By Milton Fisk
This ebook used to be written for the newbie. It is now not, in spite of 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 uncomplicated issues with which it bargains. In truth, it was
written in the conviction that one other view of those issues is extra adequate.
The fee paid for this shift is the ensuing lack of easy mechanical tests
for the correctness of rules of the common sense of sentences and of monadic
predicates. The deductive procedure of facts resorted to for constructing these
principles demanding situations ingenuity with no being so tricky as to discourage it.
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Additional info for A Modern Formal Logic
Thus 'p or q or r or s' becomes ' ( (p U q) U r) U s'. EXERCISES 1. Put these abbreviated arguments into standard form relative to S: (a) If A or B then C. If D then A, and if E then B. Not-C. :. Not both D and E, (b) A only if unless B not both C and D. Although E, A. :. B provided that C and D, (c) It is the case both that A and B and that C. :. A and both B and C, (d) A and Band C. If A and B then either C or else D or E. Neither C nor E. :. D. 2. Rewrite as unabbreviated and, as far as possible, idiomatic English sentences: (a) A U (B U C), (d) (A·B) ~ (C~ (D U E)), (b) A~(B~C), (e) ((A~B) U (C~D)) U E, (c) A U (B · C), (f) (A ~ B) ~ C.
If Smith moves then he either runs or walks. So it is not the case that Smith either runs or walks. is due solely to logical form, for no argument exemplifying 'p. if p then either q or r. so not either q or r' is valid. The conclusion of any such argument represents a denial of what is implied by the premisses. Since its invalidity is due solely to logical form, (4) is formally invalid. As a materially invalid argument we have: (5) Peter is the father of David. So David is a parent of Peter. This belongs to the invalid type describable by 'xis the father of y.
So q ,vill have a false premiss. Hence no argument of this type will have a truefalse co1nbination. Now propositions exemplifying 'both p and not-p' are called contradictions. When we arrive at a contradiction from our premisses ,ve generally stop and revamp our premisses; we do not set out from a contradiction to further conclusions. Nonetheless, if for the sheer excitement of seeing what further conclusions are contained in the original, inconsistent premisses, one were to argue: (2) This class contains itself as a member and it does not contain itself as a member.