Conditioned disjunction as a primitive connective for the erweiterter Aussagenkalkül

It has been shown that the conditioned disjunction function [X, Y, Z] with the same truth-table as (X & Y) ∨ (Z & ) together with the logical constants t and f, form a complete set of independent connectives for the 2-valued propositional calculus and that these connectives are self-dual. This has since been generalised to the theorem which states that the conditioned disjunction function [Y, X1, X2, …, Xm, Y] with the same truth-table as (X1 & J1(Y)) ∨ (X2 & J2(Y)) ∨ … ∨ (Xm & Jm(Y)) together with the logical constants 1, 2, …, m form a complete set of independent connectives for the m-valued propositional calculus and that these connectives are self-dual. It has been conjectured by Church that conditioned disjunction together with the universal and existential quantifiers form a complete set of independent connectives for the 2-valued erweiterter Aussagenkalkül. The object of the present paper is to prove a theorem for the m-valued erweiterter Aussagenkalkül which reduces, in the case m = 2, to the conjecture of Church. In the m-valued propositional calculus if the propositional variable X occurs as a free variable in the formula then (∃X) and (X) are read “there exists X such that ” and “for all X, ”, respectively. If for a given assignment of truth-values to the remaining free propositional variables occurring in , takes the truth-value f(x), where x is the truth-value of X, then (∃X) and (X) take the truth-values min (f(1), f(2), …, f(m)), max(f(1), f(2), …, f(m)), respectively. We shall prove:Theorem. The conditioned disjunction function, together with the universal and existential quantifiers, form a complete set of independent connectives for the m-valued erweiterter Aussagenkalkül.

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A formalization of an ℵ0-valued propositional calculus

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100028516 ◽

1953 ◽

Vol 49 (3) ◽

pp. 367-376

Author(s):

Alan Rose

Keyword(s):

Propositional Calculus ◽

Rational Numbers ◽

Truth Values ◽

Truth Value ◽

Image Position

In 1930 Łukasiewicz (3) developed an ℵ0-valued prepositional calculus with two primitives called implication and negation. The truth-values were all rational numbers satisfying 0 ≤ x ≤ 1, 1 being the designated truth-value. If the truth-values of P, Q, NP, CPQ are x, y, n(x), c(x, y) respectively, then

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An interpolation theorem in many-valued logic

Journal of Symbolic Logic ◽

10.1017/s0022481200031315 ◽

1986 ◽

Vol 51 (2) ◽

pp. 448-452 ◽

Cited By ~ 1

Author(s):

Masazumi Hanazawa ◽

Mitio Takano

Keyword(s):

Interpolation Theorem ◽

Propositional Variable ◽

List Type ◽

Logical Calculus ◽

Truth Values ◽

Truth Value

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ1,…, ΓM) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):(I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in A and B, and(ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.What shall be proved in this paper is the following (in representative form):(II) If a sequent ({A1}, {A2}, …, {AM}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A1,…, AM, and(ii) the following M sequents are valid:({A1},{D},…,{D}),({D},{A2},…,{D}),…,({D},{D},…,{AM}).Clearly the former can be obtained as a corollary of the latter.

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Axiom schemes for m-valued propositions calculi

Journal of Symbolic Logic ◽

10.2307/2267026 ◽

1945 ◽

Vol 10 (3) ◽

pp. 61-82 ◽

Cited By ~ 27

Author(s):

J. B. Rosser ◽

A. R. Turquette

Keyword(s):

Propositional Calculus ◽

Value Functions ◽

Truth Values ◽

Truth Value ◽

Propositional Calculi

In an m-valued propositional calculus, or a formalization of such a calculus, truth-value functions are allowed to take any truth-value t where 1 ≦ t ≦ m and m ≧ 2. In working with such calculi, or formalizations thereof, it has been decided to distinguish those truth-values which it is desirable for provable formulas to have from those which it is not desirable for provable formulas to have. The first class of truth-values is called designated and the second undesignated. This specification of certain of the m truth-values as designated and the remainder as undesignated is one of the distinguishing characteristics of m-valued propositional calculi, and it should be observed at the outset that two m-valued propositional calculi will be considered to differ even if they differ only in respect to the number of truth-values which are taken as designated.

