By Ernst Cassirer, C.S. Howe
Translated by means of Clarence Smith Howe . eightvo pp. XX + 218 Brossura (wrappers) Molto Buono (Very solid)
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This construction produces a compound sentence from two constituent sentences by applying the particle 'if': 'if p, q'. The sense of this familiar construction is not always clear. 'If Flora were fairer than Amy, Flora would be fair indeed'; 'If Flora were fairer than Amy, Amy would be plain indeed'. Commonly the force of a conditional is indeterminate except by reference to the purposes of some broader context. The conditional also has its clear and selfcontained uses, but the services rendered by these uses can be rendered as well by negation, conjunction, and existential quantification.
The negation is satisfied by just the sequences that its constituent is not satisfied by, and the conjunction is satisfied by just the sequences that its constituents are both satisfied by. Now what of the remaining construction, existential quantification? An existential quantification consists of some sentence preceded by an existential quantifier whose variable is, say, the ith variable of the alphabet. 'fhis quantification, then, is satisfied by a given sequence if and only if the constituent sentence is satisfied by some sequence that matches the given one except perhaps in its ith place.
What one was last told, then, was what it means for a sequence to satisfy a sentence of complexity O. But then the satisfaction conditions for negation, conjunction, and existential quantification tell one what it means for a sequence to satisfy a sentence of next higher complexity, once one knows what it means for a sequence to satisfy a sentence of given complexity. So, step by step, one finds out what it means for a sequence to satisfy a sentence of any preassigned complexity. Complexity n, for each n, takes n such steps.