The Socrates syllogism in modern parlance

Socrates is a man
All men are mortal
Hence, Socrates is mortal

Implied above is a basic statement of naive (non-axiomatic) set theory. The notion of Russell and Whitehead that one can dispense with classes when establishing the logical foundations of arithmetic escapes me, as the basic logic form of implication takes classes for granted. But then, I have not read Principia Mathematica in detail, and so I cannot make any dogmatic assertion.

In any case, here is how I would put the Socrates syllogism:

1. It is required that any x in A be paired with b in {b}. (Given rule)

2. x_o e A. (Given condition)

3. The ordered pair < x_o, b > is required. (By Rule 1)

4. I.e., < x_o, b > ∊ A X {b}. (Same as 3)

By substitution, A is the set of men; {b} is the set containing the property or attribute b, signifying "mortality;" x_o is an instantiation of A otherwise known as "Socrates."

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John Stuart Mill, in his A System of Logic, takes many pains to drive home the point that the syllogism proves nothing that is not already stated in the premise. Others have picked up the theme, arguing that formal logic does not really prove anything, but is a means of saying the same thing in different ways.

Doubtless, they are correct, but we must not lose sight of the fact that syllogistic reasoning is used in practical problem-solving, and seeking a solution to a problem is not necessarily a trivial endeavor.

This is how we may reason:

I know that P is true.
Now if Q is also true, then P•Q means R.
But if Q is false, then I won't be able to say that P•Q --> R,
and so R doesn't necessarily hold.
But would P•~Q --> S?

and so on...

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In response to Mill's objection that once a syllogism's premise (or its major and minor premises) has been stated, the conclusion sheds no new light, I would argue that, though that is correct if we do not consider the issue of time, in fact syllogistic logic very often is tied up with sequential arrival of information.

The specific Socrates syllogism concerns things that are so universally accepted that one can agree with Mill that it is so trivial as to be pointless, although that does not mean the general form is trivial.

As an aside, we observe here that the major and minor premises are interchangeable in the Socrates-style syllogism. That is,

both of these syllogisms are equivalent:

Socrates is a man
All men are mortal
Hence, Socrates is mortal

All men are mortal
Socrates is a man
Hence, Socrates is mortal

Now consider this example:

You learn that everyone in a town you are visiting who
 has a swastika tattoo is a member of a particular Aryan cult.
You've long known that your friend Joe Smith has a
 swastika tattoo but laughed it off as a juvenile affectation.
You suddenly remember that Joe is from the town you're visiting.
Aha! you exclaim. Joe must be in that Aryan cult!

Your logic is not quite impeccable, as Joe may no longer be connected to the Aryans, or he may have got the tattoo for some other reason, such as boyish vanity. But the syllogism has given you strong reason to believe that Joe might be among the Aryans, thus perhaps warranting some effort on your part to nail down the truth one way or the other.

In any case, what we see is that the syllogism is valuable when new information arrives. It is not necessarily dead wood for the practical human user of it.

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This last gives a good example of real-world thinking. In fact, this is indeed how scientists reason -- in which case the notion that scientific knowledge is purely inductive cannot be supported. Induction is necessary to establish some statements taken as "facts" but deduction is also required.

In other words, repeated experimentation, say, tells us that the velocity of light in a vacuum is constant, regardless of the velocity of its source. We regard this fact as inductively established. Yet, when Lorenz took this fact, he said that if it is so, then an object's length, relative to the observer, must shorten. This was a deduction.

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As Mill and others have noted, the Socrates syllogism brings in a question of proof of a reality as opposed to a logical form. That is to say, What about the resurrected Jesus? He is immortal. Oh, well, perhaps he does not really count for a man, despite orthodox theology. Well, let's leave Christ out of it (as usual). Still, as Mill points out, we are not reasoning from a generality or a universal; we are considering the fact that all our forefathers, as far as we can tell, are dead.

There is no proof that no man is immortal. As Mill would say, we are reasoning from particulars to particulars.

This little rub is handled when we change the syllogism into the if-then form.

If all men are mortal and Socrates is a man, then Socrates is mortal.

Here we are not claiming that it is generally true that all men are mortal, but merely saying that if that property holds for all men, then it holds for Socrates. We are not required to assume the truth of the premise.