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Promptness does not imply superlow cuppability

Journal of Symbolic Logic ◽

10.2178/jsl/1254748690 ◽

2009 ◽

Vol 74 (4) ◽

pp. 1264-1272 ◽

Cited By ~ 1

Author(s):

David Diamondstone

Keyword(s):

Explicit Construction ◽

Negative Answer ◽

ENRICHED INTERVAL BILATTICES AND PARTIAL MANY-VALUED LOGICS: AN APPROACH TO DEAL WITH GRADED TRUTH AND IMPRECISION

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488594000055 ◽

1994 ◽

Vol 02 (01) ◽

pp. 37-54 ◽

Cited By ~ 28

Author(s):

FRANCESC ESTEVA ◽

PERE GARCIA-CALVÉS ◽

LLUÍS GODO

Keyword(s):

Approximate Reasoning ◽

Truth Values ◽

Partial Models ◽

Truth Value ◽

The Many ◽

Available Information ◽

States Of Knowledge

Within the many-valued approach for approximate reasoning, the aim of this paper is two-fold. First, to extend truth-values lattices to cope with the imprecision due to possible incompleteness of the available information. This is done by considering two bilattices of truth-value intervals corresponding to the so-called weak and strong truth orderings. Based on the use of interval bilattices, the second aim is to introduce what we call partial many-valued logics. The (partial) models of such logics may assign intervals of truth-values to formulas, and so they stand for representations of incomplete states of knowledge. Finally, the relation between partial and complete semantical entailment is studied, and it is provedtheir equivalence for a family of formulas, including the so-called free well formed formulas.

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The cylindric algebras of three-valued logic

Journal of Symbolic Logic ◽

10.2307/2586647 ◽

1998 ◽

Vol 63 (4) ◽

pp. 1201-1217

Author(s):

Norman Feldman

Keyword(s):

Effective Procedure ◽

Cylindric Algebras ◽

Truth Values ◽

Recursive Functions ◽

Partial Recursive Functions ◽

Truth Value ◽

The One ◽

Definition Of

In this paper we consider the three-valued logic used by Kleene [6] in the theory of partial recursive functions. This logic has three truth values: true (T), false (F), and undefined (U). One interpretation of U is as follows: Suppose we have two partially recursive predicates P(x) and Q(x) and we want to know the truth value of P(x) ∧ Q(x) for a particular x0. If x0 is in the domain of definition of both P and Q, then P(x0) ∧ Q(x0) is true if both P(x0) and Q(x0) are true, and false otherwise. But what if x0 is not in the domain of definition of P, but is in the domain of definition of Q? There are several choices, but the one chosen by Kleene is that if Q(X0) is false, then P(x0) ∧ Q(x0) is also false and if Q(X0) is true, then P(x0) ∧ Q(X0) is undefined.What arises is the question about knowledge of whether or not x0 is in the domain of definition of P. Is there an effective procedure to determine this? If not, then we can interpret U as being unknown. If there is an effective procedure, then our decision for the truth value for P(x) ∧ Q(x) is based on the knowledge that is not in the domain of definition of P. In this case, U can be interpreted as undefined. In either case, we base our truth value of P(x) ∧ Q(x) on the truth value of Q(X0).

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Graphing True-False Statements

Mathematics Teacher ◽

10.5951/mt.62.7.0553 ◽

1969 ◽

Vol 62 (7) ◽

pp. 553-556

Author(s):

Margaret Wiscamb

Keyword(s):

Truth Table ◽

Graphic Representation ◽

Truth Values ◽

Truth Tables ◽

True False ◽

Tedious Process ◽

Three Components

In elementary logic the construction of truth tables, while not difficult, can be a long and tedious process. In this article I would like to present a simple graphic representation of the truth values of compound statements involving two or three components. The graph gives all the information found in a truth table and pictures the statement as an easily recognizable pattern. By using this graphing procedure, the simplifying of statements is shortened considerably. In fact, for statements involving only two components, with a little practice it can usually be done by inspection. Proving that a statement is a tautology becomes almost trivial.

