If I have an invalid syllogism, it can still be considered a syllogism, right?
Example:
All A is B
All B is C
All C is A
It is invalid. But it's still syllogism?
(I know that may seem obvious, but I see many people just considering the valid syllogisms as syllogism)
The term syllogism is due to Aristotle (originally sullogismos).
Aristotle defined it as:
an argument (logos) in which, certain things having been laid down, something different from what has been laid down follows of necessity because these things are so. (Source)
So by Aristotle's definition, all syllogisms are valid.
But consider this. In Aristotle's logic there is a valid form of syllogism called Darapti:
All C's are A's
All C's are B's
∴ Some A's are B's
But this is invalid in modern logic. (When there are no A's, B's or C's, the premises are true and the conclusion false.)
So modern usage might differ. The SEP entry on Aristotle's Logic says this:
In modern usage, ‘syllogism’ means an argument of a very specific form. Moreover, modern usage distinguishes between valid syllogisms and invalid syllogisms. The second of these is inconsistent with Aristotle’s use: since he defines a sullogismos as an argument in which the conclusion results of necessity from the premise.
If I have an invalid syllogism, it can still be considered a syllogism, right?
That's right. An argument is a syllogism
...if and only if it consists of three categorical sentences containing three terms in all, each term appearing in two different sentences.
Barker, The Elements of Logic (1965), p. 61. The example has that structure, and that is all it needs. Validity or invalidity does not enter into the definition.
The example, as written, is invalid, but a syllogism, nonetheless.
