standard interpretation of MSOL to do. I discuss here some of the major reactions to Boolos's plural interpretation, and conclude with more speculative remarks on what Quine's own response might have been. Throughout this thesis, my primary method has been one of close textual analysis. (shrink)

Under a weak set-theoretic assumption we interpret second-order logic in the monadic theory of order.

In so-called full second-order logic, the second-order variables range over all subsets and relations of the domain in question. In so-called Henkin second-order logic, every model is endowed with a set of subsets and relations which will serve as the range of the second-order variables. In our Boolean-valued second-order logic, the second-order variables range over all Boolean-valued subsets and relations on the domain. We show that (...) under large cardinal assumptions Boolean-valued second-order logic is more robust than full second-order logic. Its validity is absolute under forcing, and its Hanf and Löwenheim numbers are smaller than those of full second-order logic. (shrink)

Some widely accepted arguments in the philosophy of mathematics are fallacious because they rest on results that are provable only by using assumptions that the con- clusions of these arguments seek to undercut. These results take the form of bicon- ditionals linking statements of logic with statements of mathematics. George Boolos has given an argument of this kind in support of the claim that certain facts about second-order logic support logicism, the view that mathematics—or at least (...) part of it—reduces to logic. Hilary Putnam has offered a similar argument for the view that it is indifferent whether we take mathematics to be about objects or about what follows from certain postulates. In this paper I present and rebut these arguments. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:EXISTENTIAL PROPOSITIONS IN THE THOUGHT OF ST. THOMAS AQUINAS A REVALENT VIEW of St. Thomas Aquinas's position on the logic of propositions has been that according to him propositions of the :form, x is, hold a privileged place, that they are in a special sense " existential," and that such propositions straight.forwardly attribute the act of exi,stence to an individual or to a class of individuals.1 (...) Some texts seem to support such an interpretation. For example, in his commentary on Aristotle's Perihermeneias, Aquinas says: " To clarify this we must note that the verb 'is ' itself is sometimes predicated in an enunciation, as in ' Socrates is.' By this we intend to signify that Socrates really is." 2 This passage seems to treat the predicate " exists " or " is " as unproblematical, as though it had a straightforward meaning needing little clarification, at least in the context of a logical treatise. On the basis of such a text, one may be tempted to infer that for Aquinas " Socrates exists " has the same logical form as " Socrates sits," and that " Men exist " is of the same form as " Men are mortal.'' Peter Geach has shown, however, that Aquinas's doctrine here is not as simple as it may first appear. For Aquin,as some of the propositions which have the grammatical form of x is have 'a different logical form.3 Geach points out that for Aquinas the sentence, "Blindness exists," for example, actual1 E.g., Etienne Gilson, Being and Some Philosophers, 2nd ed. (Toronto: Pontifical Institute of Mediaeval Studies, 1952), 190-215. 2 Super Peri, Bk. II, lect. 2, # 212. Cf. Summa Contra Gentiles, Bk. II, ch. 37 at "FJ11J hoc..•." a Peter Geach, in P. T. Geach and G. E. M. Anscombe, Three Philosophers (New York: Cornell University, 1961), 88 ff; Peter Geach, "Form and Existence," in Aquinas: A Collection of Critical Essays, ed. A. Kenny (New York: Notre Dame U., 1969), pp. 41-47. 605 606 PATRICK LEE ly means, Something or other is blind. That is, blind is the real predicate rather than exists. Mo11eover, Geach argues, the logical peculiarity of this sentence is equally found in any sentence of the form, "an F exists." Such a proposition, " does not,attribute actuality (existence) to an F, but F-ness to something or other..." 4 And yet Geach will not apply rthis interpretation to all sentences of the form, " x is." He al'lgues that sentences in which " is " or " exists " is. joined to a logically pl'oper name are in a different category. Unlike in general propositions such as "an F exists," in such singular propositions, those using logically proper names, Geach,argues, actual existence is attributed to or predicated of an individual. "We have here a sense of 'is ' or 'exists ' that seems to me to be certainly a genuine predicate of individuals: the sense of ' exists ' in which one says that an individual came to exist, still exists, no longer exists, etc...."5 I believe that, while Geach's position is more accurate than the first interpretation, mentioned above, it is nevertheless still inaccurate on certain key points. While it is misleading to say that for Aquinas exists is never a genuine predicate; nevertheless, I will argue that it is his position that exists or existence is not a genuine predicate in any proposition which does not refer back to a previous proposition (or to what is known in or by previous propositional knowledge). Let us crull a proposition which refers to something known in a previous proposition a " second-order p11oposition," and one that does not thus refer a " first-order proposition." Then, I shall argue that for Aquinas exists is not a genuine predicate in any first-order proposition. Moreover, I believe that these points are crucial to understanding Aquinas's position on existence as well as his position on how knowledge is related,to reality, i.e., his realism. In this article I would like to examine Aquinas's thought on the logic of existential propositions and discuss its significance for epistemology. 4 Geach, " Form and Existence," Zoo. int., p. 45. 5 Ibid., p. 46. EXISTENTIAL PROPOSITIONS IN AQUINAS 607 In sections... (shrink)

Thus far the logic out of which mathematics has developed has been First-order Predicate Calculus with Identity, that is the logic of the sentential functors, ¬, →, ∧, ∨, etc., together with identity and the existential and universal quotifiers restricted to quotify- ing only over individuals, and not anything else, such as qualities or quotities themselves. Some philosophers—among them Quine— have held that this, First-order Logic, as it is often called, con- stitutes the whole (...) of logic. But that is a mistake. It leaves out Second-order Logic, which we need if we are to characterize the natural numbers precisely, and pays scant attention to the logic of relations, especially transitive relations, which is the key to much of modern mathematics. Quine’s argument for restricting logic to First-order Logice was based on the grounds that only First- order logical theories display “Law and Order” and himself regards modal logic as belonging with witchcraft and superstition.1 Pred- icates are ontologically more suspect than individuals, and have a different logic, which is liable to give rise to paradox and inconsis- tency. Moreover, Second-order Logic lacks the completeness that First-order Logice has, which provides a pleasing parallel between syntactic and semantic notions, and argues for the analyticity of deductive logic. (shrink)