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 \(L_{\omega, \omega}\).

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.

An Omitting Types Theorem for first order logic with infinitary relation symbols

2007 ◽  
Vol 53 (6) ◽  
pp. 564-570 ◽  
Author(s):  
Tarek Sayed Ahmed ◽  
Basim Samir
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Omitting Types

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.

Filter Logics on ω

1984 ◽  
Vol 49 (1) ◽  
pp. 241-256
Author(s):  
Matt Kaufmann
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Basic Properties ◽  
Countable Compactness ◽  
Omitting Types

AbstractLogics LF(M) are considered, in which M (“most”) is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form {(⋂F) ∪ X: ∣ω − X∣ < ω} with ∣ω − ⋂F∣ = ω. Moreover the set of validities of LF (M) and even of depends only on a few basic properties of F. Similar characterizations are given of the class of filters F for which LF (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (, where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindström's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindström's argument is applied to characterize the invariant sentences as just those of first-order logic.

scholarly journals On the multi-dimensional modal logic of substitutions

2019 ◽  
Vol 56 (4) ◽  
pp. 454-481
Author(s):  
Tarek Sayed Ahmed ◽  
Mohammad Assem Mahmoud
Keyword(s):  
Amalgamation Property ◽  
Modal Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Semantics ◽  
Infinite Dimensions ◽  
Finite Axiomatizability ◽  
First Order ◽  
Abstract Entities ◽  
Omitting Types

Abstract We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

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.

Truth Definitions, Skolem Functions and Axiomatic Set Theory

1998 ◽  
Vol 4 (3) ◽  
pp. 303-337 ◽  
Author(s):  
Jaakko Hintikka
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Negative Results ◽  
Skolem Functions ◽  
Order Language ◽  
Definition Of ◽  
Axiomatic Set Theory

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski (1935). In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore (1994, p. 635) asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable.

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
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