Although I am aware of the distinction between deduction and induction in logic, which relies on the strength of the link between premises and conclusion, with deduction a matter of necessity and induction a matter of probability, I find the distinction problematic. For instance, the argument "All men are mortal. Socrates is a man. So, Socrates is a mortal" is a classic example of a deductive argument. But the first premise is based on particular cases, so it cannot be universally guaranteed that it would be always true. But the fact that it may not always be true makes it one of probability and not necessity. Would this consideration make a difference as to the argument is deductive or inductive?

Whether an argument is deductive or inductive depends on the nature of the link between its premises and its conclusion. As you say, a deductive argument is one in which the premises entail the conclusion as a matter of necessity, i.e., that its conclusion must be true if its premises are. In contrast, an inductive argument is one in which its premises putatively support, but do not entail, its conclusion. As you say, the premises are supposed to make the conclusion more probable, but the conclusion could still be false despite the premises being true.

Deductive and inductive are therefore properties of arguments, not properties of their premises. What your example, the classic Socrates syllogism, highlights is that the premises of an argument can be justifiedin different ways. Certainly

All men are mortal

is a premise we would justify inductively: We observe that every man [sic] who's ever lived dies eventually, and so on the basis of inductive reasoning (person 1 died, person 2 died, ... person N died) conclude that 'All men are mortal.' Such a conclusion is, in light of our evidence, highly probable. The other premise

Socrates is a man

likely is not justified by induction — more like straightforward perceptual observation. It is also possible for a premise to be justified deductively. We could do so with 'Socrates is a man': Socrates is a featherless biped. All feather bipeds are men [sic]. Therefore, Socrates is a man.

The important point is that deduction and induction categorize arguments in terms of their modes of reasoning. The classic Socrates syllogism you cite is therefore deductive, regardless of how its premises might be justified (inductively, deductively, or otherwise).

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