In this article I present HYPER-REF, a model to determine the referent of any given expression in First-Order Logic. I also explain how this model can be used to determine the referent of a first-order theory such as First-Order Arithmetic. By reference or referent I mean the non-empty set of objects that the syntactical terms of a well-formed formula pick out given a particular interpretation of the language. To do so, I will (...) class='Hi'>first draw on previous work to make explicit the notion of reference and its hyperintensional features. Then I present HYPER-REF and offer a heuristic method for determining the reference of any formula. Then I discuss some of the benefits and most salient features of HYPER-REF, including some remarks on the nature of self-reference in formal languages. (shrink)

In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long ‘search’ for a purely mathematical incompleteness result in first-order Peano arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic (...) form. (shrink)

The purpose of this article is to present a new theory for fixed points over arithmetic which allows the building up of fixed points in a very nested and entangled way. But in spite of its great expressive power we can show that the proof-theoretic strength of our theory—which is intensional in a meaning to be described below—is characterized by the Feferman–Schütte ordinal Γ0. Our approach is similar to the building up of fixed points over state spaces in the (...) propositional modal μ-calculus. (shrink)

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

Theorem 1. If T is a sound formalized theory whose language contains the language of basic arithmetic, then there will be a true sentence GT of basic arithmetic such that T ￿ GT and ￿ ¬GT, so T must be negation incomplete.

A value space is a topological algebra equipped with a non-empty family of continuous quantifiers . We will describe first-order logic on the basis of . Operations of are used as connectives and its relations are used to define statements. We prove under some normality conditions on the value space that any theory in the new setting can be represented by a classical first-order theory.

We study the first-order theory of polynomial rings over a GCD domain and of the ring of formal entire functions over a non-Archimedean field in the language $\{ 1, +, \bot \}$. We show that these structures interpret the first-order theory of the semi-ring of natural numbers. Moreover, this interpretation depends only on the characteristic of the original ring, and thus we obtain uniform undecidability results for these polynomial and entire functions rings of a fixed characteristic. (...) This work enhances results of Raphael Robinson on essential undecidability of some polynomial or formal power series rings in languages that contain no symbols related to the polynomial or power series ring structure itself. (shrink)

For every sequence |p n } n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula 0 A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, 0 A is a theorem ofPA.

This paper examines the ontological commitments of the second-order language of arithmetic and argues that they do not extend beyond the first-order language. Then, building on an argument by George Boolos, we develop a Tarski-style definition of a truth predicate for the second-order language of arithmetic that does not involve the assignment of sets to second-order variables but rather uses the same class of assignments standardly used in a definition for the first- (...) class='Hi'>order language. (shrink)

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number (...) theory is forcible from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

Applying first-order logic to derive the consequences of laws that are only approximately true of empirical phenomena involves idealization of a kind that is akin to applying arithmetic to calculate the area of a rectangular surface from approximate measures of the lengths of its sides. Errors in the data, in the exactness of the lengths in one case and in the exactness of the laws in the other, are in some measure transmitted to the consequences deduced from (...) them, and the aim of a theory of idealization is to describe this process. The present paper makes a start on this in the case of applied first-order logic, and relates it to Plato's picture of a world or model of 'appearances' in which laws are only approximately true, but which to some extent resembles an ideal world or model in which they are exactly true. (shrink)

It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka’s :69–90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order (...) logic versus second-order logic divide. The descriptive approach is illustrated by Dedekind’s ‘discovery’ of the need for second-order concepts to ensure categoricity in his axiomatization of arithmetic; the deductive approach is illustrated by Frege’s Begriffsschrift project. I argue that, rather than suggesting that any use of logic in the foundations of mathematics is doomed to failure given the impossibility of combining the descriptive approach with the deductive approach, what this apparent predicament in fact indicates is that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like. I also conclude that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic for this purpose; different logical systems may be adequate for different projects. (shrink)

Elementary first-order theories of trees allowing at most, exactly $\mathrm{m}$, and any finite number of immediate descendants are introduced and proved mutually interpretable among themselves and with Robinson arithmetic, Adjunctive Set Theory with Extensionality and other well-known weak theories of numbers, sets, and strings.

