This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey’s bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize (i.e., to assess anexactquantity of the elements in a (...) collection without counting them), or on the ability to approximate quantities (i.e., to assess anapproximatequantity of the elements in a collection without counting them), or both, fail to provide a conceptual basis for bootstrapping the concept of an exact natural number. In particular, I argue that none of the existing theories explains one of the key characteristics of the natural number structure: the equidistances between successive elements of the natural numbers progression. I suggest that this regularity could be based on another innate cognitive ability, namely sensitivity to the regularity of rhythm. In the final section, I propose a new position within the core cognition paradigm, inspired by structuralist positions in philosophy of mathematics. (shrink)

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken (...) together, have substantial logical strength, amounting to a kind of abstract consistency statement. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: Adding a theory of truth to a finitely axiomatized theory T is more or less equivalent to a (...) kind of abstract consistency statement. A large part of the interest of the paper lies in the way syntactic theories are 'disentangled' from object theories. (shrink)

The semantic paradoxes are a family of arguments – including the liar paradox, Curry’s paradox, Grelling’s paradox of heterologicality, Richard’s and Berry’s paradoxes of definability, and others – which have two things in common: first, they make an essential use of such semantic concepts as those of truth, satisfaction, reference, definition, etc.; second, they seem to be very good arguments until we see that their conclusions are contradictory or absurd. These arguments raise serious doubts concerning the coherence of the concepts (...) involved. This article will offer an introduction to some of the main theories that have been proposed to solve the paradoxes and avert those doubts. Included is also a brief history of the semantic paradoxes from Eubulides to Tarski and Curry. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? It turns out that, in a wide range of cases, we can get some nice answers to this question, but only if we work in a framework that is somewhat different from those usually employed in discussions of (...) axiomatic theories of truth. These results are then used to address a range of philosophical questions connected with truth, such as what Tarski meant by "essential richness" and the so-called conservativeness argument against deflationism. -/- This draft dates from about 2009, with some significant updates having been made around 2011. Around then, however, I decided that the paper was becoming unmanageable and that I was trying to do too many things in it. I have therefore exploded the paper into several pieces, which will be published separately. These include "Disquotationalism and the Compositional Principles", "The Logical Strength of Compositional Principles", "Consistency and the Theory of Truth", and "What Is Essential Richness?" You should probably read those instead, since this draft remains a bit of a mess. Terminology and notation are inconsistent, and some of the proofs aren't quite right. So, caveat lector. I make it public only because it has been cited in a few places now. (shrink)

In his paper "Undecidability without arithmetization," Andrzej Grzegorczyk introduces a theory of concatenation $\mathsf{TC}$. We show that pairing is not definable in $\mathsf{TC}$. We determine a reasonable extension of $\mathsf{TC}$ that is sequential, that is, has a good sequence coding.

Philosopher’s judgements on the philosophical value of Tarski’s contributions to the theory of truth have varied. For example Karl Popper, Rudolf Carnap, and Donald Davidson have, in their different ways, celebrated Tarski’s achievements and have been enthusiastic about their philosophical relevance. Hilary Putnam, on the other hand, pronounces that “[a]s a philosophical account of truth, Tarski’s theory fails as badly as it is possible for an account to fail.” Putnam has several alleged reasons for his dissatisfaction,1 but one of them, (...) the one I call the modal objection (cf. Raatikainen 2003), has been particularly influential. In fact, very similar objections have been presented over and over again in the literature. Already in 1954, Arthur Pap had criticized Tarski’s account with a similar argument (Pap 1954). Moreover, both Scott Soames (1984) and John Etchemendy (1988) use, with an explicit reference to Putnam, similar modal arguments in relation to Tarski. Richard Heck (1997), too, shows some sympathy for such considerations. Simon Blackburn (1984, Ch. 8) has put forward a related argument against Tarski. Recently, Marian David has criticized Tarski’s truth definition with an analogous argument as well (David 2004, p. 389-390).2 This line of argument is thus apparently one of the most influential critiques of Tarski. It is certainly worthy of serious attention. Nevertheless, I shall argue that, given closer scrutiny, it does not present such an acute problem for the Tarskian approach to truth as many philosophers think. But I also believe that it is important to understand clearly why this is so. Moreover, I think that a careful consideration of the issue illuminates certain important but somewhat neglected aspects of the Tarskian approach. (shrink)

This article outlines what a formal theory of truth should be like, at least at first glance. As not all of the stated constraints can be satisfied at the same time, in view of notorious semantic paradoxes such as the Liar paradox, we consider the maximal consistent combinations of these desiderata and compare their relative advantages and disadvantages.

