Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

Turing and Church formulated two different formal accounts of computability that turned out to be extensionally equivalent. Since the accounts refer to different properties they cannot both be adequate conceptual analyses of the concept of computability. This insight has led to a discussion concerning which account is adequate. Some authors have suggested that this philosophical debate—which shows few signs of converging on one view—can be circumvented by regarding Church’s and Turing’s theses as explications. This move opens up the possibility that (...) both accounts could be adequate, albeit in their own different ways. In this paper, I focus on the question of whether Church’s thesis can be seen as an explication in the precise Carnapian sense. Most importantly, I address an additional constraint that Carnap puts on the explicative power of axiomatic systems—an axiomatisation explicates when it is clear which mathematical entities form the theory’s intended model—and that implicitly applies to axiomatisations of recursion theory used in Church’s account of computability. To overcome this difficulty, I propose two possible clarifications of the pre-systematic concept of “computability” that can both be captured in recursion theory, and I show how both clarifications avoid an objection arising from Carnap’s constraint. (shrink)

This essay defends the following two claims: (1) liraitation-of-size reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets).

Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “ aboutness ” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p be (...) any prime.” In this context, I will be using the term ‘p’ to reason about the primes. Although ‘p’ helps me secure the aboutness of my discourse, it may seem wrong to say that ‘p’ refers to anything. Be that as it may, this paper explores what mathematical discourse would be like if mathematicians were able to borrow freely from one another not just the reference of terms that clearly refer, but, more generally, the sort of aboutness present in a line of reasoning leading up to a universal generalization. The paper also gives reasons for believing that aboutness of this sort really is freely transferable. A key implication will be that the concept “set of natural numbers” suffers from no mathematically significant indeterminacy that can be coherently discussed. (shrink)

Stephen Yablo [23,24] introduces a new informal paradox, constituted by an infinite list of semi-formalized sentences. It has been shown that, formalized in a first-order language, Yablo’s piece of reasoning is invalid, for it is impossible to derive falsum from the sequence, due mainly to the Compactness Theorem. This result casts doubts on the paradoxical character of the list of sentences. After identifying two usual senses in which an expression or set of expressions is said to be paradoxical, since second-order (...) languages are not compact, I study the paradoxicality of Yablo’s list within these languages. While non-paradoxical in the first sense, the second-order version of the list is a paradox in our second sense. I conclude that this suffices for regarding Yablo’s original list as paradoxical and his informal argument as valid. (shrink)

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

There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...) We argue that the Fregean approach faces the same choice: abandon compositionality or abandon the centrality of truth and reference to semantic theory. Indeed, we argue that developing a fully compositional semantics in the tradition of Frege leads to a typographic variant of the most radical of Tarskian views: variabilism, the view that names should be modeled as Tarskian variables. We conclude with the consequences of this result for Frege's distinction between sense and reference. (shrink)

Otávio Bueno* * and Steven French.** ** Applying Mathematics: Immersion, Inference, Interpretation. Oxford University Press, 2018. ISBN: 978-0-19-881504-4 978-0-19-185286-2. doi:10.1093/oso/9780198815044. 001.0001. Pp. xvii + 257.

What is a minimal proof-theoretical foundation of logic? Two different ways to answer this question may appear to offer themselves: reduce the whole of logic either to the relation of inference, or else to the property of incompatibility. The first way would involve defining logical operators in terms of the algebraic properties of the relation of inference—with conjunction $$\hbox {A}\wedge \hbox {B}$$ A ∧ B as the infimum of A and B, negation $$\lnot \hbox {A}$$ ¬ A as the minimal (...) incompatible of A, etc. The second way involves introducing logical operators in terms of the relation of incompatibility, such that X is incompatible with $$\{\lnot \hbox {A}\}$$ { ¬ A } iff every Y incompatible with X is incompatible with {A}; and X is incompatible with $$\{\hbox {A}\!\wedge \!\hbox {B}\}$$ { A ∧ B } iff X is incompatible with {A,B}; etc. Whereas the first route leads us naturally to intuitionistic logic, the second leads us to classical logic. The aim of this paper is threefold: to investigate the relationship of the two approaches within a very general framework, to discuss the viability of erecting logic on such austere foundations, and to find out whether choosing one of the ways we are inevitably led to a specific logical system. (shrink)

