Hamid Alaeinejad; Morteza Hajhosseini
Abstract
According to the proof-theoretic definition of the concept of logical consequence, the sentence X is a logical consequence of the set of assumptions Γ if there is an argument ...
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According to the proof-theoretic definition of the concept of logical consequence, the sentence X is a logical consequence of the set of assumptions Γ if there is an argument from Γ for X. Tarski argues that by adding any number of rules to inferential systems, there are always cases in which a sentence is intuitively the logical consequence of a set of sentences, whereas in that system it is not possible to provide an argument for that sentence. Hence, the proof-theoretic definition cannot express the intuitive concept of logical consequence. The present article examines Tarski's critique of proof-theoretic definition. Based on our study, Tarski's critique poses a serious problem to the proof-theoretic definition, but this does not mean that the model-theoretic definition is superior to proof-theoretic definition, because there are also acceptable critiques of model-theoretic definition. It seems that neither of these approaches succeed in providing an accurate definition of the intuitive concept of logical consequence for deductive systems. However, in different deductive systems, an interpretation of each of these two approaches can be presented in a way that it is consistent with other metaphysical and epistemological perspectives related to that system, and it is acceptable according to the practical goals.
