In the article on Aristotelianism on Wikipedia, Betrand Russell criticizes Aristotle's logic:

The Aristotelian system allows formal defects leading to "bad metaphysics". For example, the following syllogism is permitted: "All golden mountains are mountains, all golden mountains are golden, therefore some mountains are golden", which insinuates the existence of at least one golden mountain.

Why does this syllogism imply that a golden mountain exists? I didn't think that it's Logic's job to say whether anything exists, only that given some premise(s), a conclusion follows (or not). Am I wrong and is Russell correct?

Hi algojack,
The proposition "All golden mountains are mountains" is tautologous, since the predicate are mountains is already given in the subject all golden mountains. Similarly, with regard to "all golden mountains are golden". The conclusion "therefore, some mountains are golden" is the apparent result of a third-figure syllogism AAI; but in fact it is already given in the subject - the compound term "golden mountains" implying that the two elements golden and mountains are compatible, i.e. that "some mountains are golden" (and vice-versa, some golden things are mountains).
All this said, just to point out that the example used here is very artificial!
But to return to the central issue - this is the issue of EXISTENTIAL IMPORT. In this regard, I invite you to read my essay "The triviality of the existential import doctrine" in my website: http://www.thelogician.net/A-FORTIORI-LOGIC/Some-Logic-Topics-of-General-Interest-Appendix7.htm#_Toc370306065
There you will see that this issue is a bit made up by modern logicians like Russell to give themselves importance.
Much more serious is the mega-error made by Russell and his ilk in the (mis)understanding of class logic, which gave rise to the RUSSELL PARADOX - and that you can find out about on the same webpage, further down, at:
http://www.thelogician.net/A-FORTIORI-LOGIC/Some-Logic-Topics-of-General-Interest-Appendix7.htm#_Toc359927315
I hope this answers your questions at a deep enough level.

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