Carlo Cellucci

This article examines the current justifications of deductive inferences, and finds them wanting. It argues that this depends on the fact that all such justification take no account of the role deductive inferences play in knowledge. Alternatively, the article argues that a justification of deductive inferences may be given in terms of the fact that they are non-ampliative, in the sense that the content of the conclusion is merely a reformulation of the content of the premises. Some possible objections to this view are discussed and found inadequate.

The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable intimations concerning the nature and constitution of the human mind” [2, p. 1]. Logic is normative, not descriptive. For, the laws of thought do not “manifest their presence otherwise than by merely prescribing the conditions of formal inference” [2, p. 419]. They are, “properly speaking, the laws of right reasoning only” [2, p. 408]. So they “form but a part of the system of laws by which the actual processes of reasoning, whether right or wrong, are governed” [2, p. 409]. Boole’s idea that logic is the study of the laws of thought was taken over by Hilbert. According to him logic is “a discipline which expresses the structure of all our thought” [31, p..