Many people are familiar with the malformed questions that, no matter which way they are answered, reflect poorly on the person to whom they were asked. For example:

  • Have you stopped beating your wife?
    • Yes, I have stopped beating my wife.
    • No, I have not stopped beating my wife.
  • Have you ever gotten away with a murder that you committed?
    • Yes, I have gotten away with a murder I committed.
    • No, I have not gotten away with a murder I committed.

These informally fallacious questions need not be answered (at least by people that do not abuse their spouses or are murderers) as they are simply malformed and create a dichotomy of an unreal scenario.

However, when we put these into a proposition, I am unsure as to whether or not they actually have truth values or if they are simply malformed propositions as well.

For example:

  • I have stopped beating my wife. (True/False)
  • I have gotten away with a murder I committed. (True/False)

Must these logically be given a truth value since they are propositions? Or are then invalid propositions altogether? Would the value be False (assuming I have not beat my wife or killed anyone) because ine cannot stop or get away with something that one has not done?

  • A related puzzle: given that no total order is defined on the complex numbers, how does one answer the question of whether 1<1+i — is the inequality true, or false? The correct answer is that it is ill-defined, and that it embeds an assumption about the ordering of numbers which is itself false. – Niel de Beaudrap Mar 2 '15 at 16:43

They can usually be put into prepositional logic. However, these phrases do demonstrate that typical conversational use of language does not adhere to first order logic.

For example, "I have stopped beating my wife" could be decomposed into a predicate "IHaveStoppedBeating(my wife)," where the predicate IHaveStoppedBeating can be applied to an individual. Following the formal (not casual) language patterns English relies on, this can be decomposed into "InThePastIBeat(my wife) AND NOT(InThePresentIBeat(my wife))," this clearly has a false value, for InThePastIBeat(my wife) is false.

The malformed question relies on a different meaning. In conversational English context adds meaning which is not directly adherent to the first order logic. saying "IHaveStoppedBeating(my wife) is false" may be a valid logical statement, but it is also a statement that is generally only made in a context. Expanding to "InThePastIBeat(my wife) and not(InThePresentIBeat(my wife)) is false" in conversational English implies that there is a reason for one of these to be brought up. If the person asking the question disconnects the answer (which is a predicate, whose truth or falsehood should be constant, regardless of whether it is associated to the question) from the question (which provided the context), it leaves an interesting situation. The conversational English listener must determine why this phrase was worth uttering.

As such, it is assumed either the value of InThePastIBeat(my wife) or InThePresentIBeat(my wife) is interesting enough to both speaking of. In the case of both of these predicates, the "interesting" bit would be if one of these is true. So we assume one or both of these must be true. This assumption does not appear anywhere in the formal statement given by the suspect.

"InThePastIBeat(my wife) and not(InThePresentIBeat(my wife)) is false" has two possible ways it can be accurate. Either InThePastIBeat(my wife) and InThePresentIBeat(my wife) are both true, or both are false. Formally this was not a problem, but in conversational English, the assumption that one or both of these statements must be true kicks in. The only resolution of this predicate with this additional assumption is InThePastIBeat(my wife) is true and so is InThePresentIBeat(my wife).

Thus, the truth statement of "I have stopped beating my wife," given the full accurate context, is certainly false, because you never started beating your wife. However, if the questioner can successfully remove the fact that the question was asked in that phrasing, conversational English adds an assumption which causes this phrase to imply you beat your wife. However, this assumption is fallacious, because it is dependent on the conversational English rule that there must be something "interesting" about a statement, and translating that into "beating must have occurred." This assumption is fallacious because it is not the only "interesting" thing about the statement. It is also interesting that it is the answer to a poignant question. Thus I believe it is reasonable to reject this line of thinking, leaving only the answer "'I have stopped beating my wife' is false."

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