In Defence of Classical Principles
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.
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.
KeywordsFormal Semantic Logical System Logical Theory Logical Truth Logical Constant
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