"Socrates is mortal" is the conclusion of a sound deductive argument but it would seem to be a synthetic proposition (as are the premises it is validly inferred from) — Hume1739

Not tellin' but observing that maybe neither premises nor conclusion are synthetic.

All Greeks wear sandals
Socrates is a Greek
Therefore Socrates wears sandals.
Clearly the subject term Greek does not contain in its concept the predicate term "wears sandals" nor does the subject term Socrates contain either the predicate term "is Greek" or "wears sandals".
The above argument is valid and if its premises were true its conclusion would be necessarily true. It is the case premise 1 is false but that just proves premise 1 is a synthetic proposition if it were analytic it could not be false.

How can the conclusion of a sound deductive argument be necessarily true if it is also the case that only analytic propositions are necessarily true? — Hume1739

I don't know where you are getting the "sound deductive arguments" are "necessarily true" from.
As you pointed out soundness uses synthetic propositions and therefore is "technically" never "necessarily true".

The most cheritable way of reeding the statement (assuming it's from a credible source) is that it tries to point out that every sound argument needs a valid structure.
Or it is persumed that there exist Premisses that are True since the syntetic basis for them is strong enough to claim it to be True. F.e. the premiss "I exist".
Maybe it is a good way to illustrate the second case with a counter example. If you assume there are no Premises that can justifiably be called True the concept of soundness falls apart.

Maybe it's helpfull to conceptualize it as synthetic Truths being held to a different Standard. If they meet this standard they are True. Once we have accepted the premisses as True it follows due to the validity that the argument is sound and that in this regard the conclussion is "necessarily True"(assuming the premisses are True).
Maybe one can further add that the Debatte if Premisses are not True and if there exist True premisses is not part of logic. So it introduces a further aspect that allows Logic to model more specific differences if one assumes the premisses to be True. Meaning if you actually believe Sokrates is a man, and that all man are mortal you necassrily have to believe the synthetic proposition that sokrates is mortal.

Thanks for the reply, my confusion arose after reading the following on the deductive reasoning page of wikipedia.
"Deductive reasoning goes in the same direction as that of the conditionals, and links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true".
The italicised part of the quote is a link which takes you to the logical truth page where it says,
"A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement".
So it sounds like whats being said is a valid argument with true premises (ie a sound argument)s conclusion is necessarily true and that necessarily true propositions are analytic propositions.
I understand if the argument is valid then it is "truth preserving" ie true premises will always give true conclusions, I guess its just the use of the term "necessarily true" especially if we are to insist (correctly) only analytic propositions are necessarily true.

We could retitle this "The dangers of learning stuff solely from Wikipedia (and similar sources)."

Re this: "A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement," what they're referring to there is a tautology. "Logical truth" is a term of art for tautologies.

Is there any missunderstanding left or did the replies answer your questions to a sufficient degree?

Btw: I forgot to mention in my last reply that the function of defining something as True can be done in a tatological system regardless of the synthetic truth.
Assume we define all man are mortal. Now let's say there is something x that is man'ish but not mortal.
In a system that puts the emphasis on the synthetic truth we would have to correct the definition to: man can be mortal or immortal. (Cmp. All swans are white example)
However we could also put the emphasis on the existing definition viewing it as analytical statement which in turn leads to the statement x is not a man since x is not mortal.
So even more synthetic seeming statements can under a certain framework be understood as analytic propositions that we call axioms. If all men are mortal and socrates is a man are axioms(maybe not the best ones) the conclusion is necessarily True since they are threated as analytic propositions (because they are definitions).

Kant defines a proposition as 'analytic' if its conclusion is 'contained in' its premise. The broad idea is that the set of analytic propositions is a proper subset of the set of all propositions that can be deductively proven.

The trouble is that the notion of 'contained in' is never defined, and unravels into a chaotic mess when one tries to pin it down.

Thanks for the reply, my confusion arose after reading the following on the deductive reasoning page of wikipedia.
"Deductive reasoning goes in the same direction as that of the conditionals, and links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true". — Hume1739

"Necessarily" is used here informally, as an amplification. This should not be confused with necessity in modal logic (which does not figure in this context). All this means is that given the truth of the premises the conclusion of a valid argument cannot be anything but true. This doesn't make the conclusion unconditionally necessary - the italicized condition still applies.

OK confusion resolved, by describing the conclusion informally as necessarily true and then equating the term via the link to the definition of an analytic proposition the absurdity arises. Seems obvious now should've recognised it myself, thanks for your patience and answers.