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

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

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 .

Beth’s theorem and Craig’s theorem

2018 ◽  
Author(s):  
Zeno Swijtink
Keyword(s):  
Sufficient Conditions ◽  
Predicate Symbol ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Infinite Length ◽  
First Order ◽  
Explicit Definition ◽  
Central Result ◽  
Definition Of

Beth’s theorem is a central result about definability of non-logical symbols in classical first-order theories. It states that a symbol P is implicitly defined by a theory T if and only if an explicit definition of P in terms of some other expressions of the theory T can be deduced from the theory T. Intuitively, the symbol P is implicitly defined by T if, given the extension of these other symbols, T fixes the extension of the symbol P uniquely. In a precise statement of Beth’s theorem this will be replaced by a condition on the models of T. An explicit definition of a predicate symbol states necessary and sufficient conditions: for example, if P is a one-place predicate symbol, an explicit definition is a sentence of the form (x) (Px ≡φ(x)), where φ(x) is a formula with free variable x in which P does not occur. Thus, Beth’s theorem says something about the expressive power of first-order logic: there is a balance between the syntax (the deducibility of an explicit definition) and the semantics (across models of T the extension of P is uniquely determined by the extension of other symbols). Beth’s definability theorem follows immediately from Craig’s interpolation theorem. For first-order logic with identity, Craig’s theorem says that if φ is deducible from ψ, there is an interpolant θ, a sentence whose non-logical symbols are common to φ and ψ, such that θ is deducible from ψ, while φ is deducible from θ. Craig’s theorem and Beth’s theorem also hold for a number of non-classical logics, such as intuitionistic first-order logic and classical second-order logic, but fail for other logics, such as logics with expressions of infinite length.

scholarly journals A FORMALIZATION OF KANT’S TRANSCENDENTAL LOGIC

2011 ◽  
Vol 4 (2) ◽  
pp. 254-289 ◽  
Author(s):  
T. ACHOURIOTI ◽  
M. VAN LAMBALGEN
Keyword(s):  
Formal Proof ◽  
Order Logic ◽  
First Order Logic ◽  
Pure Reason ◽  
Critique Of Pure Reason ◽  
Technical Application ◽  
First Order ◽  
Transcendental Logic ◽  
Definition Of ◽  
Argumentative Structure

Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kant’s logic negatively. What Kant called ‘general’ or ‘formal’ logic has been dismissed as a fairly arbitrary subsystem of first-order logic, and what he called ‘transcendental logic’ is considered to be not a logic at all: no syntax, no semantics, no definition of validity. Against this, we argue that Kant’s ‘transcendental logic’ is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kant’s Table of Judgements in Section 9 of the Critique of Pure Reason, is indeed, as Kant claimed, complete for the kind of semantics he had in mind. This result implies that Kant’s ‘general’ logic is after all a distinguished subsystem of first-order logic, namely what is known as geometric logic.

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