On the first-order logic of terms

1973 ◽  
Vol 38 (2) ◽  
pp. 177-188
Author(s):  
Lars Svenonius
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Identity Function ◽  
First Order ◽  
Elementary Condition ◽  
Combinatorial Function ◽  
The One ◽  
Definition Of ◽  
Image Position

By an elementary condition in the variablesx1, …, xn, we mean a conjunction of the form x1 ≤ i < j ≤ naij where each aij is one of the formulas xi = xj or xi ≠ xj. (We should add that the formula x1 = x1 should be regarded as an elementary condition in the one variable x1.)Clearly, according to this definition, some elementary conditions are inconsistent, some are consistent. For instance (in the variables x1, x2, x3) the conjunction x1 = x2 & x1 = x3 & x2 ≠ x3 is inconsistent.By an elementary combinatorial function (ex. function) we mean any function which can be given a definition of the formwhere E1(x1, …, xn), …, Ek(x1, …, xn) is an enumeration of all consistent elementary conditions in x1, …, xn, and all the numbers d1, …, dk are among 1, …, n.Examples. (1) The identity function is the only 1-ary e.c. function.(2) A useful 3-ary e.c. function will be called J. The definition is

On the semantics of the Henkin quantifier

1979 ◽  
Vol 44 (2) ◽  
pp. 184-200 ◽  
Author(s):  
Michał Krynicki ◽  
Alistair H. Lachlan
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Original Definition ◽  
Definition Of ◽  
Image Position

In [5] Henkin defined a quantifier, which we shall denote by QH: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of QH wasHere (QHx1x2y1y2)φ is true if for every x1 there exists y1 such that for every x2 there exists y2, whose choice depends only on x2 not on x1 and y1 such that φ(x14, x2, y1, y2). Another way of writing this isIn [5] it was observed that the logic L(QH) obtained by adjoining QH defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier “there exist infinitely many” denoted Q0 was definable from QH because

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 $.

Church’s theorem and the decision problem

2018 ◽  
Author(s):  
Rohit Parikh
Keyword(s):  
Decision Problem ◽  
Combinatorial Problem ◽  
Order Logic ◽  
First Order Logic ◽  
Negative Solution ◽  
Great Contribution ◽  
Solvable Problem ◽  
First Order ◽  
Mathematical Definition ◽  
Definition Of

Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which formulas of first-order logic are valid. Church’s paper exhibited an undecidable combinatorial problem P and showed that P was representable in first-order logic. If first-order logic were decidable, P would also be decidable. Since P is undecidable, first-order logic must also be undecidable. Church’s theorem is a negative solution to the decision problem (Entscheidungsproblem), the problem of finding a method for deciding whether a given formula of first-order logic is valid, or satisfiable, or neither. The great contribution of Church (and, independently, Turing) was not merely to prove that there is no method but also to propose a mathematical definition of the notion of ‘effectively solvable problem’, that is, a problem solvable by means of a method or algorithm.

scholarly journals Semantics for Hyperclassical Logic and the Problem of Negation in the Formal Language

2018 ◽  
Vol 16 (3) ◽  
pp. 5-15
Author(s):  
V. V. Tselishchev
Keyword(s):  
Theoretical Model ◽  
Formal Language ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Order Language ◽  
Winning Strategies ◽  
Game Theoretic ◽  
Definition Of ◽  
Game Theoretic Semantics

The application of game-theoretic semantics for first-order logic is based on a certain kind of semantic assumptions, directly related to the asymmetry of the definition of truth and lies as the winning strategies of the Verifier (Abelard) and the Counterfeiter (Eloise). This asymmetry becomes apparent when applying GTS to IFL. The legitimacy of applying GTS when it is transferred to IFL is based on the adequacy of GTS for FOL. But this circumstance is not a reason to believe that one can hope for the same adequacy in the case of IFL. Then the question arises if GTS is a natural semantics for IFL. Apparently, the intuitive understanding of negation in natural language can be explicated in formal languages in various ways, and the result of an incomplete grasp of the concept in these languages can be considered a certain kind of anomalies, in view of the apparent simplicity of the explicated concept. Comparison of the theoretical-model and game theoretic semantics in application to two kinds of language – the first-order language and friendly-independent logic – allows to discover the causes of the anomaly and outline ways to overcome it.

A probabilistic interpolation theorem

1985 ◽  
Vol 50 (3) ◽  
pp. 708-713 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Probability Logic ◽  
Admissible Set ◽  
First Order ◽  
Probability Spaces ◽  
Image Position ◽  
Dedekind Cuts ◽  
Exact Definition

The probability logic is a logic with a natural interpretation on probability spaces (thus, a logic whose model theory is part of probability theory rather than a system for putting probabilities on formulas of first order logic). Its exact definition and basic development are contained in the paper [3] of H. J. Keisler and the papers [1] and [2] of the author. Building on work in [2], we prove in this paper the following probabilistic interpolation theorem for .Let L be a countable relational language, and let A be a countable admissible set with ω ∈ A (in this paper some probabilistic notation will be used, but ω will always mean the least infinite ordinal). is the admissible fragment of corresponding to A. We will assume that L is a countable set in A, as is usual in practice, though all that is in fact needed for our proof is that L be a set in A which is wellordered in A.Theorem. Let ϕ(x) and ψ(x) be formulas of LAP such thatwhere ε ∈ [0, 1) is a real in A (reals may be defined in the usual way as Dedekind cuts in the rationals). Then for any real d > ε¼, there is a formula θ(x) of (L(ϕ) ∩ L(ψ))AP such thatand

The foundations of Suslin logic

1975 ◽  
Vol 40 (4) ◽  
pp. 567-575 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Finite Sequence ◽  
Order Logic ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Finite Sequences ◽  
Image Position

Let L be a first order logic and the infinitary logic (as described in [K, p. 6] over L. Suslin logic is obtained from by adjoining new propositional operators and . Let f range over elements of ωω and n range over elements of ω. Seq is the set of all finite sequences of elements of ω. If θ: Seq → is a mapping into formulas of then and are formulas of LA. If is a structure in which we can interpret and h is an -assignment then we extend the notion of satisfaction from to by definingwhere f ∣ n is the finite sequence consisting of the first n values of f. We assume that has ω symbols for relations, functions, constants, and ω1 variables. θ is valid if θ ⊧ [h] for every h and is valid if -valid for every . We address ourselves to the problem of finding syntactical rules (or nearly so) which characterize validity .

Cognitive Modelling Applied to Aspects of Schizophrenia and Autonomic Computing

2011 ◽  
pp. 220-234
Author(s):  
Lee Flax
Keyword(s):  
Autonomic Computing ◽  
Signs And Symptoms ◽  
Order Logic ◽  
First Order Logic ◽  
Object Language ◽  
Cognitive Modelling ◽  
Belief Contraction ◽  
Algebraic Version ◽  
First Order ◽  
The One

We give an approach to cognitive modelling, which allows for richer expression than the one based simply on the firing of sets of neurons. The object language of the approach is first-order logic augmented by operations of an algebra, PSEN. Some operations useful for this kind of modelling are postulated: combination, comparison, and inhibition of sets of sentences. Inhibition is realised using an algebraic version of AGM belief contraction (Gärdenfors, 1988). It is shown how these operations can be realised using PSEN. Algebraic modelling using PSEN is used to give an account of an explanation of some signs and symptoms of schizophrenia due to Frith (1992) as well as a proposal for the cognitive basis of autonomic computing. A brief discussion of the computability of the operations of PSEN is also given.

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