Axiom schemes for m-valued propositions calculi

1945 ◽  
Vol 10 (3) ◽  
pp. 61-82 ◽  
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.

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

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.

A formalization of an ℵ0-valued propositional calculus

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

Formalisations of further ℵ0-valued Łukasiewicz propositional calculi

1978 ◽  
Vol 43 (2) ◽  
pp. 207-210 ◽  
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Rational Numbers ◽  
Modus Ponens ◽  
Truth Values ◽  
Rules Of Procedure ◽  
C And N ◽  
Image Position ◽  
Propositional Calculi

It has been shown that, for all rational numbers r such that 0≤ r ≤ 1, the ℵ0-valued Łukasiewicz propositional calculus whose designated truth-values are those truth-values x such that r ≤ x ≤ 1 may be formalised completely by means of finitely many axiom schemes and primitive rules of procedure. We shall consider now the case where r is rational, 0≥r≤1 and the designated truth-values are those truth-values x such that r≤x≤1.We note that, in the subcase of the previous case where r = 1, a complete formalisation is given by the following four axiom schemes together with the rule of modus ponens (with respect to C),the functor A being defined in the usual way. The functors B, K, L will also be considered to be defined in the usual way. Let us consider now the functor Dαβ such that if P, Dαβ take the truth-values x, dαβ(x) respectively, α, β are relatively prime integers and r = α/β thenIt follows at once from a theorem of McNaughton that the functor Dαβ is definable in terms of C and N in an effective way. If r = 0 we make the definitionWe note first that if x ≤ α/β then dαβ(x)≤(β + 1)α/β − α = α/β. HenceLet us now define the functions dnαβ(x) (n = 0,1,…) bySinceit follows easily thatand thatThus, if x is designated, x − α/β > 0 and, if n > − log(x − α/β)/log(β + 1), then (β + 1)n(x−α/β) > 1.

Conditioned disjunction as a primitive connective for the erweiterter Aussagenkalkül

1953 ◽  
Vol 18 (1) ◽  
pp. 63-65
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Free Variable ◽  
Truth Table ◽  
Propositional Variable ◽  
Logical Constants ◽  
Truth Values ◽  
Complete Set ◽  
Truth Value

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.

On simple, weak and strong models of propositional calculi

1973 ◽  
Vol 74 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Ronald Harrop
Keyword(s):  
Simple Model ◽  
Propositional Calculus ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Structure ◽  
Propositional Calculi ◽  
Simple Models

In this paper we will be concerned primarily with weak, strong and simple models of a propositional calculus, simple models being structures of a certain type in which all provable formulae of the calculus are valid. It is shown that the finite model property defined in terms of simple models holds for all calculi. This leads to a new proof of the fact that there is no general effective method for testing, given a finite structure and a calculus, whether or not the structure is a simple model of the calculus.

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

1994 ◽  
Vol 02 (01) ◽  
pp. 37-54 ◽  
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.

The cylindric algebras of three-valued logic

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).

Contextualism, minimalism, and situationalism

2007 ◽  
Vol 15 (1) ◽  
pp. 115-137 ◽  
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.

A note on some intermediate propositional calculi

1984 ◽  
Vol 49 (2) ◽  
pp. 329-333 ◽  
Author(s):  
Branislav R. Boričić
Keyword(s):  
Positive Solution ◽  
Natural Deduction ◽  
Propositional Calculus ◽  
Conservative Extension ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Propositional Calculi

This note is written in reply to López-Escobar's paper [L-E] where a “sequence” of intermediate propositional systems NLCn (n ≥ 1) and corresponding implicative propositional systems NLICn (n ≥ 1) is given. We will show that the “sequence” NLCn contains three different systems only. These are the classical propositional calculus NLC1, Dummett's system NLC2 and the system NLC3. Accordingly (see [C], [Hs2], [Hs3], [B 1], [B2], [Hs4], [L-E]), the problem posed in the paper [L-E] can be formulated as follows: is NLC3a conservative extension of NLIC3? Having in mind investigations of intermediate propositional calculi that give more general results of this type (see V. I. Homič [H1], [H2], C. G. McKay [Mc], T. Hosoi [Hs 1]), in this note, using a result of Homič (Theorem 2, [H1]), we will give a positive solution to this problem.NLICnand NLCn. If X and Y are propositional logical systems, by X ⊆ Y we mean that the set of all provable formulas of X is included in that of Y. And X = Y means that X ⊆ Y and Y ⊆ X. A(P1/B1, …, Pn/Bn) is the formula (or the sequent) obtained from the formula (or the sequent) A by substituting simultaneously B1, …, Bn for the distinct propositional variables P1, …, Pn in A.Let Cn(n ≥ 1) be the string of the following sequents:Having in mind that the calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction (see [P]), the systems of natural deductions NLCn and NLICn (n ≥ 1), introduced in [L-E], can be identified with the calculi of sequents obtained by adding the sequents Cn as axioms to a sequential formulation of the Heyting propositional calculus and to a system of positive implication, respectively (see [C], [Ch], [K], [P]).

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