The neo-Fregean account of arithmetical knowledge is centered around the abstraction principle known as Hume’s Principle: for any concepts X and Y , the number of X ’s is the same as the number of Y ’s just in case there is a 1–1 correspondence between X and Y . The Caesar Problem, originally raised by Frege in §56 of Die Grundlagen der Arithmetik , emerges in the context of the neo-Fregean programme, because, though Hume’s Principle provides a criterion of (...) identity for objects falling under the concept of Number–namely, 1–1 correspondence—the principle fails to deliver a criterion of application. That is, it fails to deliver a criterion that will tell us which objects fall under the concept Number, and so, leaves unanswered the question whether Caesar could be a number. Hale and Wright have recently offered a neo-Fregean solution to this problem. The solution appeals to the notion of a categorical sortal. This paper offers a reconstruction of their solution, which has the advantage over Hale and Wright’s original proposal of making clear what the structure of the background ontology is. In addition, it is shown that the Caesar Problem can be solved in a framework more minimal than that of Hale and Wright, viz . one that dispenses with categorical sortals. The paper ends by discussing an objection to the proposed neo-Fregean solutions, based on the idea that Leibniz’s Law gives a universal criterion of identity. This is an idea that Hale and Wright reject. However, it is shown that a solution very much in keeping with their own proposal is available, even if it is granted that Leibniz’s Law provides a universal criterion of identity. (shrink)

Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one (...) can get some of the technical advantages of second-order axiomatizations—categoricity, in particular—while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema—a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment. (shrink)

Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality condition (...) (roughly: absence of independent explanation). The paper concludes with a moral concerning the judgment-dependence of a posteriori areas of discourse that emerges from consideration of these two a priori cases. (shrink)

In this paper, the (possible) role of model theory forstructuralism and structuralist definitions of ``reduction'' arediscussed. Whereas it is somewhat undecisive with respect tothe first point – discussing some pro's and con's ofthe model theoretic approach when compared with a syntacticand a structuralist one – it emphasizes that severalstructuralist definitions of ``reducibility'' do not providegenerally acceptable explications of ``reducibility''. This claimrests on some mathematical results proved in this paper.

From early work of N. Goodman to recent approaches by H. Field and D. Lewis, there have been attempts to combine 2nd order languages with calculi of individuals. This paper is a contribution, containing basic definitions and distinctions and some metatheorems, to the development of a general metatheory of such theories.

Deflationists argue that ‘true’ is merely a logico-linguistic device for expressing blind ascriptions and infinite generalisations. For this reason, some authors have argued that deflationary truth must be conservative, i.e. that a deflationary theory of truth for a theory S must not entail sentences in S’s language that are not already entailed by S. However, it has been forcefully argued that any adequate theory of truth for S must be non-conservative and that, for this reason, truth cannot be deflationary :493–521, (...) 1998; Ketland in Mind 108:69–94, 1999). We consider two defences of conservative deflationism, respectively proposed by Waxman :429–463, 2017) and Tennant :551–582, 2002), and argue that they are both unsuccessful. In Waxman’s hands, deflationists are committed either to a non-purely expressive notion of truth, or to a conception of mathematics that does not allow them to justifiably exclude non-conservative theories of truth. Tennant’s conservative deflationism fares no better: if deflationist truth must be conservative over arithmetic, it can be shown to collapse into a non-conservative variety of deflationism. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s quasi-categoricity (...) theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

I extend theorems due to Roy Cook on third- and higher-order versions of abstraction principles and discuss the philosophical importance of results of this type. Cook demonstrated that the satisfiability of certain higher-order analogues of Hume's Principle is independent of ZFC. I show that similar analogues of Boolos's new v and Cook's own ordinal abstraction principle soap are not satisfiable at all. I argue, however, that these results do not tell significantly against the second-order versions of these principles.

This paper is a reflexion on the computability of natural language semantics. It does not contain a new model or new results in the formal semantics of natural language: it is rather a computational analysis, in the context for type-logical grammars, of the logical models and algorithms currently used in natural language semantics, defined as a function from a grammatical sentence to a set of logical formulas—because a statement can be ambiguous, it can correspond to multiple formulas, one for each (...) reading. We argue that as long as we do not explicitly compute the interpretation in terms of possible world models, one can compute the semantic representation of a given statement, including aspects of lexical meaning. This is a very generic process, so the results are, at least in principle, widely applicable. We also discuss the algorithmic complexity of this process. (shrink)

According to Hans Kamp and Frank Vlach, the two-dimensional tense operators “now” and “then” are ineliminable in quantified tense logic. This is often adduced as an argument against tense logic, and in favor of an extensional account that makes use of explicit quantification over times. The aim of this paper is to defend tense logic against this attack. It shows that “now” and “then” are eliminable in quantified tense logic, provided we endow it with enough quantificational structure. The operators might (...) not be redundant in some other systems of tense logic, but this merely indicates a lack of quantificational resources and does not show any deep-seated inability of tense logic to express claims about time. The paper closes with a brief discussion of the modal analogue of this issue, which concerns the role of the actuality operator in quantified modal logic. (shrink)

