Finitary Research Articles

INFINITARY LOGIC HAS NO EXPRESSIVE EFFICIENCY OVER FINITARY LOGIC

Abstract We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\varphi $ (in $\mathcal {L}_{\omega ,\omega }$ ) is equivalent to a formula of the infinitary language $\mathcal {L}_{\infty ,\omega }$ with n alternations of quantifiers. We prove that $\varphi $ is equivalent to a finitary formula with n alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a simpler way.

Filter pairs and natural extensions of logics

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality \(\kappa \), where \(\kappa \) is a regular cardinal. The corresponding new notion is called \(\kappa \)-filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different \(\kappa \)-filter pairs give rise to a fixed logic of cardinality \(\kappa \). To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality \(\kappa \). Along the way we use \(\kappa \)-filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair.

On lattice-ordered algebras with infinitary operations

On lattice-ordered algebras with infinitary operations

Pure Variable Inclusion Logics

The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic.

On a Logico-Algebraic Approach to AGM Belief Contraction Theory

In this paper we investigate AGM belief contraction operators by using the tools of algebraic logic. We generalize the notion of contraction to arbitrary finitary propositional logics, and we show how to switch from a syntactic-based approach to a semantic one. This allows to build a solid bridge between the validity of AGM postulates in a propositional logic and specific algebraic properties of its intended algebraic counterpart. Such a connection deserves particular attention when we deal with maxichoice contractions, as studied in the final part of the paper.

Logics of variable inclusion and the lattice of consequence relations

In this paper, first, we determine the number of sublogics of variable inclusion of an arbitrary finitary logic with a composition term. Then, we investigate their position into the lattice of consequence relations over the language of .

Scattered products in fundamental groupoids

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. We prove that the well-definedness of products indexed by a scattered linear order in the fundamental groupoid of a first countable space is equivalent to the homotopically Hausdorff property. To prove this characterization, we employ the machinery of closure operators, on the $\pi_1$-subgroup lattice, defined in terms of test maps from one-dimensional domains.

An equational theory for $$\sigma $$-complete orthomodular lattices

The condition of $$\sigma $$-completeness related to orthomodular lattices places an important role in the study of quantum probability theory. In the framework of algebras with infinitary operations, an equational theory for the category of $$\sigma $$-complete orthomodular lattices is given. In this structure, we study the congruences theory and directly irreducible algebras establishing an equational completeness theorem. Finally, a Hilbert style calculus related to $$\sigma $$-complete orthomodular lattices is introduced and a completeness theorem is obtained.

Dense products in fundamental groupoids

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid $\Pi_1(X)$ has well-defined dense products if and only if $X$ admits a generalized universal covering space.

Addendum and corrigenda to the paper “Infinitary superperfect numbers”

Addendum and corrigenda to the paper “Infinitary superperfect numbers”

Pattern-based calculi with finitary matching

Pattern-based calculi with finitary matching

Extension Properties and Subdirect Representation in Abstract Algebraic Logic

This paper continues the investigation, started in Lavicka and Noguera (Stud Log 105(3): 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, (completely) intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics.

A New Hierarchy of Infinitary Logics in Abstract Algebraic Logic

In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively (finitely) subdirectly irreducible models. We identify two syntactical notions formulated in terms of (completely) intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, we obtain a new hierarchy of logics going beyond the scope of finitarity.

Kant and Finitism

Kant and Finitism

On Tait on Kant and Finitism

On Tait on Kant and Finitism

Finitism and Divisibility: A Reply to Puryear

Puryear (AJP, 2014) develops an objection against a prominent attempt to show that the universe must have a temporal beginning. Here I formulate a reply.

Almost structurally complete infinitary consequence operations extending S4.3

Almost structurally complete infinitary consequence operations extending S4.3

A Note on Natural Extensions in Abstract Algebraic Logic

Transfer theorems are central results in abstract algebraic logic that allow to generalize properties of the lattice of theories of a logic to any algebraic model and its lattice of filters. Their proofs sometimes require the existence of a natural extension of the logic to a bigger set of variables. Constructions of such extensions have been proposed in particular settings in the literature. In this paper we show that these constructions need not always work and propose a wider setting (including all finitary logics and those with countable language) in which they can still be used.

Kronecker in Contemporary Mathematics, General Arithmetic as a Foundational Programme

Kronecker called his programme of arithmetization “General Arithmetic” (Allgemeine Arithmetik). In his view, arithmetic is the building block of the whole edifice of mathematics. The aim of this paper is to show that Kronecker’s arithmetical philosophy and mathematical practice have exerted a permanent influence on a long tradition of mathematicians from Hilbert to Weil, Grothendieck and Langlands. The conclusion hints at a constructivist finitist stance in contemporary mathematical logic, especially proof theory, beyond Hilbert’s programme of finitist foundations which can be seen as the continuation of Kronecker’s arithmetization programme by metamathematical or logical means. It is finally argued that the introduction of higher-degree polynomials by Kronecker inspired Hilbert’s notion of functionals, which in turn influenced G¨odel’s functional Dialectica interpretation for his intuitionistic proof of the consistency of arithmetic.

The Proof by Cases Property and its Variants in Structural Consequence Relations

This paper is a contribution to the study of the role of disjunction in Abstract Algebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic.