Logical consequence is the relation that holds between a set of sentences and a sentence when the latter "follows from" the former. A formally specified logical consequence relation may be characterized model-theoretically or proof-theoretically (or both). In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...
Logical consequence can also be expressed as a function from sets of sentences to sets of sentences (Tarski's preferred formulation), or as a relation between two sets of sentences (multiple-conclusion logic). Alfred Tarski (January 14, 1901 in Warsaw – October 26, 1983 in Berkeley, USA) was a Polish mathematician, and widely considered one of the four greatest logicians of all time, along with Aristotle, Gottlob Frege, and Kurt Gödel. ...
The concept of logicalconsequence is one of those whose introduction into a field of strict formal investigation was not a matter of arbitrary decision on the part of this or that investigator; in defining this concept efforts were made to adhere to the common usage of the language of everyday life.
Such constants are called logical constants, and we say that the logical form of a sentence is a function of the logical constants that occur in the sentence and the pattern of the remaining expressions.
The status of the deductive-theoretic approach to logic is not clear for, as Tarski argues in his (1936), deductive-theoretic accounts are unable to reflect the fact that, according to the common concept, logicalconsequence is not compact.
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