Attitudinal propositionalism is the view that all mental attitude content is truth-evaluable. While attitudinal propositionalism is still silently assumed in large parts of analytic philosophy, recent work on objectual attitudes (i.e. attitudes like ‘fearing Moriarty’ and ‘imagining a unicorn’ that are reported through intensional transitive verbs with a direct object) has put attitudinal propositionalism under explanatory pressure. This paper defends propositionalism for a special subclass of objectual attitudes, viz. experiential attitudes. The latter are attitudes like seeing, remembering, and imagining whose (...) grammatical objects intuitively denote (events or) scenes. I provide a propositional analysis of experiential attitudes that preserves the merits of propositionalism. This analysis uses the possibility of representing the target-scenes of experiential attitudes by the intersection of all propositions that are true in these scenes. I show that this analysis makes available the usual (Russellian) account of intensionality and the common (Boolean) logic for entailments. (shrink)

In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a reconstruction of this (...) New Science that meets modern standards and to examine possible problems surrounding Frege’s original proposal. The paper is organized as follows: the first two sections summarize the main points of the Frege–Hilbert controversy and discuss some issues surrounding the problem of independence proofs. Section 3 contains an informal presentation of Frege’s proposal. In section 4 a more detailed reconstruction of Frege’s New Science is set out while section 5 examines what is left out. The concluding section is devoted to a discussion of Frege’s strategy and its significance from a broader perspective. (shrink)

SummaryMetatheory may be pursued immanently, i.e., within the object language, or transcendently in metalanguages. Immanently, the hierarchy of metalanguages gives way to a hierarchy of predicates. The immanent approach accentuates the symmetry between Russell's paradox and Cantor's theorem: class shortage versus predicate shortage. Appeal to metatheoretic models, in defining logical truth, gives way to appeal to substitutions of expressions of the object language. Can this be said also of set‐theoretic truth, despite predicate shortage? Equivalently: is substitutional quantification unscathed by predicate (...) shortage? Anyway substitutional quantification balks at impredicative cases. (shrink)

For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson (...) generalization in the case of disjoint languages. We show that the Andréka–Németi generalization is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability, which is another definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories in non-disjoint languages. (shrink)

The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts of \ ’s, (...) as well. We prove that the equational theory of \ is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. (shrink)

The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second‐order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing (...) a direct logical characterization of rudimentary languages and a representation result of second‐order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity. (shrink)

