Categorical abstract algebraic logic categorical algebraization of first-order logic without terms

2004 ◽  
Vol 44 (4) ◽  
pp. 473-491 ◽  
Author(s):  
George Voutsadakis
Keyword(s):  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Abstract Algebraic Logic ◽  
First Order

scholarly journals Omitting types in fragments and extensions of first order logic

2021 ◽  
Author(s):  
Tarek Sayed Ahmed
Keyword(s):  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Negative Results ◽  
Omitting Types

Fix \(2 < n < \omega\). Let \(L_n\) denote first order logic restricted to the first $n$ variables. Using the machinery of algebraic logic, positive and negative results on omitting types are obtained for \(L_n\) and for infinitary variants and extensions of \(L_{\omega, \omega}\).

Logic in the early 20th century

2018 ◽  
Author(s):  
Gregory H. Moore
Keyword(s):  
Twentieth Century ◽  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Axiomatic Method ◽  
Symbolic Logic ◽  
Modern Logic ◽  
Genetic Method ◽  
First Order ◽  
Modern Development

The creation of modern logic is one of the most stunning achievements of mathematics and philosophy in the twentieth century. Modern logic – sometimes called logistic, symbolic logic or mathematical logic – makes essential use of artificial symbolic languages. Since Aristotle, logic has been a part of philosophy. Around 1850 the mathematician Boole began the modern development of symbolic logic. During the twentieth century, logic continued in philosophy departments, but it began to be seriously investigated and taught in mathematics departments as well. The most important examples of the latter were, from 1905 on, Hilbert at Göttingen and then, during the 1920s, Church at Princeton. As the twentieth century began, there were several distinct logical traditions. Besides Aristotelian logic, there was an active tradition in algebraic logic initiated by Boole in the UK and continued by C.S. Peirce in the USA and Schröder in Germany. In Italy, Peano began in the Boolean tradition, but soon aimed higher: to express all major mathematical theorems in his symbolic logic. Finally, from 1879 to 1903, Frege consciously deviated from the Boolean tradition by creating a logic strong enough to construct the natural and real numbers. The Boole–Schröder tradition culminated in the work of Löwenheim (1915) and Skolem (1920) on the existence of a countable model for any first-order axiom system having a model. Meanwhile, in 1900, Russell was strongly influenced by Peano’s logical symbolism. Russell used this as the basis for his own logic of relations, which led to his logicism: pure mathematics is a part of logic. But his discovery of Russell’s paradox in 1901 required him to build a new basis for logic. This culminated in his masterwork, Principia Mathematica, written with Whitehead, which offered the theory of types as a solution. Hilbert came to logic from geometry, where models were used to prove consistency and independence results. He brought a strong concern with the axiomatic method and a rejection of the metaphysical goal of determining what numbers ‘really’ are. In his view, any objects that satisfied the axioms for numbers were numbers. He rejected the genetic method, favoured by Frege and Russell, which emphasized constructing numbers rather than giving axioms for them. In his 1917 lectures Hilbert was the first to introduce first-order logic as an explicit subsystem of all of logic (which, for him, was the theory of types) without the infinitely long formulas found in Löwenheim. In 1923 Skolem, directly influenced by Löwenheim, also abandoned those formulas, and argued that first-order logic is all of logic. Influenced by Hilbert and Ackermann (1928), Gödel proved the completeness theorem for first-order logic (1929) as well as incompleteness theorems for arithmetic in first-order and higher-order logics (1931). These results were the true beginning of modern logic.

Provability with Finitely Many Variables

2002 ◽  
Vol 8 (3) ◽  
pp. 348-379 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Predicate Logic ◽  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Alternative Semantics ◽  
Relation Symbol ◽  
First Order ◽  
Binary Predicate ◽  
Standard Set ◽  
Binary Relation Symbol

AbstractFor every finite n ≥ 4 there is a logically valid sentence φn with the following properties: φn contains only 3 variables (each of which occurs many times); φn contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φn has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φn has a proof with only n variables. To show that φn has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

Omitting types for finite variable fragments and complete representations of algebras

2008 ◽  
Vol 73 (1) ◽  
pp. 65-89 ◽  
Author(s):  
Hajnal Andréka ◽  
István Németi ◽  
Tarek Sayed Ahmed
Keyword(s):  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
First Order ◽  
Complete Representations ◽  
Atomic Relation ◽  
Representations Of Algebras ◽  
Omitting Types ◽  
Binary Relation Symbol

AbstractWe give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

scholarly journals GAMES AND CARDINALITIES IN INQUISITIVE FIRST-ORDER LOGIC

2021 ◽  
pp. 1-28
Author(s):  
IVANO CIARDELLI ◽  
GIANLUCA GRILLETTI
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order

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.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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