in: Infinity and Truth, Edited by: Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin, Lecture Notes Series of the Institute for Mathematical Sciences, World Scientific, vol. 25, 2014, 171-186
I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc {\em large domain assumptions}. In this paper I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist approach to mathematics characteristic of second order logic. As a model-theoretic logic sort logic is the strongest logic. In fact, every model class definable in set theory is the class of models of a sentence of sort logic. Because of its strength sort logic can be used to formulate particularly strong reflection principles in set theory.
In this paper we prove a Robinson consistency theorem for a class of many-sorted hybrid logics as a consequence of an Omitting Types Theorem. An important corollary of this result is an interpolation theorem.
We present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic.
We consider a many-sorted variant of Japaridze's polymodal provability logic $\mathsf{GLP}$. In this variant, which is denoted $\mathsf{GLP}^\ast$, propositional variables are assigned sorts $α\leq ω$, where variables of finite sort $n < ω$ are interpreted as $Π_{n+1}$-sentences of the arithmetical hierarchy, while those of sort $ω$ range over arbitrary ones. We prove that $\mathsf{GLP}^\ast$ is arithmetically complete with respect to this interpretation. Moreover, we relate $\mathsf{GLP}^\ast$ to its one-sorted counterpart $\mathsf{GLP}$ and prove that the former inherits some well-known prop…
First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved for DFOL. The semantics is functorial in the …
Within the program of finding axiomatizations for various parts of computability logic, it was proved earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting's intuitionistic calculus. That sort of reduction permits unlimited reusage of the computational resource represented by the antecedent. An at least equally basic and natural sort of algorithmic reduction, however, is the one that does not allow such reusage. The present article shows that turning the logic of the first sort of reduction into the logic of the second sort of reduction takes not…
The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality ($\text{FOL}_=$) allows us to define structures on sets. We develop the syntax, semantics and deductive system for such a logic, which we call first-order logic with isomorphism ($\text{FOL}_{\cong}$). The syntax of $\text{FOL}_{\cong}$ extends $\text{FOL}_{=}$ in two ways. First, by incorporating into its signatures a notion of dependent sorts along the lines of Makkai's FOLDS as well as a notion of an $h$-level of each sort. Second, by specifying th…
Building on previous work by Mummert, Saadaoui and Sovine, we study the logic underlying the web of implications and nonimplications which constitute the so called reverse mathematics zoo. We introduce a tableaux system for this logic and natural deduction systems for important fragments of the language.
This is a set of 288 questions written for a Moore-style course in Mathematical Logic. I have used these (or some variation) four times in a beginning graduate course. Topics covered are: propositional logic axioms of ZFC wellorderings and equivalents of AC ordinal and cardinal arithmetic first order logic, and the compactness theorem Lowenheim-Skolem theorems Turing machines, Church's Thesis completeness theorem and first incompleteness theorem undecidable theories second incompleteness theorem
In the present paper syntax and semantics will be presented for an expansion of ordinary n-agent QML with constant domain, non-rigid constants, rigid variables and including both functions, relations, and equality. Further, the number of agents will be specified axiomatically thus ensuring maximal flexibility wrt. the cardinality of the set of agents. Domain, variables, and constants will be partitioned in an agent-part and an object-part and the syntax will be expanded to include strings in which indexes of modal operators are quantified over as wff's of the language. This will enhance expres…
Classical logic (the logic of non-constructive mathematics) is stronger than intuitionistic logic (the logic of constructive mathematics). Despite this, there are copies of classical logic in intuitionistic logic. All copies usually found in the literature are the same. This raises the question: is the copy unique? We answer negatively by presenting three different copies.
We show that numerous distinctive concepts of constructive mathematics arise automatically from an "antithesis" translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically "constructivize" classi…
This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Los--Tarski and the Chang--Los--Suszko preservation theorems follow.
The first seeds of mathematical intuitionism germinated in Europe over a century ago in the constructive tendencies of Borel, Baire, Lebesque, Poincaré, Kronecker and others. The flowering was the work of one man, Luitzen Egbertus Jan Brouwer, who taught mathematics at the University of Amsterdam from 1909 until 1951. By proving powerful theorems on topological invariants and fixed points of continuous mappings, Brouwer quickly build a mathematical reputation strong enough to support his revolutionary ideas about the nature of mathematical activity. These ideas influenced Hilbert and Gödel and…
This paper considers the problem of building saturated models for first-order graded logics. We define types as pairs of sets of formulas in one free variable which express properties that an element is expected, respectively, to satisfy and to falsify. We show, by means of an elementary chains construction, that each model can be elementarily extended to a saturated model where as many types as possible are realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability, and a gener…
The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the "logical" content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system for s-logic, and explore in detail two fragmen…
We characterise non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by a new notion of simulation called a meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs of states from one model to single states from another. En route to this result we use a more traditional notion of simulations and prove a Hennessy-Milner style theorem for it, using an analogue of modal saturation called meet-compactness.
At a first glance the Theory of computation relies on potential infinity and an organization aimed at solving a problem. Under such aspect it is like Mendeleev theory of chemistry. Also its theoretical development reiterates that of this scientific theory: it makes use of doubly negated propositions and its reasoning proceeds through ad absurdum proofs; a final, universal predicate of equivalence of all definitions of a computations is translated into an equality one, and at the same time intuitionist logic into classical logic. Yet, the last step of this development of current theory includes…
This article introduces the conjecture that "mathematics, logic and related disciplines may usefully be understood as information compression (IC) by 'multiple alignment', 'unification' and 'search' (ICMAUS)". As a preparation for the two main sections of the article, concepts of information and information compression are reviewed. Related areas of research are also described including IC in brains and nervous systems, and IC in relation to inductive inference, Minimum Length Encoding and probabilistic reasoning. The ICMAUS concepts and a computer model in which they are embodied are briefly …
In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers. We first define the free clone T(L, C) of terms of a first order language L over a set C of parameters in a standard way. The free right algebra F(L, C) of formulas over T(L, C) is then generated by atomic formulas. Structures for L over C are represented as perfect valuations of F(L, C), and theories of L are represented as filters of F(L). Finally Godel's completeness theorem and first incompleteness theorem are stated as expected.
