If q, and (if p, then q), then p.
That is saying that if q were the case, and if p entailed or implied q, then p would have to be the case too. That is, if q then p (in short-hand). In order for q to be the case, p would need to be the case whether or not q is actually the case.
However, we can split the compound
If q, and (if p, then q), then p.
It may well be the case that this implication is correct (or, in its shorter version, that p implies q) even if p (the antecedent) is false. That is, a false proposition can still imply things. It still has implications. After all, the compound formula begins with the locution “If q” and also contains “if p”. In other words, it's an implicational conditional. It tells us what is implied by what even if both p and q are false.
That could also mean that if the antecedent, p, is false, then the entire implication (or conditional) must also be false. That is, if p is false, even though it implies q, then the whole conditional implication must be false.... but wait. In addition, the entire conditional above may be false, and p is false, yet the implied q may still be true!
Clearly it's useful here to distinguish truth from correctness. That is,
If q, and (if p, then q), then p.
is a correct implicational conditional; though it's not a true one. It's false because p is false, and p implies q. Indeed the conditional is correct (the inference pattern is correct); thought false (one or more propositions are false) even if the consequent, q, is true (when taken on its own).
However, logicians also say that the conditional itself is true (as Arthur Pap does) and not simply correct. Here the notions of truth and correctness are fused. That means that that the overall conditional is deemed true, despite one of its propositions being false. It's the inference pattern that's true. Not necessarily one or all of its propositions.
Here's another tautology:
If not-q and (if p, then q), then not-p
It's not the case that q is the case. Therefore q is false. However, p implies q. Yet q, as we've said, isn't the case. Thus if p implies something that's not the case (q), then p implies q. Therefore p mustn't be the case either. That is, if p were true, it would imply q. Yet it has already been stated that q is false or not-q. Thus not-q itself implies not-p (though only in conjunction with the parenthesized conditional (i.e., “if p, then q”).
Inferences Involving Classes
Instead of using p's and q's (propositions) in logic, logicians sometimes think in terms of classes instead, as with the Aristotelian syllogism. Take this example:
i) For all classes AM, BS, and A:
ii) if all AM are BS
iii) and some A are AM
iv) then some A are BS.
i) If all African-Americans (AM) have black skin (BS)
ii) and some Americans (A) are African-Americans (AM)
iii) then some Americans (A) have black skin (BS).
However, don't make the mistake of illicit conversion. That is, from
All AM are BS
it doesn't follow that
All BS are AM.
It doesn't follow that because
All African-Americans (AM) have black skin (BS)
All those who have black skin (BS) are African-Americans (AM).
Clearly, there are many people who have black skin who aren't also American.
Illicit conversations can happen when logical matters are discussed as well.
For example, it's often said that
i) All tautologies are formal truths, hence logical truths.
However, that doesn't mean that
ii) All logical truths are tautologies.
The inversion of “All tautologies are logical/formal truths” is “All logical/formal truths are tautologies” is illegitimate for the simple reason that not all logical/formal truths are tautologies. There is more to logic than tautologies (as against the position advanced in Wittgenstein’s Tractatus).
It other words, it must be said that some logical truths, or inferences (as with mathematics), supply us with additional information (though not facts). Indeed with additional truths.
Here's another example of illicit conversion. Take the statement
All logical truths are true by sole virtue of the meanings of their constituent terms (in particular, their logical constants).
All statements that are true just by virtue of the meanings of the terms (hence, that don't require empirical verification and that can't be falsified) are logical truths.
For example, the well-known example of
All bachelors are unmarried.
This is indeed true because the interpretations of its constituent meanings make it true. Nonetheless, the words “bachelors” and “unmarried” are hardly logical in nature. And if they aren't logical in nature, then “All bachelors are unmarried” can't simply be true because of logic (or it can't solely be a logical truth).
Nonetheless, as Quine made very clear, “All bachelors are unmarried” can be made logical; or it can be translated into a logically true statement, simply by writing
All unmarried men are unmarried.
You don't need to understand what “married”, and therefore “unmarried”, means. All you need to understand are the strictly logical terms: “all” and “not” (or “un”). (Though “are” is not strictly speaking a logical word... then again, in a certain sense, neither is “all”.)
And just as it was said that not all logical, or formal, truths are tautologies, it can also be said that not all necessary, or a priori, truths are strictly analytic. (Of course analytic truths have been seen as tautological too; though this this was largely discounted earlier on.)
Basically, the argument is that some truths are necessary, or can be known a priori, because of the way the world is; or, on some accounts, because of the way all (human) minds work.
Take this example from Arthur Pap:
No event precedes itself.
This isn't true because of its “constituent meanings” and it's not a tautology (as such). However, it is about world; though it can also be said that one need not consult the world in order to know that it's true. Though surely if that's the case, then how can it be about the world at all? Perhaps the best that can be said that once one gets to know the world (as it were), and one gets to understand and use the words contained in the statement, then one knows that it's necessarily true and that it can be known a priori. That is, it requires previous experience of the world and of language. Yet once that experience and knowledge is in, you don't need to check to see that it's true – indeed that it's necessarily true.
Despite saying all that, the “meanings of the constituent words” can still be said to make it true. After all, once we understand the word “precede”, then it will become clear that an event cannot preceded itself or that x cannot precede x. However, meanings are still important (as they are to most logical truths). Nonetheless, as said earlier, the word “precede” isn't logical in nature.
Defining the Logical Constants
Wittgenstein once said that “some things can only be seen, not said”. This is true of the logical constants. When you define the logical constants, you do so by using other logical constants.
All things have property P.
That can be defined as:
Not-(some things don't have P).
Strictly speaking, “all things” is a use of the universal quantifier (which isn't a logical constant). Anyway, this universal quantifier too is defined by using the logical constant known as negation (or “not-” in this case).
p and q
This is a propositional conjunction which uses the logical constant “and”. It can be defined in this way:
Not-(not-p or not-q).
To say that both p and q are true is also to say that it's not the case that both p and q are false. Or that “not-p or not-q” is false because both p and q are true. So here a compound proposition which uses the logical constant “and” is defined in terms of the logical constant of negation ( or “not-” in this case).
Finally we have:
p or q
This is a compound proposition which uses the logical constant of disjunction (or “or” in this case). That in turn is defined in this way:
If not-p, then q
Here again the constant of negation is used to define another constant. That is, if either p or q is true (though not both), and p is false (or “not-p” in this case), then q must be true because the compound proposition (“p or q”) says that either p or q is true. So if p is false, then q must be true.