Year: 1975
Type: article
Abstract: In this paper, we characterize the logical consequence relation as the minimal relation in a class of 'adequate 7 consequence relations for quantification theory.These adequate relations are easily seen to include the usual syntactic relations of deductive consequence and thus our results can, perhaps, be considered an abstract version of the completeness theorem.Our work is closely related to the methods of Hintikka [1] and Smullyan [2], [3].We adopt [3] as our standard reference and we assume the reader is acquainted with Smullyan's marvellous a, β, γ, 6 notation for sentences of quantification theory 1 as well as his Unifying Principle [3], p. 67.In what follows, Kt-X will stand for an abstract binary relation between finite sets of sentences K and single sentences I.We allow K = 0, the empty set, in which case we write, hi.Also if Y is a sentence, we will write K, Y \-X for K U {FJhl, etc.Definition We say that h is an adequate consequence relation (for quantification theory) if it satisfies the following conditions: I. K, A, ~A y-X, A atomic.II.IfK,~X hi, then Kt-X.III.//if, α^ hi or K, a 2 hi, then K, a hi.IV. // K, β 1 hi and K, β 2 hi, then K, β hi, V. // K, γ(a) HI, then K, γ HI.VI.IfK, δ(a) hi, then K, δ hi, if a is new to K, δ, and I.If ϋf f=I means I is true whenever all sentences in K are true, then 1.By a sentence of quantification theory we mean what Smullyan [3] calls an unsigned sentence, that is, one not preceded by a T or F.
Author: Robert H. Cowen
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2004813709