Charles E. Caton. A stipulation of logical truth in a modal propositional calculus. Synthese, vol. 14 (1962), pp. 196–199. - Charles E. Caton. A stipulation of a modal propositioned calculus in terms of modalized truth-values. Notre Dame journal of formal logic, vol. 4 no. 3 (1963), pp. 224–226.

1974 ◽  
Vol 39 (3) ◽  
pp. 611-611 ◽  
Author(s):  
Gerald J. Massey
Keyword(s):  
Propositional Calculus ◽  
Logical Truth ◽  
Formal Logic ◽  
Truth Values

Toward a calculus of concepts

1936 ◽  
Vol 1 (1) ◽  
pp. 2-25 ◽  
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.

scholarly journals A logic for approximate reasoning

1994 ◽  
Vol 59 (3) ◽  
pp. 830-837 ◽  
Author(s):  
Mingsheng Ying
Keyword(s):  
Artificial Intelligence ◽  
Set Theory ◽  
Theorem Proving ◽  
Intuitionistic Fuzzy Set ◽  
Propositional Calculus ◽  
Forthcoming Paper ◽  
Approximate Reasoning ◽  
Approximate Methods ◽  
Truth Values ◽  
Antecedent Clause

Classical logic is not adequate to face the essential vagueness of human reasoning, which is approximate rather than precise in nature. The logical treatment of the concepts of vagueness and approximation is of increasing importance in artificial intelligence and related research. Consequently, many logicians have proposed different systems of many-valued logic as a formalization of approximate reasoning (see, for example, Goguen [G], Gerla and Tortora [GT], Novak [No], Pavelka [P], and Takeuti and Titani [TT]). As far as we know, all the proposals are obtained by extending the range of truth values of propositions. In these logical systems reasoning is still exact and to make a conclusion the antecedent clause of its rule must match its premise exactly. In addition. Wang [W] pointed out: “If we compare calculation with proving,... Procedures of calculation... can be made so by fairly well-developed methods of approximation; whereas... we do not have a clear conception of approximate methods in theorem proving.... The concept of approximate proofs, though undeniably of another kind than approximations in numerical calculations, is not incapable of more exact formulation in terms of, say, sketches of and gradual improvements toward a correct proof” (see pp, 224–225). As far as the author is aware, however, no attempts have been made to give a conception of approximate methods in theorem proving.The purpose of this paper is. unlike all the previous proposals, to develop a propositional calculus, a predicate calculus in which the truth values of propositions are still true or false exactly and in which the reasoning may be approximate and allow the antecedent clause of a rule to match its premise only approximately. In a forthcoming paper we shall establish set theory, based on the logic introduced here, in which there are ∣L∣ binary predicates ∈λ, λ ∈ L such that R(∈, ∈λ) = λ where ∈ stands for ∈1 and 1 is the greatest element in L, and x ∈λy is interpreted as that x belongs to y in the degree of λ, and relate it to intuitionistic fuzzy set theory of Takeuti and Titani [TT] and intuitionistic modal set theory of Lano [L]. In another forthcoming paper we shall introduce the resolution principle under approximate match and illustrate its applications in production systems of artificial intelligence.

On an algebra of lattice-valued logic

2005 ◽  
Vol 70 (1) ◽  
pp. 282-318
Author(s):  
Lars Hansen
Keyword(s):  
Boolean Algebra ◽  
Complete Lattice ◽  
Algebraic Theory ◽  
Propositional Calculus ◽  
Boolean Lattice ◽  
Finite Chain ◽  
Axiomatic Theory ◽  
Truth Values ◽  
Algebraic Generalization

AbstractThe purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.

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

Alternative postulate sets for Lewis's S5

1956 ◽  
Vol 21 (4) ◽  
pp. 347-349
Author(s):  
E. J. Lemmon
Keyword(s):  
Propositional Calculus ◽  
Formal Logic ◽  
Propositional Formula ◽  
Modal Operator ◽  
Special Rules ◽  
Definition Of ◽  
Derived Rules

Professor Prior (Formal logic (1955), pp. 305–307) gives alternative postulate sets for Lewis's S5, one substantially Lewis's own set (p. 305, 6.1) and another, considerably simpler, due to Prior himself (pp. 306–307, 6.5). This note aims at showing that the systems based on these two postulate sets are in fact equivalent. We label the system based on Lewis's postulates ‘S5’, as usual, and that on Prior's postulates ‘P’.The form of the postulate set for P considered here is as follows. We add to the full propositional calculus one definition:Df. M: M = NLN;and two special rules:L1: ⊦Cαβ → ⊦CLαβ;L2: ⊦Cαβ → ⊦CαLβ, if α is fully modalized (a propositional formula is fully modalized, if and only if all occurrences of propositional variables in it lie within the scope of a modal operator).Prior's text (pp. 201–205) indicates sufficiently how it could be shown that any thesis of S5 is a thesis of P and that the rules and definitions of S5 are obtainable as derived rules of P. We turn, therefore, to the problem of showing that any thesis of P is a thesis of S5 and that the rules and definition of P are obtainable as derived rules of S5.

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.

A stipulation of logical truth in a modal propositional calculus

Synthese ◽  
1962 ◽  
Vol 14 (2-3) ◽  
pp. 196-199 ◽  
Author(s):  
Charles E. Caton
Keyword(s):  
Propositional Calculus ◽  
Logical Truth
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