Start with the premises P1 = “All men are mortal” and P2 = “Socrates is a man” and from them we want to know the status of the conclusion C = “Socrates is mortal.”
Well, we all know this:
(1) Pr( C | P1 & P2 ) = 1.
Because, of course, C is validly inferred from these premises. Now let’s think of a truth; any will do. How about T1 = “All bachelors are unmarried”? If you don’t love that one, substitute in another; again, any will do. Then let’s append this truth to our list of premises:
(2) Pr( C | P1 & P2 & T1) = 1.
It is a fact (a truth) of logic that adding a truth to a list of premises (or to the conclusion) does not change the status of a complete argument1. This works in math, too, and for the same reason. There is, further, no controversy about this. Just as there is no controversy that adding not just one, but any number of truths, to a list of premises does not change the status of the argument. So:
(3) Pr( C | P1 & P2 & T1 & T2) = 1,
where T2 is another truth.
A (logical, not grammatical2) tautology is a truth; it is a statement, perhaps about contingent events,
which is true no matter what; true “in all possible worlds” as some philosophers might say. Examples: T2 = “Tomorrow it will rain or not rain”, T3 = “You either have lung cancer or you don’t”, T4 = “The coin will land heads or it won’t.” No matter what happens, it will be true that tomorrow (anywhere, anytime) it will rain or it won’t. And it is perfectly true that you (and everybody) has lung cancer or not. I could have substituted in “bladder cancer” or “ebola sickness” or any other malady and the tautology would still be just as true.
I like Stanford’s Encyclopedia of Philosophy’s words about tautologies: “A tautology is a truth, and as certain as anything can be, but it is never the answer to any interesting inquiry outside mathematics and logic.”
Tautologies are not interesting because they are truths like those we appended to our first argument. If we swap in T2 in (2) nothing happens because this tautology is a truth and adding a truth to a list of premises does not change status of an argument.
Rain or its absence can have something to say about the mortality of men, but the truth that it will rain or it won’t has none. We could have substituted any truth, or any tautology and (2) and (3) would remain the same.
Let’s swap P1 for P3 = “Most men are mortal”. Eq. (2) becomes
(4) 0.5 ≤ Pr( C | P3 & P2 & T) < 1.
Again, any truth will do, and we infer from the English meaning of “most” that “at least half” of men are mortal.
Let’s change the argument entirely. You want to know the probability that C = “I have bladder cancer” is true. Well, what premises do you offer as probative? How about (a modified) T3 = “You have bladder cancer or you don’t.” Then this is the best we can do:
(5) 0 < Pr( C | T3 ) < 1.
In particular, the probability of C given T3 is not 1/2. Many (oh, but many) people have thought it was.
Actually, to get (5) we must supply a premise which is hidden in the words of C. If we did not supply this hidden premise, (5) would technically be “unknown” and that would have been the best we could do. The hidden premise is evidence that C is contingent; that is, it is not (logically) necessarily true that you have cancer or that you do not have cancer. But we do know that having cancer is contingent. So it would be better to un-hide this as E = “Having cancer is contingent” and re-write the result:
(6) 0 < Pr( C | E & T3 ) < 1.
The lesson is: never let just the words or notation used fool you into thinking that that is all there is to a problem. Now, because any truth can be substituted, we could re-write (6) as
(7) 0 < Pr( C | E & T1 ) < 1,
where now it should be obvious that the probability is not 1/2.
One last example. Let C = “The coin will land heads” and ask:
(8) Pr( C | T4 ) = ?
where recall that T4 = “The coin will land heads or it won’t.” This probability is not 1/2 and for the same reasons as last time. But we can also see that we have forgotten to un-hide our evidence that a coin flip is contingent. This leads us to:
(9) 0 < Pr( C | E & T4 ) < 1.
If you are unhappy about this and insist that (9) is 1/2, then I invite you to swap in another truth for T4, such as “All bachelors are unmarried.” Some will still be left with the feeling that (9) just is 1/2, but that will be because they are mentally supplying a premise which is not given; a premise like, “There is a coin, just one side of which is labeled ‘head’, which will be thrown once and which can land ‘head’ or ‘not head’ (‘tail’).” That premise (which also contains E) does give 1/2.
To see the difference between implied premises, I’ll leave you as homework the problem of a die throw. C = “A six will show” and let T5 = “A six will show on a die or it won’t.” Why do you feel the probability of C is 1/6 and not 1/2? What premises are you using?
1Of course, irreplaceable truths begin many arguments (especially in math), but once the argument is complete—has all its premises and conclusions—adding a truth changes nothing. (2) is easily proved: have a go.
2A grammatical tautology is a needless repetition of words such as (as from Rumpole case) “enthroning coronation”, all brought to you by the Department of Redundancy Department.