By W. H. Newton-Smith
An entire advent to good judgment for first-year collage scholars without history in good judgment, philosophy or arithmetic. In simply understood steps it exhibits the mechanics of the formal research of arguments.
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Extra resources for Logic: An Introductory Course
Assuming that the premises used (excepting A) are true and that the other rules used are truth-preserving, we can infer that A is true. For if A were false, A would be true and given the proof of B & B from A, both B and B would be true and this is impossible. This rule is used to derive the sequent P ( P& Q): Prem Prem 1 1,2 2 (1) (2) (3) (4) (5) P& Q P P P& P ( P& Q) 1 &E 2,3 &I 1,4 I As illustrated in the above derivation, in citing the rule of I, we give the line number of the premise which is negated in the application of the rule and the number of the line on which the derived contradiction occurs.
The conclusion will rest on whatever premises the formula A ↔ B rests on and in citing the rule we cite the line on which that formula occurs writing for the rule ↔E. The truth-table for the bi-conditional shows that individual applications of this rule to be truthpreserving. 6 (P v Q) & (P v R) P v (Q & R) Prem(1) (P v Q) & (P v R) 1 (2) (P v Q)1 &E Prem(5) Prem(3) P 1 (6) 3 (4) P v (Q 3 vI & R) Prem(7) 7 (8) Q (P v 1 &E R) P P v 7 vI (Q & R) Prem(9) R 5,9 (10)(Q & 5,9 R) &I 5,9 (11)P v 10 (Q & vI R) 1,5 (12)P v 6,7,8,9,11 (Q & vE Logic 46 R) 1 (13)P v (Q 2,3,4,5,12, & R) vE Bi-conditional Introduction (↔I) The bi-conditional introduction rule licenses us to derive a formula of the form A ↔ B from formulae of the form A → B and B → A.
Therefore, Icabod is a Balliol student. Interpretation P : Icabod is a Balliol student. Q : Icabod is clever. Formalization P → Q, Q P Circumstance surveyor P Q T T F F T F T F P → Q, T F T T Q T F T F P T T F F The argument is displayed to be invalid. For the circumstance surveyor gives a condition under which the premises are true and the conclusion false. Some invalid argument forms which are sometimes confusedly taken to be valid have been given names. This form is called the fallacy of affirming the consequent.