Where is the logic fallacy here, and which is it?

A. Whatever is not normal is abnormal.

B. Sleeping is normal.

Conclusion: Being awake is abnormal.

Better example:

A. Whatever is not normal is abnormal.

B. Being an Asian is normal.

Conclusion: Being a Caucasian is abnormal.

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    It is called equivocation. If "sleeping" means "sleeping all the time" then it is not normal, and B is false. And if it means "sleeping sometimes" than "not sleeping sometimes" is "being awake all the time", and there is no paradox in the conclusion. – Conifold May 28 at 4:37
  • Thank you. But what if I replace B with, say, "Being an Asian is normal" and the conclusion with "Being a Caucasian is abnormal", where will the fallacy be then? Will it also be a case of equivocation? – brilliant May 28 at 4:53
  • @brilliant To reason that properties of opposites are necessarily the same is a false analogy that does revolve around an equivocation of the copula. See below. – J D May 28 at 5:11
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    Now you are simply denying the antecedent:"if Asian then normal" does not entail "if not Asian then not normal". You'll need the converse, "if normal then Asian", for that, i.e. one has to be Asian to be normal, which is false. In contrast, "if normal then sleeps sometimes" was true at least on one of equivocal readings. – Conifold May 28 at 5:42
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    It could also be just a sort of implicit assumption "only one member of a given conceptual category, like states of consciousness or ethnic groups, can be normal". Either way, the conclusion does not follow logically from the premises as stated, so it's just a non sequitur argument or "formal fallacy". – Hypnosifl May 28 at 15:15

This looks like a fallacy of equivocation, exploiting the ambiguity between the "is" of predication and the "is" of identity, as in the following argument:

  1. Alice is human
  2. Bob is not Alice
  3. Therefore, Bob is not human

The ambiguity here is between interpreting premise 1 as "Alice has the property of being human" (predication) and interpreting it as "Alice is identical to 'human'" (identity). Clearly, the former is the correct interpretation, but the argument proceeds as if it were the latter.

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Conifold is right to draw attention to the fact it involves an equivocation of the copula 'is' for starters.

Let's look at the argument with enthymemes and some clarification

P1 The opposite of 'normal' IS-BY-DEFINITION abnormal. (a priori claim)
P2 'Sleeping' IS-BY-FREQUENCY normal. (a posteriori claim)
E1 [The 'opposite' of sleeping IS-BY-DEFINITION being awake.] (a priori claim)
C If sleeping is normal, and then it's opposite being awake must have the opposite of its properties. (a priori claim)

But in fact, the property of normality applied to either state is a question of frequency, that is to say, it is an empirical fact that is normal for humans to sleep and to be awake. The properties aren't analogues.

But the equivocation only undermines the rule for drawing the conclusion, which itself would seem to be a false analogy. That is to reason that both terms have opposites means that their opposites share the same properties.

Let's look at another example:

P1. Not in the kitchen is the opposite of being in the kitchen. (a priori claim)
P2. Socrates IS-BY-EXPERIENCE in the kitchen. (a posteriori claim)
E1. [Socrates IS-BY-DEFINITION the opposite of his wife.] (a priori claim)
C. Therefore, Socrates's wife is not in the kitchen.

Here again, the copula is used in two different senses, and we cannot presume that the location of Socrates's wife is the opposite of his, because men and women are in a sense paired in a marriage.

It's a false analogy to presume that two different entities that are opposites have opposite properties.

You've added a second example, and stated correctly, it's not a fallacy.

P1. Whatever is not normal IS-BY-DEFINITION abnormal. (a priori claim)
P2. Being Asian IS-BY-FREQUENCY normal (say, in Asia). (a posteriori claim)
E1. [Among a mixed group of Asians and Caucasians, if you are not Asian, you ARE-BY-DEFINITION Caucasian and vice versa.] (a priori claim)
C. Being a Caucasian (say, in Asia) IS-BY-FREQUENCY abnormal. (a priori claim).

In this case, the argument is correctly analogical because the frequency of two groups DO have opposite properties. In this case, the majority population by definition has the opposite property of the minority population.

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  • I am a little bit puzzled over why you needed to add "say in Asia" and "Among a mixed group of Asians and Caucasians". It looks like you have changed my second example. My original thought in the statement "Being an Asian is normal" was not limited to some geographical region. In other words, it meant that if one was born Asian into this world - regardless of what continent, region or country - it is just natural, that is, there is nothing wrong with that. – brilliant May 28 at 11:51
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    @brilliant Normal has no less than 8 meanings. As you didn't specify, I just selected the probabilistically oriented definition. It seems you meant definition 2, "occuring naturally", so I apologize for disambiguating your meaning incorrectly. – J D May 28 at 14:40
  • Before I read this link I didn't even realize that "normal" had so many meanings. So, is it possible to readjust your answer now for the "occurring naturally" meaning in the second example? – brilliant May 28 at 15:11
  • @brilliant It's good practice when critical thinking to look at dictionaries to probe for contradictions and equivocations in one's thoughts. The linguistic turn is an important moment in the history of philosopy, and understanding language and how it is used clears up a lot of philosophical quandries. – J D May 29 at 15:08
  • Thank you. But can you, please, edit the second example in your answer? I really want to see how it will look like with the meaning for the word "normal" in it being "occurring naturally" instead of "frequently found in Asia". Also, would you, please, explain to me what "P1", "P2", "E" and "C" stand for? – brilliant May 29 at 22:02

The underlying assumption you need to close the argument seems to be that

if A is B, then the contrary of A is not B

which is clearly not valid (even if I am not sure about how we may define in general the contrary of)

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