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third (...) author in [V. G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 ]. (shrink)

We consider one aspect of the problem of specification and verification of reactive real-time systems which involve operations and constraints concerning time. Time is continuous what is motivated by specifications of hybrid systems. Our goal is to try to find a framework that is based on applied first order logic that permits to represent the verification problem directly, completely and conservatively , and that is apt to describe interesting decidable classes, maybe showing way to feasible algorithms. To achieve (...) this goal we use a first order timed logic that is an extension of a decidable theory of reals with timed functions. This logic permits, on the one hand, to rewrite directly and completely requirements and, on the other hand, to describe executions of various timed algorithms—here we consider block Gurevich abstract state machines because of their theoretical clarity and sufficient expressive power. Then we describe one decidable class of the verification problem that is based on notions reflecting finiteness properties of systems of control. These notions may be of independent interest, as, in particular, they give a way to describe a limited usage of arithmetics preserving decidability that is not covered by existing model-theoretic approaches. As an example we consider the generalized railroad crossing problem that we analyze in its entirety. (shrink)

Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic, using the notion of a first-order hyperdoctrine from categorical logic, to a structure which we (...) will call the hyperdoctrine of mass problems. We study the intermediate logic that the hyperdoctrine of mass problems gives us, and we study the theories of subintervals of the hyperdoctrine of mass problems in an attempt to obtain an analogue of Skvortsova’s result that there is a factor of the Medvedev lattice characterising intuitionistic propositional logic. Finally, we consider Heyting arithmetic in the hyperdoctrine of mass problems and prove an analogue of Tennenbaum’s theorem on computable models of arithmetic. (shrink)

In this dissertation, we construct a consistent, complete quotational logic G$\sb1$. We first develop a semantics, and then show the undecidability of circular quotation and anaphorism . Next, a complete axiom system is presented, and completeness theorems are shown for G$\sb1$. We show that definable truth exists in G$\sb1$. ;Later, we replace equality in G$\sb1$ with an equivalence relation. An axiom system and completeness theorems are provided for this equality-free version of G$\sb1$, which is useful in program verification. ;Interpolation (...) and definability are discussed in intensional and extensional contexts. We show that G$\sb1$ contains the complete first order theory of arithmetic by implicit definitional extension. ;The last chapter deals with representability and fixed points. Derived induction schemata are produced, and G$\sb1$ is shown to be a recursion theory. We give an example of how G$\sb1$ may be used to solve domain equations of Scott. (shrink)

In this paper the modal operator "x is provable in Peano Arithmetic" is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.

What we call the Hilbert‐Bernays (HB) Theorem establishes that for any satisfiable first‐order quantificational schema S, there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S. Our goals here are, first, to explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first‐order logical validity of a schema in terms of predicate substitution; second, to (...) clarify the theorem by sketching an accessible and illuminating new proof of it; and, third, to explain how Quine's substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians. (shrink)

We introduce the notion of bisimulation for first-order Kripke models. It is defined as a relation that satisfies certain zig-zag conditions involving back-and-forth moves between nodes of Kripke models and, simultaneously, between the domains of their underlying structures. As one of our main results, we prove that if two Kripke models bisimulate to a certain degree, then they are logically equivalent with respect to the class of formulae of the appropriate complexity. Two applications of the notion introduced in (...) the paper are given. First, we consider bisimulations in the context of Kripke models of Heyting Arithmetic. Next, we discuss the notion of elementary submodel of a Kripke model and provide a suitable example. (shrink)

We define the notion of a combinatorics of a first order structure, and a relation of covering between first order structures and propositional proof systems. Namely, a first order structure M combinatorially satisfies an L-sentence Φ iff Φ holds in all L-structures definable in M. The combinatorics Comb(M) of M is the set of all sentences combinatorially satisfied in M. Structure M covers a propositional proof system P iff M combinatorially satisfies all Φ for (...) which the associated sequence of propositional formulas 〈Φ〉 n , encoding that Φ holds in L-structures of size n, have polynomial size P-proofs. That is, Comb(M) contains all Φ feasibly verifiable in P. Finding M that covers P but does not combinatorially satisfy Φ thus gives a super polynomial lower bound for the size of P-proofs of 〈Φ〉 n . We show that any proof system admits a class of structures covering it; these structures are expansions of models of bounded arithmetic. We also give, using structures covering proof systems R * (log) and PC, new lower bounds for these systems that are not apparently amenable to other known methods. We define new type of propositional proof systems based on a combinatorics of (a class of) structures. (shrink)

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted (...) by finite sets of natural numbers. (shrink)

