I have a question about the identity of a certain kind of fallacy, namely: A = C B = C therefore A = C Confusingly, I have read that the above syllogism is valid; and yet consider this argument I've heard recently: Obama = Good speaker Hitler = Good speaker therefore Obama = Hitler Clearly the latter is a fallacy. So, I have two questions, really: 1) What is the name of this fallacy? 2) How can it be a fallacy if the first syllogism (A = C, B = C, therefore A = C), whose form it follows, is considered to be valid . . . or am I wrong about it being valid?

Well, yes and no. What you've got here is a tangle, just the sort of tangle that actually does lead to serious philosophical problems. You see, what you've got in the first place isn't exactly a syllogism. So, it's neither a valid nor an invalid syllogism. It looks a lot like the following syllogistic form (which is invalid): "All P are M. All S are M. Therefore, All S are P." You can see that this invalid by plugging in the following terms. "All Pigs are Mammals. All Siberian Huskies are Mammals. Therefore, all Siberian Huskies are Pigs." While the two premises are true, the conclusion is clearly false--and that doesn't happen in valid arguments. This invalid form doesn't have a specific name, really, but it does commit the fallacy of "undistributed middle." I think one reason you may have lost your way here is because you use equal signs in your presentation. If you intend to use the equal sign as short hand for the verb "to be" ("are") in the same what as I have used "to be" ("are") in my example, you've got a case of invalid reasoning. BUT, if you intend to use the equal sign to indicate that A and B are "identical" to C, then you've got something else going on. If A and B are both identical to C and you conclude therefore that A is identical to B, you may be reasoning validly. (The reason I'm hedging my claim here is that there's controversy about the concept of identity, and by some accounts your reasoning would be valid and by other other accounts it would not be valid.) So, for example, if Bat Man is (identical to) Bruce Wayne, and Alfred's employer is (identical to) Bruce Wayne, then by valid reasoning Alfred's employer is (identical to) Bat Man. Note, however, that this doesn't mean your example works. You see, in your example the use of "good speaker" is misleading. Being a good speaker is a property that can apply to many things, while "Obama" and "Hitler" are proper names that pick out only one thing each. So, it makes sense to say that "Hitler was a good speaker" or that "Obama is a good speaker." (In a similar way it makes sense to say that "My car is blue" and "Your car is blue," even if my car is not the same car as your car). But it doesn't make sense to say, "Obama is good speaker" or that "Hitler is good speaker"; and it certainly doesn't make sense to say that "Obama is identical to good speaker" and "Hitler is identical to good speaker." Those are grammatically (philosophers like to say "syntactically") malformed sentences and really don't make proper sense in this context. Note, finally, that your example also looks similar to the way one can reason about transitive relations. If A is bigger than B and B is bigger than C, then validly A is bigger than C. But while "bigger than" is a transitive relation, many relations are not transitive. So A might love B and B might love C, but A might not love C. That's because, alas, love is not a transitive relation. Like love, I'm' afraid, logic can be a tangled affair, and it's easy to get all mixed up.

And, to add to all the confusion, one can say: Obama is identical to a good speaker; and also: Hitler is identical to a good speaker. But it certainly doesn't follow that Obama is Hitler. The reason, in this case, is because what stands on the right-hand side of the identity here is not a name, but what philosophers and linguists call an "indefinite". Exactly how indefinites work is a matter of some controversy, but one (older) way to resolve this puzzle is to treat "Obama is identical to a good speaker" as meaning: There is a good speaker with whom Obama is identical. Or, in logical symbolism:

(∃x)(good-speaker(x) & x = Obama)

Now the fallacy should be clear.

The crucial point is that the only really well-defined notion of validity in logic is one that applies only to formal, logical representations. To apply the notion of formal validity to arguments in ordinary language, one has to "translate" the ordinary arguments into logical notation, and it is not always clear how this is to be done. Indeed, precisely what the relationship is between ordinary language and logical representation is a contentious philosophical issue. But (almost) everyone would agree that, as Peter said, one has to be very careful about how the English verb "to be" gets represented.

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