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This question is, admittedly, a little less mathematical than what I normal ask. I seemed to remember that the compound sentence $A\wedge \neg A$ has an official name (maybe even "contradiction" but I seem to remember a different name) just like $A\wedge B$ is "conjunction" and $\neg A$ is "negation".

So, first, what is the classical name of $A\wedge \neg A$?

Second, is there a nice resource to find a list of all classical names for compound sentences?

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3 Answers

I don't know that there is a widely accepted name. Certainly the formula $A \wedge \neg A$ is a contradiction. The law of non-contradiction is $\neg\left(x \wedge \neg x\right)$ so that does give some privileged status to that sentence in the universe of contradictions. But I wouldn't call it contradiction -- if so then $x \vee \neg x$ should be called something like tautology whereas it is instead called the law of excluded middle.

I would think of conjunction as the name of the of the logical operator $\wedge$ while $A \wedge \neg A$ is the conjunction of $A$ and its negation.

You might peruse the Stanford Encyclopedia of Philosophy for careful terminological distinctions made by some. For example some might say that $x \wedge \neg x$ is not an actual formula in a particular language, it is a schema.

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I think Pace wants a term like reductio ad absurdum, or something else from the middle ages or from Roman times. I would suggest a web search on things like a fortiori and so on till he finds what he wants. Evem better would be to ask why he wants it, and find an answer for the want. Gerhard "Ask Me About System Design" Paseman, 2012.05.17 –  Gerhard Paseman May 17 '12 at 19:42
    
Why do you think that? His examples were not in Latin. –  Aaron Meyerowitz May 17 '12 at 22:34
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I wouldn't be opposed to Latin terms, but I wouldn't require it either. (By the way, the closest thing I could find so far is "antinomy".) –  Pace Nielsen May 17 '12 at 22:49
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From his use of the word classical. That is my take though. Gerhard "Ask Me About System Design" Paseman, 2012.05.18 –  Gerhard Paseman May 19 '12 at 1:18
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In our CMSC A course at UP Open University (i.e. Discrete Structures in Computer Science), after we had covered logic fundamentals, I found out that your compound sentence (after suffixing $=False$ to the sentence) is referred to as the "Inverse Rule". (Of course, this is to be interpreted to mean that "True" is the identity element for the OR operation, while "False" is the identity element for the AND operation.)

I also did some research on antimony, and while it does point to a connection between "a thesis and its antithesis", its etymological roots lie more likely in philosophy-based logic rather than mathematical logic as we know it today. (See this Wikipedia article for more information.)

So I guess "logical inversion" might be a good term to use, but then again, it's not necessarily classical.

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The term you are looking for may be absurdity. In logic, the inference of $\neg A$ from $A \to (B \wedge \neg B)$ is called the "law of absurdity".

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