We now consider the logic associated with predicate wffs, including a new set of derivation rules for demonstrating validity (the analogue of tautology in the propositional calculus).
Caveat: t must not already appear as a variable in the expression for P(x): in the equation above, (1), it would not do to use P(y) or P(z), as they appear in the expression already.
Example: Practice 21, p. 47
Caveat: t must be introduced for the first time (so do these early in proofs). You can do a universal instantiation which also uses t after an existential instantiation with t, but not vice versa.
Caveats:
Example: Ex. #11, p. 57
Caveat: x must not appear in P(a).
Example: Ex. #3, p. 56.
as shown on page 51, and
as they suggest. This means that we can ``pass over'' predicates outside our own scope, or include them within our own scope. This is similar to what we do with summation notation, when, for example, we can write
Note also their proof method: they introduce a ``temporary hypothesis''. If you think about the deduction method, it takes a conclusion which is an implication and rewrites it so that the implication disappears (the antecedent becomes one of the hypotheses). Similarly, we can take an hypothesis (in this case, one which we introduce) and turn a conclusion into an implication. This is the deduction method backwards!
Look at the three proofs using a temporary hypothesis (Examples #30, and 31(a,b)). Notice how the introduction of the temporary hypothesis ends with an implication, which is then useful for the continuation of the proof.
Example: Practice 24, p. 47
Example: Ex. #13, p. 57 (part of your homework set) illustrates this well. Let's try that one together.