Download PDF by John Mueller Anderson, Henry W. Johnstone Jr.: Natural Deduction

By John Mueller Anderson, Henry W. Johnstone Jr.

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P V (q & r) Y r) Associative Laws (Assoc) (a) p &(q & r) : . (p&q) &r (b) (p &q) & r : . p &(q & r) (c) p V (q V r) : . °. ·. ,_,q -:::, ,_,p (b) ,_,q -:::, ,_,p :. '. "-Ip V q Law of Idempo tence (Id) (a) p :. p &p Law of Exporta tion (Exp) (p &q)-=> r : . p-=> (q-=> r) (b) p&p :. p (b) ,_,p V q : . p-:::, q (c) pvp :. '. p V p Law of Importa tion (Imp) p -=> (q-=> r) : . = I, we could summarize the statement of such rules as DM, CM, Dist, Assoc, Cp, Jmpl, Exp, and Imp. Consider, for example, (a) of DM: Let us discharge Jine I: -,- 1 .

_,(,_,N & ,_,T) 2. ,_,,_,N v ,_,,_,T 3. ,T 4. N vT I, DM 2, DN, Rep 3, DN, Rep Here, Rep stands for the Rule of Replacement. Another instance of the use of this rule (as a preliminary to stating it) may begin with a consideration of the sentence "It is not the case both that today is not Tuesday and that today is not Wednesday". We write (4) - - 1 . ,T & ,_,W) 2. ,_,,_,(T v W) 3. TvW I, DM, Rep 2, ON, Rep (or 2, ,_,,_,£) r- The second line is reached by noting that "-{p v q) - (,_,p & ,_,q) and replacing the instance of the right side in line I by its equivalent.

5. " Tom is tall and handsome" is a sentence. 6. "Tom" is a shorter word than is "Thomas". 7. " 'Tom is tall and handsome' is a sentence" is itself a sentence. 8. That "Tom" is shorter than "Ed" is false. 9. "'Tom' is shorter than 'Ed' "is false. "' "Tom" is shorter than "Ed" ' is false" is a sentence. 10. Tom is shorter than Ed. 11. "Tom is shorter than Ed" is true. 12. If "'Tom' is shorter than 'Ed'" is symbolized as A, then A is false. 13. (1) through (10) above are all false. 14. Statement ( l l) is false.

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Natural Deduction by John Mueller Anderson, Henry W. Johnstone Jr.

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