In this paper, an objective conception of contexts based loosely upon situation theory is developed and formalized. Unlike subjective conceptions, which take contexts to be something like sets of beliefs, contexts on the objective conception are taken to be complex, structured pieces of the world that (in general) contain individuals, other contexts, and propositions about them. An extended first-order language for this account is developed. The language contains complex terms for propositions, and the standard predicate "ist" that expresses the relation (...) that holds between a context and a proposition just in case the latter is true in the former. The logic for the objective conception features a global classical predicate calculus, a local logic for reasoning within contexts, and axioms for propositions. The specter of paradox is banished from the logic by allowing "ist" to be nonbivalent in problematic cases: it is not in general the case, for any context c and proposition p, that either ist(c,p) or ist(c, ¬p). An important representational capability of the logic is illustrated by proving an appropriately modified version of an illustrative theorem from McCarthy's classic Blocks World example. (shrink)

Bob Hale’s distinguished record of research places him among the most important and influential contemporary analytic metaphysicians. In his deep, wide ranging, yet highly readable book Necessary Beings, Hale draws upon, but substantially integrates and extends, a good deal his past research to produce a sustained and richly textured essay on — as promised in the subtitle — ontology, modality, and the relations between them. I’ve set myself two tasks in this review: first, to provide a reasonably thorough (if not (...) exactly comprehensive) overview of the structure and content of Hale’s book and, second, to a limited extent, to engage Hale’s book philosophically. I approach these tasks more or less sequentially: Parts I and 2 of the review are primarily expository; in Part 3 I adopt a somewhat more critical stance and raise several issues concerning one of the central elements of Hale’s account, his essentialist theory of modality. (shrink)

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe. In this paper, I am going to challenge this claim by taking seriously the idea that we can (...) talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched. (shrink)

I provide a tableau system and completeness proof for a revised version of Carnap's semantics for quantified modal logic. For Carnap, a sentence is possible if it is true in some first order model. However, in a similar fashion to second order logic, no sound and complete proof theory can be provided for this semantics. This factor contributed to the ultimate disappearance of Carnapian modal logic from contemporary philosophical discussion. The proof theory I discuss comes close to Carnap's semantic vision (...) and provides an interesting counterpoint to mainstream approaches to modal logic. Despite its historical origins, my intention is to demonstrate that this approach to modal logic is worthy of contemporary attention and that current debate is the poorer for its absence. (shrink)

We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman.

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for (...) examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored. (shrink)

According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second-order logic is set theory in disguise. (shrink)

Whether a predicate is a referential expression depends upon what reference is conceived to be. Even if it is granted that reference is a relation between words and worldly items, the referents of expressions being the items to which they are so related, this still leaves considerable scope for disagreement about whether predicates refer. One of Frege's great contributions to the philosophy of language was to introduce an especially liberal conception of reference relative to which it is unproblematic to suppose (...) that predicates are referring expressions. According to this liberal conception, each significant expression in a language has its own distinctive semantic role or power, a power to effect the truth-value of the sentences in which it occurs. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

There are three different degrees to which we may allow a systematic theory of the world to embrace the idea of relatedness—supposing realism about non-symmetric relations as a background requirement. (First Degree) There are multiple ways in which a non-symmetric relation may apply to the things it relates—for the binary case, aRb ≠ bRa. (Second Degree) Every such relation has a distinct converse—for every R such that aRb there is another relation R* such that bR*a. (Third Degree) Each one of (...) them applies in an order to the things it relates—with regard to the state that results from R's applying to a and b, either R applies to a first and b second, or it applies to b first and a second. Whereas the first degree is near-indubitable, embracing the second or third generates unwholesome consequences. The second degree embodies a commitment to the existence of a superfluity of distinct converses and states to which such relations give rise. The third degree embodies commitment to recherché facts of the matter about how the states that arise from the application of one non-symmetric relation compare to any other. It is argued that accounts that purport to offer an analysis of the first degree generate unwelcome second or third degree consequences. This speaks in favour of our adopting an account of the application of relations that's not an analysis at all, an account that takes the first degree as primitive. (shrink)

In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, (...) and we show that the two approaches are equivalent. (shrink)

Putnam argues that, by ‘reinterpretation’, the Axiom of Constructibility can be saved from empirical refutation. This paper contends that this argument fails, a failure which leaves Putnam's sweeping appeal to the Lowenheim –Skolem Theorem inadequately motivated.

