A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...) of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has (...) decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles. (shrink)

Of the various notions of reduction in the logical literature, relative interpretability in the sense of Tarskiet al. [6] appears to be the central one. In the present note, this syntactic notion is characterized semantically, through the existence of a suitable reduction functor on models. The latter mathematical condition itself suggests a natural generalization, whose syntactic equivalent turns out to be a notion of interpretability quite close to that of Ershov [1], Szczerba [5] and Gaifman [2].

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

How can we be certain that software is reliable? Is there any method that can verify the correctness of software for all cases of interest? Computer scientists and software engineers have informally assumed that there is no fully general solution to the verification problem. In this paper, we survey approaches to the problem of software verification and offer a new proof for why there can be no general solution.

We introduce a notion of complexity for interpretations, which is used to prove some new results about interpretations of sequential theories. In particular, we give a new, elementary proof of Pudlák's theorem that sequential theories are connected. We also demonstrate a counterexample to the infinitary distributive law $a \vee \bigwedge_{i \in I} b_i = \bigwedge_{i \in I} (a \vee b_i)$ in the lattice of chapters, in which the chapters a and b i are compact. (Counterexamples in which a is not (...) compact have been found previously.). (shrink)

We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and (...) purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable. (shrink)

We provide and interpret a new measure independent characterization of the Gödel speed-up phenomenon. In particular, we prove a theorem that demonstrates the indifference of the concept of a measure independent Gödel speed-up to an apparent weakening of its definition that is obtained by requiring only those measures appearing in some fixed Blum complexity measure to participate in the speed-up, and by deleting the “for all r” condition from the definition so as to relax the required amount of speed-up. We (...) interpret our results as correlating the relative difficulty of mechanically recognizing theories with the relative power and the relative abstractness of the theories. We conclude by providing two open problems concerning possible similarities and relationships between the Blum speedability and Gödel speed-up phenomena. MSC: 03D15, 03F20. (shrink)

In this paper we study several translations that map models and formulae of the language of second-order arithmetic to models and formulae of the language of truth. These translations are useful because they allow us to exploit results from the extensive literature on arithmetic to study the notion of truth. Our purpose is to present these connections in a systematic way, generalize some well-known results in this area, and to provide a number of new results. Sections 3 and 4 contain (...) some recursion- and proof-theoretic results about Kripke-style fixed-point theories of truth. Section 5 shows how to derive full second-order arithmetic from principles of truth. Section 6 investigates the proof-theoretic strength of disquotation without an arithmetical base theory. (shrink)

Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: terminological and conceptual preliminaries, Tarski's definition of the concept of logical consequence, Tarski's discussion of omega-incomplete theories, and concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as (...) Tarski's discussion concerning the distinction between material and formal consequence and the important question ofthe criterion for distinguishing between logical and non-logical terms.§1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P and a single proposition c. The expression ‘c is a [logical] consequence of P’ is used with the same meaning as the expression ‘c is [logically] implied by P’. The expressions ‘is a logical consequence of’ and the converse ‘implies’ are relational. Often, I shall be talking in the same sense of validity of an argument. Validity is a property of arguments; an argument with premise-set P and conclusion c is valid if and only if P implies c; i.e., c is a logical consequence of P. Notice that this notion of argument is strictly ontic; it does not involve any agent that thinks, determines or establishes that a given proposition is or is not a consequence of a given set of propositions. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

In this paper, we study finitely axiomatizable conservative extensions of a theoryUin the case whereUis recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.Consider a finite expansion of the signature ofUthat contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extensionαofUin the expanded language that is conservative overU, there is a conservative extensionβofUin the expanded language, such that$\alpha (...) \vdash \beta$and$\beta \not \vdash \alpha$. The result is preserved when we consider eitherextensionsormodel-conservative extensionsofUinstead ofconservative extensions. Moreover, the result is preserved when we replace$\dashv$as ordering on the finitely axiomatized extensions in the expanded language by a relevant kind of interpretability, to witinterpretability that identically translates the symbols of the U-language.We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions ofUordered by interpretability that preserves the symbols ofU. (shrink)

The problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo-Fraenkel which is already complete with respect to the ∀*∀ class of sentences. NP-hardness and NP-completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented.

We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.

Vann McGee has argued that, given certain background assumptions and an ought-implies-can thesis about norms of rationality, Bayesianism conflicts globally with computationalism due to the fact that Robinson arithmetic is essentially undecidable. I show how to sharpen McGee's result using an additional fact from recursion theory—the existence of a computable sequence of computable reals with an uncomputable limit. In conjunction with the countable additivity requirement on probabilities, such a sequence can be used to construct a specific proposition to which Bayesianism (...) requires an agent to assign uncomputable credence—yielding a local conflict with computationalism. (shrink)

Intertheoretic relations are an important topic in the philosophy of science. However, since their classical discussion by Ernest Nagel, such relations have mostly been restricted to relations between pairs of theories in the natural sciences. This paper presents a case study of a new type of intertheoretic relation that is inspired by Montague’s analysis of the linguistic syntax-semantics relation. The paper develops a simple model of this relation. To motivate the adoption of our new model, we show that this model (...) extends the scope of application of the Nagelian models and that it shares the epistemological advantages of the Nagelian model. The latter is achieved in a Bayesian framework. (shrink)

A new logic is presented without predicates—except equality. Yet its expressive power is the same as that of predicate logic, and relations can faithfully be represented in it. In this logic we also develop an alternative for set theory. There is a need for such a new approach, since we do not live in a world of sets and predicates, but rather in a world of things with relations between them.

