1. If [A] or , and [not-A], then .
This is also not a truth-functor. That is because of the relationship A has to B. I could write If [the Blue Jays won the series in 1992] or [3 Independent MP's were elected in the last Canadian federal election], and [the Blue Jays did not win in 1992], then [3 Independent MP's were elected in the last Canadian federal election]. The relationship between A, B simply does not follow that in the negation of A, B is a necessary consequent. The sentence is entirely dependent upon my choice of constituent sentences and where I choose sentences that have only 2 possibilities If [the Republican's took control of the Senate in 2006] or [the Democrats took control of the Senate in 2006], and [the Republicans did not take control of the Senate in 2006], then [the Democrats took control of the Senate in 2006]. Because of our knowledge we know it can either be the Democrats or the Republicans at this moment in time, but there is nothing in the sentence alone that justifies this. What if there was a third party as there is in Canada. What then? This cannot be a truth-functor.
Your justification for #1 seems convincing, but I remember the definition of a truth-functor as being a sentence-functor which has a full truth table.
[(A v B) & ~A] -> B
is a tautology. It seems to have a full truth table:
[(A v B) & ~A] -> B
T T T F FT T T
T T F F FT T F
F T T T TF T T
F F F F TF T F
The whole sentence is true regardless of the truth values of
I suppose it could be argued that if the sentence variables were replaced with declarative sentences, that the whole sentence would be true not because of the truth-values of those declarative sentences, but rather because of the truth-functional connectives of the tautology. But I still think it is a truth-functor. I could be wrong.
Also, since the whole sentence is a conditional (if/then) sentence, consider the logical force of material implication. '(A v B) & ~A' is the antecedent. The meaning of the sentence (and material implication) is IF
the antecedent condition were met, it would be sufficient for the consequent. You ask "What if there was a third party as there is in Canada. What then?"
. Well, that condition is not the antecedent condition that is contained in the antecedent. If you suppose some other antecedent and conclude that the consequent must be false, you're committing the fallacy of denying the antecedent.
Some background on sentence-functors and truth-functors:
A sentence-functor is defined to be a string of English words and sentence variables, such that if the sentence variables are replaced by declarative sentences, then the whole becomes a declarative sentence with the inserted sentences as constituents.
A sentence-functor which has a full (as opposed to a partial) truth-table is called a truth-functor. Some sentence-functors are not truth-functors because sometimes the truth-values of the constituent sentences are not enough by themselves to determine the truth-value of the whole. For example:
I know that [A].
has only a partial truth-table:
A | I know that [A]
T | ?
F | F
Suppose that [A] is the declarative sentence 'The sky is green'. Whatever the truth-value of 'The sky is green', such truth-value by itself is not enough to determine the truth-value of the whole sentence 'I know that the sky is green', because if 'The sky is green' is true, such truth has no impact on the truth value of knowing that the sky is green.