Truth Value of Sentences Containing Logical Contradictions

Do propositions containing logical contradictions have truth values, or are they meaningless? For example:

A) Some married bachelors exist.

B) 95% of married bachelors live in Maryland.

C) There is a window behind the spot where the married bachelor stood.

D) Shane opened the window to the left of the painting of a married bachelor.

E) No married bachelors exist.

F) Necessarily, no married bachelors exist.

Similarly, do yes/no questions containing logical contradictions have an answer, or are they meaningless? Translating the above examples into questions:

G) Do any married bachelors exist?

H) Do 95% of married bachelors live in Maryland?

I) Is there a window behind the spot where the married bachelor stood?

J) Did Shane open the window to the left of the painting of a married bachelor?

K) Is it false that any married bachelors exist?

L) Is it necessarily false that any married bachelors exist?

G, K, and L each seems to have a valid answer, while it's not so clear that H, I, and J are meaningful.

– user20153
Commented May 28, 2016 at 22:57
• I mean logical, not material contradictions.
– user20153
Commented May 28, 2016 at 23:00
• "meaningful" is not synonymous with "has a truth value". all of your examples are meaningful, otherwise we would not be able to understand them.
– user20153
Commented May 28, 2016 at 23:04
• put it this way: if it's a sentence, it's meaningful. compare "bachelors exist married some".
– user20153
Commented May 28, 2016 at 23:21
• How are E and F contradictory? Married bachelors cannot exist, and so, they don't. "No married bachelors exist" seems to be a true observation. Commented May 29, 2016 at 20:28

A compound phrase like `married bachelors` is not a logical contradiction](https://en.wikipedia.org/wiki/Contradiction), it is a contradiction-in-terms.

In this case `married` and `bachelors` are sub-terms. Combining the sub-terms gives a contradiction-in-terms. By the definition of the sub-terms -- that is to say, going by what it means for something to be a married thing and going by what it means for something to be a bachelor -- one can't have a married bachelor. If the meanings of the sub-terms of a compound term contradict each other then it is fair to say that the compound term is meaningless, one can talk about such things using natural language but one can never actually construct or instantiate such things in the “real world”. Another example would be spherical cubes.

A logical contradiction a "logical incompatibility between two or more propositions". This is the given definition of a logical contradiction. None of the examples you gave are logical contradictions. But if you had given any examples of a logical contradiction we could be assured that they have one and only representation†, that of (falsum).

I think it is fair to say that a contradiction-in-terms is semantically meaningless. Let's analyse your assertions `A)`..`F)` and your questions `G)`..`L)` in light of this.

Assuming we are not talking about fictional scenarios -- I could imagine some kind of Woody Allen surreal comedy playing with these kinds of entities :) -- none of `A)` through `F)` could articulate actual states of affairs.

Something like `G) Do any married bachelors exist?` does make a kind of sense. It could be answered with no. And this hinges on a fine distinction. Not because there happen to be no married bachelors at this point in time and/or in this particular place and/or in a certain way -- that is to say, that there could be but there happen in this instance not to be -- but because there could never be by virtue of the semantic meaninglessness of the compound term.

Something like `H) Do 95% of married bachelors live in Maryland?` is more nonsensical because we're not merely speaking about existence now but sort of assuming the existence and asking for additional information. Valid response are both no and the question is meaningless. It could be answered like, “The answer is no but not because of anything to do with quirky facts about percentages and Maryland but because it doesn't even make sense to talk about married bachelors so the whole question is a quite meaningless.” `I)` and `J)` amount to the same kind of thing as `H)`.

For `K)` answer is yes for the reasons given for `G)`.

The answer to `L) Is it necessarily false that any married bachelors exist?` is a simple yes because that necessarily acknowledges the contradiction-in-terms.

In classical logic, particularly in propositional and first-order logic, a proposition φ is a contradiction if and only if φ ⊢ ⊥ . Since for contradictory φ it is true that ⊢ φ → ψ for all ψ (because ⊥ → ψ ), one may prove any proposition from a set of axioms which contains contradictions. This is called the "principle of explosion" or "ex falso quodlibet" ("from falsity, whatever you like").

In a complete logic, a formula is contradictory if and only if it is unsatisfiable.

As far as I understand, a proposition cannot have internal logical contradictions; if a sentence is logically contradictory, then it contains more than one proposition.

For instance,

"There is a window behind the spot where the married bachelor stood".

This implies that,

1. someone stood with a window behind him/her;
2. the person referred in 1. is a bachelor;
3. the person referred in 1. is married.

So that, given the definitions of "bachelor" and "married" (which in turn probably require propositions of their own), either 2. or 3. above are false (or perhaps both if the person in question is a widower).

It is a (in my opinion, nasty) trend to call sentences that incur in such contradictions "meaningless", but to me it is evident that they have meaning, just a contradictory meaning.

(In practice, it may be that no utterable sentence is actually a proposition, for the meaning of each word in there is based upon a non-explicit proposition of itself.)

H) Do 95% of married bachelors live in Maryland?

True. There are no married bachelors in Maryland, and no married bachelors outside Maryland. So the total population of married bachelors is zero; 95% of zero is... zero. So 95% (or 50%, or 0.001%, or any other porcentage to your liking) of married bachelors live in Maryland (or Kentucky, or Kenya, or the Kamtchaka peninsula).

I) Is there a window behind the spot where the married bachelor stood?

False. There is no spot where a married bachelor may have stood (or sat, or laid) because there are no married bachelors. So, no; no windows can be "behind" an inexistent place.

J) Did Shane open the window to the left of the painting of a married bachelor?

Again false; the painting does not exist, so there can be no window to the left of it for Shane to open.

In a sense - I remember one book which treated contradictions as such, though it was in the context of zeroth-order logic -, contradictions are the opposite of tautologies, i.e. sentences which are always false, regardless of the truth values of the propositions.

The first sentence translates to "there exists an x such that Mx & not(Mx)" (M being the 'is married' predicate), which is false because there doesn't exist such x

• right, but a false sentence is ipso facto meaningful.
– user20153
Commented May 28, 2016 at 23:06
• right now this does not answer `Do propositions containing logical contradictions have truth values, or are they meaningless?` Maybe you can connect it better? Commented May 29, 2016 at 3:35
• Excuse me? "contradictions are (...) always false, regardless of the truth values of the propositions." That is, yes, they have truth values, and it is always false.
– wet
Commented May 29, 2016 at 19:00
• @mobileink: I don't recall mentioning contradictions were meaningless in my post.
– wet
Commented May 29, 2016 at 19:02
• the OP's question was "do logical contradictions have truth values or are they meaningless?" which as I suggested in my other comments is a misuse of "or". so even though the question is rather ill-formed I think my comment is a propos. saying something about the truth value of contradictions or paradoxes does not address the question of meaning.
– user20153
Commented May 29, 2016 at 19:57