By Milton Fisk
This e-book was once written for the newbie. It is now not, despite the fact that, a survey of
modern formal good judgment, for it does now not try out to characterize the present opinion of
logicians on the easy concerns with which it offers. In truth, it was
written in the conviction that one other view of those issues is extra adequate.
The cost paid for this shift is the ensuing lack of uncomplicated mechanical tests
for the correctness of ideas of the common sense of sentences and of monadic
predicates. The deductive procedure of evidence resorted to for developing these
principles demanding situations ingenuity with out being so tricky as to discourage it.
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Extra info for A Modern Formal Logic
Sa. Since we shall be concerned exclusively with proof in a system, we shall henceforth mean proof in a system when we speak simply of proof. We proceed to set out in detail the conditions under which a sequence of formulas is a proof of a principle. Consider the sequence: /1/ P·q p~r p r We shall say that the formula 'p' forms together with the formula 'P · q' the principle 'p · q :. p', for writing 'p' to the right and 'p · q' to the left of a ' :. ' gives this principle. Likewise, 'r' forms together with 'p ~ r' and 'p' the principle 'p ~ r, p :.
So if not-p then not-q' rather than by the logical form 'if jx then gx. so if not1x then not gx'. Both fonns contain all of the argument's logical constants, but the former is simpler in that it contains fewer variables. The full type of (3) contains invalid arguments ; however, that of: (4) If Smith both runs and leaps then Smith both leaps and runs. So if Smith docs not both run and leap then Smith does not both leap and run contains only formally valid arguments and is describable by the logical form 'if both p and q then both q and p.
Thus we reduce the materially valid argument: (8) The tie is red. Hence the tie is colored to the formally valid one : (9) If the tie is red then the tie is colored. The tie is red. Hence the tie is colored. By a reduction we mean here the construction of a new argument by the addition to the old one of a conditional premiss whose antecedent contains a conjunction of the premisses and whose consequent contains the conclusion of the old argument. Reduction is then merely a mechanical routine. Because reduction is always possible, one might be tempted to think that materially valid arguments are not really valid.