A number of general points behind the story of this paper may be worth setting out separately, now that we have come to the end.There is perhaps one obvious omission to be addressed right away. Although the word “information” has occurred throughout this paper, it must have struck the reader that we have had nothing to say on what information is. In this respect, our theories may be like those in physics: which do not explain what “energy” is (a notion (...) which seems quite similar to “information” in several ways), but only give some basic laws about its behaviour and transmission.The eventual recommendation made here has been to use a broad type-theoretic framework for studying various more classical and more dynamic notions of proposition in their interaction. This is not quite the viewpoint advocated by many current authors in the area, who argue for a whole-sale switch from a ‘static’ to a ‘dynamic’ perspective on propositions. This is not the place, however, to survey the conceptual arguments for and against such a more radical move.This still leaves many questions about possible reductions from one perspective to another. For instance, it would seem that classical systems ought to serve as a ‘limiting case’, which should still be valid after procedural details of some cognitive process have been forgotten. There are various ways of implementing the desired correspondence: e.g. by considering extreme cases with ⫅ equal to identity, or, in the pure relational algebra framework by considering only pairs (x, x). What we shall want then are reductions of dynamic logics, in those special cases, to classical logic. But perhaps also, more sophisticated views are possible. How do we take a piece of ‘dynamic’ prose, remove control instructions and the like, and obtain a piece of ‘classical’ text, suitable for inference ‘in the light of eternity’?There is also a more technical side to the matter of ‘reduction’. By now, Logic has reached such a state of ‘inter-translatability’ that almost all known variant logics can be embedded into each other, via suitable translations. In particular, once an adequate semantic has been given for a new system, this usually induces an embedding into standard logic: as we know, e.g., for the case of Modal Logic. Like-wise, all systems of dynamic interpretation or inference proposed so far admit of direct embedding into an ordinary ‘static’ predicate logic having explicit transition predicates (cf. van Benthem 1988b). Thus, our moral is this. The issue is not whether the new systems of information structure or processing are essentially beyond the expressive resources of traditional logical systems: for, they are not. The issue is rather which interesting phenomena and questions will be put into the right focus by them.The next broad issue concerns the specific use of the perspective proposed here, vis-à-vis concrete proposals for information-oriented or dynamic semantics. The general strategy advocated here is to locate some suitable base calculus and then consider which ‘extras’ are peculiar to the proposal. For instance, this is the spirit in which modal S4 would be a base logic of information models, and intuitionistic logicthe special theory devoting itself to upward persistent propositions. Or, with the examples in Section 4.1, the underlying base logic is our relational algebra, whereas, say, ordinary updates then impose special properties, such as ‘idempotence’: $$xRy \Rightarrow yRy$$ Does this kind of application presuppose the existence of one distinguished base logic, of which all others are extensions? This would be attractive-and some form of relational algebra or linear logic might be a reasonable candidate. Nevertheless, the enterprise does not rest on this outcome. What matters is an increased sensitivity to the ‘landscape’ of dynamic logics, just as with the ‘Categorial Hierarchy’ in Categorial Grammar (cf. van Benthem 1989a, 1991) where the family of logics with their interconnections seems more important than any specific patriarch.Finally, perhaps the most important issue in the new framework is the possibility of new kinds of questions arising precisely because of its differences from standard logic. Notably, given the option of regarding propositions as programs, it will be of interest to consider systematically which major questions about programming languages now make sense inside logic too.EXAMPLE. Correctness. When do we have $$\left[\kern-0.15em\left[ \pi \right]\kern-0.15em\right](\left[\kern-0.15em\left[ A \right]\kern-0.15em\right]) \subseteq \left[\kern-0.15em\left[ B \right]\kern-0.15em\right]$$ for (s, t) propositions A, B and a dynamic (s, (s, t)) proposition π?Program Synthesis. Which dynamic proposition will take us from an information state satisfying A to one satisfying B? (This question needs refinement, lest there be trivial answers.)Determinism. Which propositions as programs are deterministic, in the sense of defining single-valued functions from states to states?Querying. What does it mean to ask for information in the present setting? (Again, individual types referring to e will be crucial here.)This is not merely an agenda for wishful thinking. Within Logic, there are various ways of introducing such concerns into semantics, especially, using tools from Automata Theory. (See van Benthem 1989c for further discussion of such computational perspectives in ‘cognitive programming’.)At least if one believes that ‘dynamics’ is of the essence in cognition (rather than a mere interfacing problem between the halls of eternal truth and the noisy streets of reality), the true test for the present enterprise is the development of a significant new research program not merely copying the questions of old. (shrink)

Recent work on formal theories of truth has revived an approach, due originally to Tarski, on which syntax and truth theories are sharply distinguished—‘disentangled’—from mathematical base theories. In this paper, we defend a novel philosophical constraint on disentangled theories. We argue that these theories must be epistemically stable: they must possess an intrinsic motivation justifying no strictly stronger theory. In a disentangled setting, even if the base and the syntax theory are individually stable, they may be jointly unstable. We contend (...) that this flaw afflicts many proposals discussed in the literature; we defend a new, stable disentangled theory, double second-order arithmetic. (shrink)

An elementary theory of concatenation,QT+, is introduced and used to establish mutual interpretability of Robinson arithmetic, Minimal Predicative Set Theory, quantifier-free part of Kirby’s finitary set theory, and Adjunctive Set Theory, with or without extensionality. The most basic arithmetic and simplest set theory thus turn out to be variants of string theory.

There's Something about Gödel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical.The first part, which stays close to Gödel's original proofs, strikes a nice balance, giving enough details that the reader understands what is going on in the proofs, without (...) giving so many that the reader feels overburdened. Perhaps he skimps too much on details, as when he decides not to explain how to convert recursive definitions into explicit ones. Also, I wish he had talked about Löb's theorem. But these are small complaints.The second half discusses a sampling of what one reads about Gödel's theorems in philosophy journals and in the popular press, and here Berto often finds himself exasperated, especially by …. (shrink)

