An Observation on Deduction

Consider the following syllogism:

Major Premise: All elephants are grey.

Minor Premise: Nellie is an elephant.

Conclusion: Therefore Nellie is grey.

This is a formally valid syllogism, but it is unsound. The reason why it is unsound is that the major premise is false: while most elephants are grey, some albino elephants exist. If Nellie were a real life elephant that we have never seen before, there is the possibility that Nellie is an albino elephant, and not grey.

But how do we interpret the epistemological status of major premise?:

All elephants are grey.

If this is meant to assert information about the real world (that is, real world elephants), it must be synthetic a posteriori. We can establish its truth by experience, empirical evidence and inductive arguments.

In this case, as we saw, it is false, because albino elephants exist, and it is also logically possible, though improbable, that a genetic mutation might occur which makes an elephant some colour other than grey or white.

Nevertheless, one could also take “all elephants are grey” as an analytic a priori proposition. We can assert it as true, but only as an empty and hypothetical statement where elephants are arbitrarily defined as having the property “grey.” In this case, we do not care about real world elephants, because they are irrelevant to a purely tautologous proposition we have devised that asserts something merely as an imaginary definition, true by stipulation.

One could even claim that asserted of purely imaginary elephants that are grey by definition, even the conclusion is true, and the argument has necessary a priori truth in our imaginary world of purely grey elephants.

But, of course, what has happened is that we have rendered all the propositions and the whole argument vacuous, empty and tautologous, so that it says nothing necessary of the real world. The propositions and inference would be all true by arbitrary definition of terms, and necessary truth is a purely verbal construct.

And once applied to real world elephants, it all collapses and empirically we know the argument is unsound.

But we can even apply a similar type of analysis to a syllogism that is both valid and sound:

Major premise: All humans are mortal.

Minor premise: Socrates is a human.

Conclusion: Socrates is mortal.

Here both the major and minor premises are synthetic a posteriori and true in the sense that we can construct a set of inductive arguments, from empirical evidence, to the effect that it is extremely probable that all human beings are mortal, and it is extremely probable that the historical person we know as Socrates was human.

The conclusion is necessarily true, but only if the premises are true.

But, when the conclusion is applied to the historical person we know as Socrates as a synthetic a posteriori statement about him, the necessary, apodictic truth is not preserved. Why? The reason is that both the major and minor premises are synthetic a posteriori, and can only ever be highly probable but still fallible.

In order to question the truth of the premises, we have to think of some possibilities that are ridiculous or highly improbable, but nevertheless they are logically possible. Suppose it is possible for a genetic mutation to make a human being immortal in the sense of living without aging. Say Socrates was such a person. Say, his death was staged and he still lives to this day.

Suppose, as an even more outlandish idea, that Socrates was secretly visited by aliens or humans from the far future. Using science, they made him immortal and his death was only staged, so that he still lives.

Of course, these are all either outrageous or extremely unlikely possibilities, but the fact remains that synthetic a posteriori propositions can never be apodictically true: some doubt must remain.

When the syllogism above asserts that “Socrates is mortal” as its conclusion, one can speak of this truth as necessary, only if the premises are taken to be absolutely true.

But, just as in the first case above, we can see how this creates a necessary truth that is ultimately a logical construct (or de dicto necessity) because it banishes all doubt about the truth of the empirical premises and renders them analytic a priori.

Of course, these days, after the work of Putnam and Kripke, analytic philosophers are willing to recognise the existence of certain metaphysical (or de re) necessities, such as (1) identity statements involving proper names or definite descriptions and (2) scientific essences of certain natural kind phenomena, but I do not think this is inconsistent with my comments above on deduction.