By Nick Earle (auth.)
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Additional info for Logic
Therefore it is not the case both that Cis B's father and that A is married to C. 20 'If p then q else r' is defined as (p' V q) /\ (p V r). Using n for a contradiction, (s /\ s'), and u for a tautology, (s V s'), state which of the following are tautologies: (a) 'if p then q else u' =' (q =? 6 Axioms This section may be omitted at first reading without loss of continuity. 16) of the principle which states that any proposition can be expressed in full conjunctive normal form you may have felt a little uneasy; in fact you ought to have done so!
They are indeed no more than an arrangement of symbols on a piece of paper and as such have no more meaning than we choose to give them. So, if we can show that for some meaning of the word 'true' the substitution for p, q and r of any propositions whatever yields axioms that are 'true', and if, further, we can show that any manipulation of 'true' axioms according to the rules (of substitution and inference) yields only 'true' assertions, ·then the possibility of any 'false' assertion being brought about by such manipulation is excluded at the outset; and the axioms are said to be consistent.
Why not use a simple proposition, say x E Y for this town has a tavern and express the quantified proposition by (x)(x E Y)? The answer is that in this case x would not have any precise significance, even as a place-keeper. , it is a genuine dummy, then the proposition (x)(x E Y) means everything is a town, which though not exactly nonsense is not the sense intended. If on the other hand, x stands for that town and nothing else it has ceased to be a place-keeper at all. The overwhelming advantage of the symbolism here adopted is that although the significance of x can range freely under the quantifier (since it is assumed that its place can be taken by any subject at am yet it has a precise and precisely the same meaning in each part of any proposition which it is used to form.