Is there any phrase of the form "A and not-A" that is meaningful?

We can imagine vernacular expressions with that form that carry meaning. I could say, "I do like France and at the same time I don't like France ... I like it as a place to visit but I wouldn't want to live there." Perhaps the truth of the terms used in a statement can vary in the time it takes to make the utterance. If I'm flicking a switch, I could say, "The light is on and ... the light is off (hence not on)" and say something not only meaningful but true.

But here we are talking about the sense of a proposition not statements made in ordinary language, which have looser requirements for carrying meaning. If we are speaking of the logical sense of a proposition, we strip away this vagueness and the contingencies of expression. Both instances of the term are supposed to carry precisely the same sense. The "indexicals" in the proposition, such as I, here, and now are translated into their referents. The above examples would be rendered as something like:

"I (Person A) like France (in the sense of being a place to visit) and I don't like France (in the sense of being a place to live)"

"The light is on (at t1) and the light is off (at t2)"

So they are not cases of "A and not-A".

Let's take the last example and change it so that we are forming an 'A and not-A' proposition. "The light is on and the light is off". Is it meaningful or just false?

I am assuming that different theories of language and meaning will have different answers to this so I am open to all approaches.


7 Answers 7


The expression is meaningful, because the meaning of a sentence is a function of the meaning of its parts, as long as those parts are put together in the right way. "A and not-A" is a conjunction of "A" and "not A". A is a meaningful sentence, not-A is a meaningful sentence, and if you conjoin two meaningful sentences, you still get a meaningful sentence in return; hence "A and not-A" is meaningful, even when we mean exactly the same thing by "A" in both hands of the conjunction (avoiding the worries about indexicality you mention), it just happens to be false. Indeed, the sentence is necessarily false.

But there's no general rule to the effect that necessary falsehoods are meaningless. Sometimes mathematical conjectures turn out to be false. But if a mathematical conjecture is false, it is necessarily false. But the research mathematicians who were working on the conjecture were clearly treating it as meaningful all along.

A final example, consider a sentence that is a tautology rather than a contradiction: "A or not A." This sentence is true, and indeed necessarily true. But the fact that it is a necessary truth still entails that it is meaningful utterance to make, even if we don't often have occasion to utter tautologies in ordinary language very often.

  • Thanks, yes I can see the force behind the suggestion that "if you conjoin two meaningful sentences, you still get a meaningful sentence in return". I was pulled away from that conclusion by the idea that meaning implies conceivability. A meaningful proposition P must refer to a state of affairs that could conceivably be the case. But "A and not-A" is conceptually incoherent so couldn't possibly refer to a state of affairs in any logically possible world. And couldn't we say that mathematicians reveal certain conjectures to be, in fact, meaningless when placed under scrutiny? Jun 6, 2016 at 19:32
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    Whether or not meaning is compositional (sum of parts) is a matter of considerable debate.
    – user20153
    Jun 6, 2016 at 22:49
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    "A and not A" is not conceptually incoherent. Schroedinger's cat is dead and not dead. Perplexing, yes; incoherent, not so much.
    – user20153
    Jun 6, 2016 at 22:52
  • Aha, I thought of Schroedinger's cat as a counterexample, but I am not convinced. The cat in that experiment, being alive and dead at the same time in a quantum state, is neither alive in the same sense in which my cat at home is alive, nor dead in the sense in which her predecessor is deceased. Whatever the status of Schroedinger's cat (both before and after peeking), it is a state-of-affairs that is not (logically) self-contradictory. Jun 6, 2016 at 23:15
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    @mobileink I'm not aware of any controversy about the compositionality of meaning. Could you point me to the relevant literature? I'm quite skeptical, since any good argument to think that meaning isn't compositional would be tantamount to a good argument that there could be no formal semantics for language. I think the linguists would be quite surprised to hear that!
    – user5172
    Jun 6, 2016 at 23:56

In dealing with 3-valued logic, statements of the form "A and Not A" can be judged either false or equivocal. It is easy to imagine states of affairs "The light is and not on", when it is flickering, "the door is open and not open" when it is slightly ajar, It is "raining and not raining" during a drizzly fog. Such transitional states, ambiguous statements, and conflicting definitions are uncomfortable, and we frequently create definitions in order to resolve the questions, but sometimes those definitions are highly arbitrary and contradictory. Consider "Socrates is dead", or "Elvis lives" when there are multiple definitions of "dead" and "live" and Socrates and Elvis meet one set of criteria but not another. It is sometimes technically advantageous to relax the "law" of the excluded middle and allow statements of the form "A and not A" to be equivocal and suspend judgement on them, even if we cannot allow them as true.


Is there any phrase of the form "A and not-A" that is meaningful?

I suppose the phrase is meaningful in the sense that it is describing an empty set. When the observer reaches the conclusion, "A and not-A is true", then the observer knows that something went wrong in the original premise, in the individual observations, or both. The ability to understand that contradiction is meaningful.


Yes and no. ;) "A and not A" is obviously meaningful, because if it were not we would even be able to ask whether it is true or not. But it is also contradictory, which strictly speaking means something like absurd or ridiculous, but not meaningless, in spite of the fact that ordinary language often makes no distinction between meaningless, absurd, false, etc.


Values are contingent and temporal at best and may represent an intrinsic thing like France, but are not themselves France as those values can change.

That which is essential and intrinsic is eternal and gives rise to itself. i.e., meaning.

In other words, meaning is self-sustaining and cyclic, whereas values of said meaning are temporal.


Heraclitus states 'We are and are-not'. This statement is meaningful because it defines our existential status as having two aspects neither of which is exclusive. This is the language of nondualism, which uses contradiction to avoid endorsing positive metaphysical theories. Hence the convoluted nature of the language of mysticism, which relies on apophasis and contradiction.

But there's a twist. His statement does not take the form of 'A/not-A' because he indicates a third option, such that both halves of his statement would be false on their own. Thus the use of contradiction in this language is in the eye of the beholder, not in the eye of the speaker or of Aristotle.

So, when looking at seemingly-contradictory statements it would be important to be sure that they do take the form 'A/not-A', and I'm not sure any meaningful statement does.


You're only talking language preferences in this thread. A sequence of alphabet letters with some spaces interspersed, whether it contains any of the things labeled "words" or not, either MAKES SENSE or it DOESN'T MAKE SENSE period. If it MAKES SENSE it is either empirically TRUE or empirically FALSE. If it doesn't make sense, then it might as well be "Bxnbmzm jgtq kpfzbmq". Contradictions don't make sense regardless of whether you like to label them with the row of letters "meaningful" the row of letters "meaningless" or the row of letters and one space "logically false". Reality is what it is regardless of anything labeled "philosophy", "linguistics" or "formal logic". I hear you asking me "What about 'A bachelor is an unmarried man?" It's empirically TRUE that the English word "bachelor" is used in English synonymously with "unmarried man", that's the only way to take that sentence.

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