By Paul Teller
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Extra info for A modern formal logic primer: sentence logic
In practice, to show that a formula is a tautology, or that two formulas are logically equivalent, we have several methods available. First of all, we could use truth tables, but this is no longer viable once the number of variables exceeds 3 or 4. In certain cases, we could have recourse to what might be called 'economical truth tables': this consists in discussing the values taken by a restricted number of variables; in a way, we are treating several lines of the truth table in a single step.
Let V be an initial segment of H. 8). • So we see that in all cases, o[V] 2: el V]. first symbol is an opening parenthesis and for every word W E W(A) which is a proper initial segment of F, we have (strict inequality) o[W] > e[W]. Proof For once, the proof is not by induction! Consider a formula F which can be written as F = (G a H) where G and Hare arbitrary formulas and a is a symbol for a binary connective. Let W be a proper initial segment of F. There are two possible cases. 9), which leads once again to o[W] > e[W].
Because, although it is doubtful that anyone would be surprised that a formula F will receive the value I if and only if the formula -oF receives the value 0, the decision to attribute the value I to the formula (F :=} G) when the formulas F and G each have the value 0 will perhaps give rise to more uneasiness (at least at first sight). One way to dissipate this uneasiness is to ask ourselves under what circumstances the formula (F :=} G) could be considered false: we would probably agree that this would only happen in the case where F were true without G being true, which leads us to attribute the value I to (F :=} G) in the other three possible cases.
A modern formal logic primer: sentence logic by Paul Teller