Mereological universalists and nihilists disagree on the conditions for composition. In this paper, we show how this debate is a function of one’s chosen semantics for plural quantifiers. Debating mereologists have failed to appreciate this point because of the complexity of the debate and extraneous theoretical commitments. We eliminate this by framing the debate between universalists and nihilists in a formal model where these two theses about composition are contradictory. The examination of the two theories in the model brings clarity (...) to a debate in which opponents frequently talk past one another. With the two views stated precisely, our investigation reveals the dependence of the mereologists’ ontological commitments on the semantics of plural quantifiers. Though we discuss the debate with respect to a simplified and idealized model, the insights provided will make more complex debates on composition more productive and deflationist criticisms of the debate less substantial. (shrink)

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", (...) or turning meta-mathematical analyses of logical concepts into "philosophical" ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down. (shrink)

Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be (...) expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory, and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)

In this paper we discuss two approaches to the axiomatization of scientific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppes’ and to da Costa and Chuaqui’s works. We argue that theories can be developed both in a way more akin to the usual mathematical practice (Suppes), in an informal set theoretical environment, writing the set theoretical predicate (...) in the language of set theory itself or, more rigorously (da Costa and Chuaqui), by employing formal languages that help us in writing the postulates to define a class of structures. Both approaches are called internal , for we work within a mathematical framework, here taken to be first-order ZFC. We contrast these approaches with an external one, here discussed briefly. We argue that each one has its strong and weak points, whose discussion is relevant for the philosophical foundations of science. (shrink)

Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...) From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. (shrink)

The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on (...) diagonal construction are caused by superficial understanding of what a name is. (shrink)

We will formulate some analogous higher-order versions of Skolem’s paradox and assess the generalizability of two solutions for Skolem’s paradox to these paradoxes: the textbook approach and that of Bays (2000). We argue that the textbook approach to handle Skolem’s paradox cannot be generalized to solve the parallel higher-order paradoxes, unless it is augmented by the claim that there is no unique language within which the practice of mathematics can be formalized. Then, we argue that Bays’ solution to the original (...) Skolem’s paradox, unlike the textbook solution, can be generalized to solve the higher-order paradoxes without any implication about the possibility or order of a language in which mathematical practice is to be formalized. (shrink)

The following inference is valid: There are exactly 101 dalmatians, There are exactly 100 food bowls, Each dalmatian uses exactly one food bowl Hence, at least two dalmatians use the same food bowl. Here, “there are at least 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x100) and “there are exactly 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ (...) x100) & Ø"x1"x2…."x101$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x101). This is abbreviated $101xDx. The validity of the above inference corresponds to the valid formula, PHP(100): [$101xDx & $100xFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). More generally, for variable n, the formula PHP(n) is PHP(n): [$n+1xDx & $nxFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). A mathematical proof that PHP(n) is valid, for all n > 0, is quite short (less than a page), but refers to numbers, functions and sets. It uses the Pigeonhole Principle. This explains why PHP(n) is valid, for all n>0. However, I estimate that a predicate calculus derivation of PHP(100), using natural deduction, say, would require around 107 symbols. Unfeasibility Problem: nominalism is the radical anti-realist view that there are no numbers, functions or sets. So, how could a nominalist know that PHP(100) is valid, without directly performing the rather long derivation? Can the nominalist “ride piggyback” on the standard mathematical proof? If so, how is this justified? (shrink)

The notion of strict identity is sometimes given an explicit second-order definition: objects with all the same properties are identical. Here, a somewhat different problem is raised: Under what conditions is the identity relation on the domain of a structure first-order definable? A structure may have objects that are distinct, but indiscernible by the strongest means of discerning them given the language (the indiscernibility formula). Here a number of results concerning the indiscernibility formula, and the definability of identity, are collected (...) and a number of applications discussed. (shrink)

In the paper the following questions are discussed: What is logical consequence? What are logical constants? What is a logical system? What is logical pluralism? What is logic? In the conclusion, the main tendencies of development of modern logic are pointed out.

Semantic theories based on a hierarchy of types have prominently been used to defend the possibility of unrestricted quantification. However, they also pose a prima facie problem for it: each quantifier ranges over at most one level of the hierarchy and is therefore not unrestricted. It is difficult to evaluate this problem without a principled account of what it is for a quantifier to be unrestricted. Drawing on an insight of Russell’s about the relationship between quantification and the structure of (...) predication, we offer such an account. We use this account to examine the problem in three different type-theoretic settings, which are increasingly permissive with respect to predication. We conclude that unrestricted quantification is available in all but the most permissive kind of type theory. (shrink)

This paper explores the impact of quantification into predicate position on the metaphysics of properties, arguing that two familiar debates about properties are fundamentally altered by recasting them in a second-order setting. Two theories of properties are outlined, differing over whether the existence of properties is expressed using first-order or second-order quantifiers. It is argued that the second-order theory: provides good reason to regard debate about the locations of properties as contentless; resolves debate about whether properties are particulars or universals (...) in favour of universals. (shrink)

This paper uses the resources of higher-order logic to articulate a Fregean conception of predicate reference, and of word-world relations more generally, that is immune to the concept horse problem. The paper then addresses a prominent style of expressibility problem for views of broadly this kind, versions of which are due to Linnebo, Hale, and Wright.

From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...) which several proofs of the Fundamental Theorem of Arithmetic are compared, provides a miniature case study. (shrink)

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...) fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink)