scholarly journals Maximality of bi-intuitionistic propositional logic

2021 ◽  
Author(s):  
Grigory Olkhovikov ◽  
Guillermo Badia
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Expressive Power ◽  
Formal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Intuitionistic Propositional Logic ◽  
Abstract Logic ◽  
First Order ◽  
Work Done

Abstract In the style of Lindström’s theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of the previous work done by the authors for intuitionistic logic (Badia and Olkhovikov, 2020, Notre Dame Journal of Formal Logic, 61, 11–30).

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

Lifting propositional proof compression algorithms to first-order logic

2020 ◽  
Author(s):  
Jan Gorzny ◽  
Ezequiel Postan ◽  
Bruno Woltzenlogel Paleo
Keyword(s):  
Propositional Logic ◽  
Empirical Evaluation ◽  
Order Logic ◽  
Automated Deduction ◽  
First Order Logic ◽  
Compression Algorithms ◽  
International Conference ◽  
First Order ◽  
Theorem Provers ◽  
Resolution Proofs

Abstract Proofs are a key feature of modern propositional and first-order theorem provers. Proofs generated by such tools serve as explanations for unsatisfiability of statements. However, these explanations are complicated by proofs which are not necessarily as concise as possible. There are a wide variety of compression techniques for propositional resolution proofs but fewer compression techniques for first-order resolution proofs generated by automated theorem provers. This paper describes an approach to compressing first-order logic proofs based on lifting proof compression ideas used in propositional logic to first-order logic. The first approach lifted from propositional logic delays resolution with unit clauses, which are clauses that have a single literal. The second approach is partial regularization, which removes an inference $\eta $ when it is redundant in the sense that its pivot literal already occurs as the pivot of another inference in every path from $\eta $ to the root of the proof. This paper describes the generalization of the algorithms LowerUnits and RecyclePivotsWithIntersection (Fontaine et al.. Compression of propositional resolution proofs via partial regularization. In Automated Deduction—CADE-23—23rd International Conference on Automated Deduction, Wroclaw, Poland, July 31–August 5, 2011, p. 237--251. Springer, 2011) from propositional logic to first-order logic. The generalized algorithms compresses resolution proofs containing resolution and factoring inferences with unification. An empirical evaluation of these approaches is included.

scholarly journals Taking Plurals at Face Value

2021 ◽  
pp. 8-30
Author(s):  
Salvatore Florio ◽  
Øystein Linnebo
Keyword(s):  
Philosophy Of Mathematics ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Logical System ◽  
Plural Logic ◽  
First Order ◽  
Philosophical Significance ◽  
Pure Logic

Plural logic is a logical system in which plural terms and predicates figure as primitive expressions alongside the singular resources of ordinary first-order logic. The philosophical significance of this system depends on two of its alleged features: being pure logic and providing more expressive power than first-order logic. This chapter first introduces the language and axioms of plural logic and then analyzes this logic’s main philosophical applications in metaphysics, philosophy of mathematics, and semantics.

scholarly journals Local Normal Forms for First-Order Logic with Applications to Games and Automata

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Thomas Schwentick ◽  
Klaus Barthelmann
Keyword(s):  
Normal Forms ◽  
Winning Strategy ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Relational Structures ◽  
Second Order Logic ◽  
Monadic Second Order Logic

International audience Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

scholarly journals Subformula Linking for Intuitionistic Logic with Application to Type Theory

2021 ◽  
pp. 200-216
Author(s):  
Kaustuv Chaudhuri
Keyword(s):  
Theorem Proving ◽  
Intuitionistic Logic ◽  
Type Theory ◽  
Linear Logic ◽  
Interactive Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
Direct Manipulation ◽  
First Order

AbstractSubformula linking is an interactive theorem proving technique that was initially proposed for (classical) linear logic. It is based on truth and context preserving rewrites of a conjecture that are triggered by a user indicating links between subformulas, which can be done by direct manipulation, without the need of tactics or proof languages. The system guarantees that a true conjecture can always be rewritten to a known, usually trivial, theorem. In this work, we extend subformula linking to intuitionistic first-order logic with simply typed lambda-terms as the term language of this logic. We then use a well known embedding of intuitionistic type theory into this logic to demonstrate one way to extend linking to type theory.

scholarly journals Successor-Invariant First-Order Logic on Classes of Bounded Degree (Extended Abstract)

2021 ◽  
Author(s):  
Julien Grange
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Successor Relation

We study the expressive power of successor-invariant first-order logic, which is an extension of first-order logic where the usage of a successor relation on the vertices of the graph is allowed, as long as the validity of formulas is independent on the choice of a particular successor. We show that when the degree is bounded, successor-invariant first-order logic is no more expressive than first-order logic.

A First Course in Logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Propositional Logic ◽  
Proof Theory ◽  
Order Logic ◽  
First Order Logic ◽  
University Of Maryland ◽  
First Order ◽  
Science Philosophy ◽  
The University ◽  
Second Order Logic ◽  
Theory And Model

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course.

Proof theory

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Propositional Logic ◽  
Completeness Theorem ◽  
Formal Proof ◽  
Order Logic ◽  
Truth Table ◽  
First Order Logic ◽  
Formal Proofs ◽  
Systematic Method ◽  
First Order

As with any logic, the semantics of first-order logic yield rules for deducing the truth of one sentence from that of another. In this chapter, we develop both formal proofs and resolution for first-order logic. As in propositional logic, each of these provides a systematic method for proving that one sentence is a consequence of another. Recall the Consequence problem for propositional logic. Given formulas F and G, the problemis to decide whether or not G is a consequence of F. From Chapter 1, we have three approaches to this problem: • We could compute the truth table for the formula F → G. If the truth values are all 1s then we conclude that F → G is a tautology and G is a consequence of F. Otherwise, G is not a consequence of F. • Using Tables 1.5 and 1.6, we could try to formally derive G from {F}. By the Completeness Theorem for propositional logic, G is a consequence of F if and only if {F} ├ G. • We could use resolution. By Theorem1.76, G is a consequence of F if and only if ∅ ∈ Res(H) where H is a formula in CNF equivalent to (F ∧￢G). Using these methods not only can we determine whether one formula is a consequence of another, but also we can determine whether a given formula is a tautology or a contradiction. A formula F is a tautology if and only if F is a consequence of (A∨￢A) if and only if ￢F is a contradiction. In this chapter, we consider the analogous problems for first-order logic. Given formulas φ and ψ, how can we determine whether ψ is a consequence of φ? Equivalently, how can we determine whether a given formula is a tautology or a contradiction? We present three methods for answering these questions. • In Section 3.1, we define a notion of formal proof for first-order logic by extending Table 1.5. • In Section 3.3, we “reduce” formulas of first-order logic to sets of formulas of propositional logic where we use resolution as defined in Chapter 1.

Structures and first-order logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Propositional Logic ◽  
Function Symbol ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Ternary Relation ◽  
Specific Element ◽  
First Order ◽  
Lower Case ◽  
Second Order Logic

First-order logic is a richer language than propositional logic. Its lexicon contains not only the symbols ∧, ∨, ￢, →, and ↔ (and parentheses) from propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all,” along with various symbols to represent variables, constants, functions, and relations. These symbols are grouped into five categories. • Variables. Lower case letters from the end of the alphabet (. . . x, y, z) are used to denote variables. Variables represent arbitrary elements of an underlying set. This, in fact, is what “first-order” refers to. Variables that represent sets of elements are called second-order. Second-order logic, discussed in Chapter 9, is distinguished by the inclusion of such variables. • Constants. Lower case letters from the beginning of the alphabet (a, b, c, . . .) are usually used to denote constants. A constant represents a specific element of an underlying set. • Functions. The lower case letters f, g, and h are commonly used to denote functions. The arguments may be parenthetically listed following the function symbol as f(x1, x2, . . . , xn). First-order logic has symbols for functions of any number of variables. If f is a function of one, two, or three variables, then it is called unary, binary, or ternary, respectively. In general, a function of n variables is called n-ary and n is referred to as the arity of the function. • Relations. Capital letters, especially P, Q, R, and S, are used to denote relations. As with functions, each relation has an associated arity. We have an infinite number of each of these four types of symbols at our disposal. Since there are only finitely many letters, subscripts are used to accomplish this infinitude. For example, x1, x2, x3, . . . are often used to denote variables. Of course, we can use any symbol we want in first-order logic. Ascribing the letters of the alphabet in the above manner is a convenient convention. If you turn to a random page in this book and see “R(a, x, y),” you can safely assume that R is a ternary relation, x and y are variables, and a is a constant.

SKEPTICAL ABDUCTION

1993 ◽  
Vol 02 (04) ◽  
pp. 511-540 ◽  
Author(s):  
P. MARQUIS
Keyword(s):  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Real Problem ◽  
Logical Framework ◽  
First Order ◽  
Selective Generation ◽  
The Given ◽  
Reasonable Prospect

Abduction is the process of generating the best explanation as to why a fact is observed given what is already known. A real problem in this area is the selective generation of hypotheses that have some reasonable prospect of being valid. In this paper, we propose the notion of skeptical abduction as a model to face this problem. Intuitively, the hypotheses pointed out by skeptical abduction are all the explanations that are consistent with the given knowledge and that are minimal, i.e. not unnecessarily general. Our contribution is twofold. First, we present a formal characterization of skeptical abduction in a logical framework. On this ground, we address the problem of mechanizing skeptical abduction. A new method to compute minimal and consistent hypotheses in propositional logic is put forward. The extent to which skeptical abduction can be mechanized in first—order logic is also investigated. In particular, two classes of first-order formulas in which skeptical abduction is effective are provided. As an illustration, we finally sketch how our notion of skeptical abduction applies as a theoretical tool to some artificial intelligence problems (e.g. diagnosis, machine learning).

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