We consider weak theories of concatenation, that is, theories for strings or texts. We prove that the theory of concatenation WTC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WTC}^{-\varepsilon}}$$\end{document}, which is a weak subtheory of Grzegorczyk’s theory TC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{TC}^{-\varepsilon}}$$\end{document}, is a minimal essentially undecidable theory, that is, the theory WTC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WTC}^{-\varepsilon}}$$\end{document} is essentially undecidable and if one omits an axiom scheme from WTC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WTC}^{-\varepsilon}}$$\end{document}, then the resulting theory is no longer essentially undecidable. Moreover, we give a positive answer to Grzegorczyk and Zdanowski’s conjecture that ‘The theory TC-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{TC}^{-\varepsilon}}$$\end{document} is a minimal essentially undecidable theory’. For the alternative theories WTC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WTC}}$$\end{document} and TC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{TC}}$$\end{document} which have the empty string, we also prove that the each theory without the neutrality of ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon}$$\end{document} is to be such a theory too. (shrink)

We prove that a variant of Robinson arithmetic $\mathsf{Q}$ with nontotal operations is interpretable in the theory of concatenation $\mathsf{TC}$ introduced by A. Grzegorczyk. Since $\mathsf{Q}$ is known to be interpretable in that nontotal variant, our result gives a positive answer to the problem whether $\mathsf{Q}$ is interpretable in $\mathsf{TC}$. An immediate consequence is essential undecidability of $\mathsf{TC}$.

We study a hierarchy of Kalmàr elementary functions on integers based on a classification of LOOP programs of limited complexity, namely those in which the depth of nestings of LOOP commands does not exceed two. It is proved that -place functions in can be enumerated by a single function in , and that the resulting hierarchy of elementary predicates (i.e., functions with 0,1-values) is proper in that there are predicates that are not in . Along the way the rudimentary predicates (...) of Smullyan are classified as. (shrink)

We investigate quantifier-free induction for Lisp-like lists constructed inductively from the empty list $$ nil $$ nil and the operation $${\textit{cons}}$$ cons, that adds an element to the front of a list. First we show that, for $$m \ge 1$$ m ≥ 1, quantifier-free $$m$$ m -step induction does not simulate quantifier-free $$(m + 1)$$ ( m + 1 ) -step induction. Secondly, we show that for all $$m \ge 1$$ m ≥ 1, quantifier-free $$m$$ m -step induction does not (...) prove the right cancellation property of the concatenation operation on lists defined by left-recursion. (shrink)

We define a new theory of concatenation WTC which is much weaker than Grzegorczyk's well-known theory TC. We prove that WTC is mutually interpretable with the weak theory of arithmetic R. The latter is, in a technical sense, much weaker than Robinson's arithmetic Q, but still essentially undecidable. Hence, as a corollary, WTC is also essentially undecidable.

Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature \ be given. For a set \ of \-formulas, we introduce a corresponding set \ of new relation symbols and a set of explicit definitions of the \ in terms of the \. (...) This is called a definition system, denoted \. A definition system \ determines a translation function\. Any \-structure A can be uniquely definitionally expanded to a model \, called \. The reduct \ to the Q-symbols is called the definitional image\ of A. Likewise, a theory T in \ may be extended a definitional extension \; the restriction of this extension \ to \ is called the definitional image\ of T. If \ and \ are in disjoint signatures and \, we say that \ and \ are definitionally equivalent and \). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories. (shrink)

Following recent developments in the literature on axiomatic theories of truth, we investigate an alternative to the widespread habit of formalizing the syntax of the object-language into the object-language itself. We first argue for the proposed revision, elaborating philosophical evidences in favor of it. Secondly, we present a general framework for axiomatic theories of truth with theories of syntax. Different choices of the object theory O will be considered. Moreover, some strengthenings of these theories will be introduced: we will consider (...) extending the theories by the addition of coding axioms or by extending the schemas of O, if present, to the entire vocabulary of our theory of truth. Finally, we touch on the philosophical consequences that the theories described can have on the debate about the metaphysical status of the truth predicate and on the formalization of our informal metatheoretic reasoning. (shrink)

The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications.

This paper traces the development of Quine's ontological ideas throughout his early logical work in the period before 1948. It shows that his ontological criterion critically depends on this work in logic. The use of quantifiers as logical primitives and the introduction of general variables in 1936, the search for adequate comprehension axioms, and problems with proper classes, all forced Quine to consider ontological questions. I also show that Quine's rejection of intensional entities goes back to his generalisation of Principia (...) Mathematica in 1932. (shrink)

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.