We introduce a first order extension of GL, called ML 3 , and develop its proof theory via a proxy cut-free sequent calculus GLTS. We prove the highly nontrivial result that cut is a derived rule in GLTS, a result that is unavailable in other known first-order extensions of GL. This leads to proofs of weak reflection and the related conservation result for ML 3 , as well as proofs for Craig’s interpolation theorem for GLTS. Turning (...) to semantics we prove that ML 3 is sound with respect to arithmetical interpretations and that it is also sound and complete with respect to converse well-founded and transitive finite Kripke models. This leads us to expect that a Solovay-like proof of arithmetical completeness of ML 3 is possible. (shrink)

In the tech report Artemov and Yavorskaya [4] an elegant formulation of the first-order logic of proofs was given, FOLP. This logic plays a fundamental role in providing an arithmetic semantics for first-order intuitionistic logic, as was shown. In particular, the tech report proved an arithmetic completeness theorem, and a realization theorem for FOLP. In this paper we provide a possible-world semantics for FOLP, based on the propositional semantics of Fitting [5]. We also give (...) an Mkrtychev semantics. Motivation and intuition for FOLP can be found in Artemov and Yavorskaya [4], and are not fully discussed here.This paper is dedicated to Sergei Artemov, an honored colleague and friend, who has made wonderful things for the rest of us to play with. (shrink)

We introduce a notion of syntactical truth predicate (s.t.p.) for the second order arithmetic PA 2 . An s.t.p. is a set T of closed formulas such that: (i) T(t = u) if and only if the closed first order terms t and u are convertible, i.e., have the same value in the standard interpretation (ii) T(A → B) if and only if (T(A) $\Longrightarrow$ T(B)) (iii) T(∀ x A) if and only if (T(A[x ← t]) (...) for any closed first order term t) (iv) T(∀ X A) if and only if (T(A[X ←▵]) for any closed set definition $\triangle = \{x \mid D(x)\}$ ). S.t.p.'s can be seen as a counterpart to Tarski's notion of (model-theoretical) validity and have main model properties. In particular, their existence is equivalent to the existence of an ω-model of PA 2 , this fact being provable in PA 2 with arithmetical comprehension only. (shrink)

It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.

We prove that PTCN cannot be a model of U12. This implies that there exists a first order sentence of bounded arithmetic which is provable in U12 but does not hold in PTCN.

BAT is a logic built to capture the inferential behavior of informal provability. Ultimately, the logic is meant to be used in an arithmetical setting. To reach this stage it has to be extended to a first-order version. In this paper we provide such an extension. We do so by constructing non-deterministic three-valued models that interpret quantifiers as some sorts of infinite disjunctions and conjunctions. We also elaborate on the semantical properties of the first-order system and (...) consider a couple of its strengthenings. It turns out that obtaining a sensible strengthening is not straightforward. We prove that most strategies commonly used for strengthening non-deterministic logics fail in our case. Nevertheless, we identify one method of extending the system which does not. (shrink)

Summary We have considered complete consistent systems in the first‐oder predicate calculus with identity, and have studied the set of the models of such a system by means of the maximal consistent condition‐sets associated with the system. The results may be summarized thus: (a) A complete consistent system is no‐categorical (= categorical in the denumerable domain) if and only if for every n, the number of different conditions in n variables is finite (T10). (b) If a complete consistent system (...) has a model M with finite character (i.e. a model M such that every maximal consistent condition‐set satisfied in M has a finite basis), then this model M is uniquely characterized by the property that every other model is an arithmetic extension of M (T 5). (c) Every complete consistent system, which has only a denumerable number of different associated maximal consistent condition‐sets, has a model with finite character (T8). (shrink)

LetZ2,Z3, andZ4denote 2nd, 3rd, and 4thorder arithmetic, respectively. We let Harrington’s Principle, HP, denote the statement that there is a realxsuch that everyx-admissible ordinal is a cardinal inL. The known proofs of Harrington’s theorem “$Det\left$implies 0♯exists” are done in two steps: first show that$Det\left$implies HP, and then show that HP implies 0♯exists. The first step is provable inZ2. In this paper we show thatZ2+ HP is equiconsistent with ZFC and thatZ3+ HP is equiconsistent with ZFC + there (...) exists a remarkable cardinal. As a corollary,Z3+ HP does not imply 0♯exists, whereas Z4+ HP does. We also study strengthenings of Harrington’s Principle over 2ndand 3rdorder arithmetic. (shrink)