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)

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. IfDis an ultrafilter over a setI, andis a structure, the ultrapower ofmoduloDis denoted byD-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure. Our ultimate aim is to find out what kinds of structure are ultrapowers of. We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis, for (...) each cardinalαthere is an ultrafilterDover a set of powerαsuch that for all structures,D-prodisα+-saturated. (shrink)

In this paper a new property of theories, called effective retractability is introduced and used to obtain a characterization for the degrees of subtheories of arithmetic and set theory. By theory we understand theory in standard formalization as defined by Tarski [10]. The word degree refers to the Kleene-Post notion of degree of recursive unsolvability [2]. By the degree of a theory we mean, of course, the degree associated with its decision problem via Gödel numbering.

Maddy gave a semi-formal account of restrictiveness by defining a formal notion based on a class of interpretations and explaining how to handle false positives and false negatives. Recently, Hamkins pointed out some structural issues with Maddy's definition. We look at Maddy's formal definitions from the point of view of an abstract interpretation relation. We consider various candidates for this interpretation relation, including one that is close to Maddy's original notion, but fixes the issues raised by Hamkins. Our work brings (...) to light additional structural issues that we also discuss. (shrink)

Several philosophers of science construe models of scientific theories as set-theoretic structures. Some of them moreover claim that models should not be construed as structures in the sense of model theory because the latter are language-dependent. I argue that if we are ready to construe models as set-theoretic structures (strict semantic view), we could equally well construe them as model-theoretic structures of higher-order logic (liberal semantic view). I show that every family of set-theoretic structures has an associated language of higher-order (...) logic and an up to signature isomorphism unique model-theoretic counterpart, which is able to serve the same purposes. This allows to carry over every syntactic criterion of equivalence for theories in the sense of the liberal semantic view to theories in the sense of the strict semantic view. Taken together, these results suggest that the recent dispute about the semantic view and its relation to the syntactic view can be resolved. (shrink)

A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Hartig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition (...) of these results is needed. The aim of this paper is to give an overview of the present knowledge about the language LI and list a selection of open problems concerning it. After the Introduction $(\S1)$ , in $\S\S2$ and 3 we give the fundamental results about LI. In $\S4$ the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories in LI. In $\S6$ the spectra of sentences of LI are discussed, and $\S7$ is devoted to properties of LI which depend on set-theoretic assumptions. The paper finishes with a list of open problem and an extensive bibliography. The bibliography contains not only papers we refer to but also all papers known to us containing results about the language with Hartig quantifier. Contents. $\S1$ . Introduction. $\S2$ . Preliminaries. $\S3$ . Basic results. $\S4$ . Model-theoretic properties of $LI. \S5$ . Decidability of theories with $I. \S6$ . Spectra of LI- sentences. $\S7$ . Independence results. $\S8$ . What is not yet known about LI. Bibliography. (shrink)

About a month after his 83rd birthday Raphael Robinson was almost wholly incapacitated by a massive stroke, and 8 weeks later, on January 27, 1995, he died of ensuing complications. Mathematics was his life. He was always working on problems—those brought to him in journals or by colleagues, and others that he invented. Just three days before his death he received word that a paper of his, originating in a published problem, was accepted for publication. His 64 publications spanned a (...) full 6 decades, and included significant papers on number theory, combinatorics, complex analysis, and geometry, in addition to logic and set theory.Robinson was born on November 2, 1911, in National City, California. He came to Berkeley to study mathematics, obtaining an A.B. in 1932, an M.A. in 1933, and his Ph.D. in December, 1934. His dissertation was in complex analysis. He took 12 graduate courses in a variety of fields, but not one was close to logic or set theory—which was later the setting of about a quarter of his publications.After completing his studies Robinson served as Instructor for two years at Brown University, then returned for good to Berkeley in 1937, reaching the rank of full Professor in 1949. He taught a great variety of courses and was known as an excellent teacher, but was most interested in problemsolving and research and in 1972, at age 61, he elected early retirement, at considerable financial sacrifice, in order to concentrate on these pursuits. (shrink)

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic ILM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ILM}$$\end{document} of Peano Arithmetic. It is well-known that any theories extending PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PA}$$\end{document} have a supremum in the interpretability ordering. While provable in PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PA}$$\end{document}, this fact is not reflected in the theorems of the modal (...) system ILM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ILM}$$\end{document}, due to limited expressive power. Our goal is to enrich the language of ILM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ILM}$$\end{document} by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a provability predicate satisfying the axioms of the provability logic GL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {GL}$$\end{document}. (shrink)

Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...) proves, without any use of arithmetical induction, that Frege's definition is extensionally correct and so answers the circularity objections. (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)

A substitutional account of logical validity for formal first‐order languages is developed and defended against competing accounts such as the model‐theoretic definition of validity. Roughly, a substitution instance of a sentence is defined as the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category and possibly relativizing quantifiers. In particular, predicate symbols can be replaced with formulae possibly containing additional free variables. A sentence is defined to be logically true iff all its substitution (...) instances are satisfied by all variable assignments. Logical consequence is defined analogously. Satisfaction is taken to be a primitive notion and axiomatized.For every set‐theoretic model in the sense of model theory there exists a corresponding substitutional interpretation in a sense to be specified. Conversely, however, there are substitutional interpretations – in particular the ‘intended’ interpretation – that lack a model‐theoretic counterpart. The substitutional definition of logical validity overcomes the weaknesses of more restrictive accounts of substitutional validity; unlike model‐theoretic logical consequence, the substitutional notion is trivially and provably truth preserving. In Kreisel's squeezing argument the formal notion of substitutional validity naturally slots into the place of intuitive validity. (shrink)