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The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested; the logician traditionally is not interested in the sentence as uttered but in the proposition, an idealised sentence suitable for logical manipulation.
The book later influenced A. J. Ayer's Language, Truth, and Logic, an introduction to logical positivism, and both the Richards–Ogden book and the Ayer book in turn influenced Alec King and Martin Ketley in the writing of their book The Control of Language, which appeared in 1939, and which influenced C. S. Lewis in the writing of his defence ...
The truth value of an arbitrary sentence is then defined inductively using the T-schema, which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, φ ∧ ψ is satisfied if and only if both φ and ψ are satisfied.
Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within a system of inference.
Meaning and Necessity: A Study in Semantics and Modal Logic (1947; enlarged edition 1956) is a book about semantics and modal logic by the philosopher Rudolf Carnap.The book, in which Carnap discusses the nature of linguistic expressions, was a continuation of his previous work in semantics in Introduction to Semantics (1942) and Formalization of Logic (1943).
Semantic-referential meaning refers to the aspect of meaning, which describes events in the world that are independent of the circumstance they are uttered in. An example would be propositions such as: "Santa Claus eats cookies." In this case, the proposition is describing that Santa Claus eats cookies. The meaning of the proposition does not ...
Example b. Many girls has a truth value of true iff there are many girls who are tall. This quantifier is satisfied with more than 1 instance of a girl being tall. Example c. Every girl has a truth value of true iff every girl is tall. This quantifier requires for all girls, that every instance of a person being female, she must be tall. Example d.
Indeed, modal logic was the basis of one of the most popular and rigorous formulations in modern semantics called the Montague grammar. The successes of such systems naturally give rise to the argument that these systems have captured the natural meaning of connectives like if-then far better than an ordinary, truth-functional logic ever could.