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A Probabilistic Propositional Logic System is an Event Semantics for Classical Formal System of Propositional Calculus

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.427-429.1917 ◽

2013 ◽

Vol 427-429 ◽

pp. 1917-1923

Author(s):

Hong Lan Liu ◽

De Zheng Zhang

Keyword(s):

Formal System ◽

Propositional Calculus ◽

Semantic Interpretation ◽

Probabilistic Logic ◽

Value Functions ◽

Event Semantics ◽

Set Inclusion ◽

Set Operations ◽

Truth Value ◽

Logic System

The well formed formulas (wffs) in classical formal system of propositional calculus (CPC) are only some formal symbols, whose meanings are given by an interpretation. A probabilistic logic system, based on a probabilistic space, is an event semantics for CPC, in which set operations are the semantic interpretations for connectives, event functions are the semantic interpretations for wffs, the event (set) inclusion is the semantic interpretation for tautological implication, and the event equality = is the semantic interpretation for tautological equivalence. CPC is applicable to probabilistic propositions completely. Event calculus instead of truth value (probability) calculus can be performed in CPC because there arent truth value functions (operators) to interpret all connectives correctly.

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Contextualism, minimalism, and situationalism

Pragmatics & Cognition ◽

10.1075/pc.15.1.09cor ◽

2007 ◽

Vol 15 (1) ◽

pp. 115-137 ◽

Cited By ~ 12

Author(s):

Eros Corazza

Keyword(s):

Semantic Content ◽

Truth Values ◽

Context Dependent ◽

Truth Value ◽

True False

After discussing some difficulties that contextualism and minimalism face, this paper presents a new account of the linguistic exploitation of context, situationalism. Unlike the former accounts, situationalism captures the idea that the main intuitions underlying the debate concern not the identity of propositions expressed but rather how truth-values are situation-dependent. The truth-value of an utterance depends on the situation in which the proposition expressed is evaluated. Hence, like in minimalism, the proposition expressed can be truth-evaluable without being enriched or expanded. Along with contextualism, it is argued that an utterance’s truth-value is context dependent. But, unlike contextualism and minimalism, situationalism embraces a form of relativism in so far as it maintains that semantic content must be evaluated vis-à-vis a given situation and, therefore, that a proposition cannot be said to be true/false eternally.

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Toward a calculus of concepts

Journal of Symbolic Logic ◽

10.2307/2269323 ◽

1936 ◽

Vol 1 (1) ◽

pp. 2-25 ◽

Cited By ~ 8

Author(s):

W. V. Quine

Keyword(s):

Formal System ◽

Propositional Calculus ◽

Common Procedure ◽

Truth Values ◽

The Common

By concepts will be meant propositions (or truth-values), attributes (or classes), and relations of all degrees. The degree of a concept will be said to be 0, 1, or n (> 1), and the concept will be said to be medadic, monadic, or n-adic, according as the concept is a proposition, an attribute, or an n-adic relation. The common procedure in systematizing logistic is to treat these successive degrees as ultimately separate categories. The partition is not rested upon properties of the thus classified elements within the formal system, but is imposed rather at the metamathematical level, through stipulations as to what combinations of signs are to be accorded or denied meaning. Each function of the formal system is restricted, thus metamathematically, to one degree for its values and to one for each of its arguments. The theory of types imports a further scheme of infinite partition, imposed by metamathematical stipulations as to the relative types of admissible arguments of the several functions and stipulations as to the types of the values of the functions relative to the types of the arguments.The elaborateness of the metamathematical grillwork which thus underlies formal logistic accounts in part for the tendency of those interested in logistic less for the matters treated than for the structures exemplified to limit their attention to the propositional calculus and the Boolean calculus of attributes (or classes), which, taken separately, are independent of the partitioning. A second reason for the algebraic appeal of these departments is their freedom from bound (apparent) variables: for use of bound variables fuses systematic considerations with notational or metamathematical ones in a way which resists ordinary formulation in terms of fixed functions and their arguments. Freedom from bound variables may be regarded, indeed, as the feature distinguishing algebra from analysis.

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