Abstract
In contemporary logic the presentation of a logical theory follows, in general, the line of thought given in (A) and (B) below:
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(A)
The language (grammar) of the system is described. First the primitive expressions of the language—among them logical constants, auxiliary signs (e.g. brackets), variables bindable by operators, and nonlogical (descriptive) constants—are enumerated, sometimes grouped according to their category (logical type), allowing for extra-categorical signs, too. Then rules are specified by which from certain expression(s) compound expressions can be generated. Generally in these rules it is explicitly stated to which category the input(s) and the output can belong. As a result the class of the well-formed expressions of the language subdivided according to the different categories is defined by an inductive syntactical definition. Among the syntactical categories of the language there is one which is analogous to the category of the indicative sentence in natural language, which is called ‘formula’, or ‘well-formed formula’, or simply ‘sentence’. A more general method is to define a family of languages instead of a single language, where the members of the family can differ only in the class of non-logical constants.
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(B)
The relation of consequence (i.e. the relation ‘A follows from Γ’ where A is a sentence and Γ is a set of sentences) and as a special case the notion of logical truth or validity is defined. The definition of consequence can be either syntactical or semantical. In the case of a syntactical definition we speak about deductibility (A is deductible from Γ), and the relation of deductibility is introduced by an inductive syntactical definition. In a semantical definition of consequence first the class of admissible interpretations is given (most often these interpretations are set-theoretical entities), and it is specified what semantic value the well-formed expressions of the language can take. The possible semantic values of sentences are normally called truth values. In the classical case there are two truth values, Truth and Falsity, in other cases it is postulated that among the truth values there is one or more which is a ‘distinguished’ value. Then the definition of ‘A is a semantic consequence of I (or ‘A follows semantically from Γ’) may be formulated as ‘A is true (A bears a distinguished value) in all interpretations where every member of Γ is true (bears a distinguished value)’. This definition may be strengthened by some additional clauses.
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References
R. Routley and R. K. Meyer, The semantics of entailment. In: H. Leblanc (ed.), Truth, Syntax and Modality, pp. 199–243, Amsterdam, 1973.
See, e.g. B. Russell, On denoting. In R. C. Marsh (ed.), Logic and Knowledge,London, 1956; and W. V. O. Quine, Methods of Logic,revised edition, 1963, sections 36–37.
Admitting truth value gaps does not conflict with the principle of tertium non datur—in the case of the ontologie approach. For, this principle holds for statements, not for sentences. If an indicative sentence is—due to the facts—without a truth value, it does not express a statement, and whenever it expresses—due to the facts, again—a statement, it must be either true or false.
See, e.g. A. Church, A formulation of the logic of sense and denotion. In Henle, Kallen and Langer (eds), Structure, Method and Meaning: Essays in Honor of H.F. Sheffer, New York, 1951, pp 3–24.
If F is false for some object, then obviously, `d(F)’ must be false. Concerning the truth condition of `b’(F)’,there might be different views. A fair condition: let F be true of all objects for which it is defined (assuming, of course, that it is defined for some objects).
See, e.g., I. Ruzsa, Intensional logic and semantic value gaps, Logique et Analyse,29, pp. 187206, 1986. A more detailed report on this subject is I. Ruzsa, intensional Logic Revisited,budabest 1991 (published by the author).
R. Montague, Universal Grammar, 1970, and The proper treatment of quantification in ordinary English, 1973. In: R. H. Thomason (ed), Formal Philosophy: Selected Papers of R. Montague, Yale Univ. Press, New Haven London, 1974.
That is: given an interpretation, an assignment of the variables, and an index where...
The notion of a week consequence might be the classical one: the truth of the premises must exclude the falsity of the conclusion (it is not excluded the conclusion without a truth value).
Never’, i.e. at no indices of any interpretation with no assignment of the variables.
Always’, i.e. at all indices of all interpretations and all assignments of the variables.
e., a formula involving no intensional functors.
Note the strong condition: the truth of A together with the non-truth (i.e. both the falsity and the truthvaluelessness) of B must be impossible.
However, ‘−B →’R −A’ follows from ‘A → R B’
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Ruzsa, I. (1996). In Defence of Classical Principles. In: Bystrov, P.I., Sadovsky, V.N. (eds) Philosophical Logic and Logical Philosophy. Synthese Library, vol 257. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8678-8_9
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