In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies associated to finite models is not recursively enumerable. In 1999, P. Hájek generalized this result to the first-order versions of Łukasiewicz, Gödel and Product logics, w.r.t. their standard algebras. In this paper we extend the analysis to the first-order versions of axiomatic extensions of MTL. Our main result is the following. Let \ be a class of MTL-chains. Then the set of all (...) class='Hi'>first-order tautologies associated to the finite models over chains in \, fTAUT\, is \ -hard. Let TAUT\ be the set of propositional tautologies of \. If TAUT\ is decidable, we have that fTAUT\ is in \. We have similar results also if we expand the language with the Δ operator. (shrink)

We describe a decidable class of formulas in a first order timed logic that covers a good amount of properties of real-time distributed systems. Earlier we described a decidable class based on some finiteness properties, and sketched a decidable class in a weaker logic that captures periodicity properties, though without complete proof. The new feature of the decidable class presented here is to be able to treat parametric properties, in particular, properties that concern an arbitrary number of processes (...) as compared to the just mentioned class that could express only properties for a fixed, explicitly given number of processes. As before the properties may use non-trivial arithmetics. We also state another important feature of the class: if a verification formula is not true then our algorithm gives a quantifier-free description of all its counter-models of the complexity involved in the definition of the decidable class and that is quite representative. In conclusion we discuss theoretical and practical issues of the complexity and some open questions. (shrink)

The incompleteness of formal systems for arithmetic has been a recognized fact of mathematics. The term “incompleteness” suggests that the formal system in question fails to offer a deduction which it ought to. This chapter focuses on the status of a formal system, Peano Arithmetic, and explores a viewpoint on which Peano Arithmetic occupies an intrinsic, conceptually well-defined region of arithmetical truth. The idea is that it consists of those truths which can be perceived directly from the (...) purely arithmetical content of a categorical conceptual analysis of the notion of natural number. The chapter explores its conceptual stability in light of the apparently destabilizing effect, through extensibility, of the phenomenon of incompleteness. It also offers heuristic and conceptual support for the viewpoint that Peano Arithmetic is the strongest natural first-order system for arithmetic, and that there is a sense in which it is complete with respect to purely arithmetical truth. (shrink)

We propose a rudimentary formal framework for reasoning about recursion equations over inductively generated data. Our formalism admits all equational programs , and yet singles out none. While being simple, this framework has numerous extensions and applications. Here we lay out the basic concepts and definitions; show that the deductive power of our formalism is similar to that of Peano's Arithmetic; prove a strong normalization theorem; and exhibit a mapping from natural deduction derivations to an applied λ -calculus, à (...) la Curry–Howard, which provides a transparent proof of Gödel's result that the provably recursive functions of PA are definable by primitive recursion in finite types. Also of interest is the extent, uncommon among similar constructions, to which our mapping is oblivious to first order terms, including first order quantification and equality. Additional applications and extensions of the method will be presented in sequel papers. In particular, the methodology introduced here lends itself to systematic links between natural sub-formalisms and computational complexity classes. Since these sub-formalisms are largely transparent, allowing one to reason formally in the full system while leaving to automatic software tools the task of verifying that restrictions are complied with, the framework is particularly attractive for the development of Feasible Mathematics. (shrink)

Via the formulas-as-types embedding certain extensions of Heyting Arithmetic can be represented in intuitionistic type theories. In this paper we discuss the embedding of ω-sorted Heyting Arithmetic HA ω into a type theory WL, that can be described as Troelstra's system ML 1 0 with so-called weak Σ-elimination rules. By syntactical means it is proved that a formula is derivable in HA ω if and only if its corresponding type in WL is inhabited. Analogous results are proved for (...) Diller's so-called restricted system and for a type theory based on predicate logic instead of arithmetic. (shrink)

In this article we study linear temporal logics with team semantics that are novel logics for defining hyperproperties. We define Kamp-type translations of these logics into fragments of first-order team logic and second-order logic. We also characterize the expressive power and the complexity of model-checking and satisfiability of team logic and second-order logic by relating them to second- and third-order arithmetic. Our results set in a larger context the recent results of Lück showing that (...) the extension of TeamLTL by the Boolean negation is highly undecidable under the so-called synchronous semantics. We also study stutter-invariant fragments of extensions of TeamLTL. (shrink)

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that it is trivial that relevant arithmetic is absolutely consistent, but classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under, I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, (...) of course, is exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under, I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment. (shrink)

What seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher- (...) class='Hi'>order logic, with formal derivations closer to one hundred steps in length and up to four nested temporary assumptions with their scope indicated by vertical intermittent lines. (shrink)

(Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are (...) applied to answer a question posed by Bacon and Dorr (2024). (shrink)