Truth conditional semantics and wffs

Question

How specifically does truth conditional semantics separate syntactically well formed sentences (wffs) with semantic meaning from syntactically well formed sentences without semantic meaning? An example of a grammatically non-well formed English sentence is:

(1) "A a believe the into moon."

An example of a syntactically well formed sentence that has no semantic meaning is:

(2) "Colorless green ideas sleep furiously."

An example of a semantically well founded sentence according to the truth conditional theory of meaning is:

(3) "The cat is on the mat."

(3) has semantic meaning because we understand what conditions are sufficient and necessary to give the sentence a "true" truth condition. "The cat is on the mat," is true if and only if the cat is indeed on the mat. Truth conditional theories of meaning allow us to assign either a "true" or a "false" value to (3), but they assign meaning to the sentence because it is well formed to the extend that we understand what its necessary and sufficient conditions are.

How does truth conditional semantics treat (2)? Does it say that there are no possible conditions under which it has a "true" truth value? Meaning that in all possible interpretations (2) will be false and therefore it is a contradiction? Or does it say that there is no way that it could have any truth value, because it is nonsensical? Or does it say something entirely different?

(1) is nonsensical because it is not grammatically well structured but (2) is grammatically well structured. How does truth conditional semantics treat (2) in light of how it treats (1)?

Answer 1

As the name already suggests, truth conditional semantics is only interested in the truth of a statement, not so much in whether or not that statement makes sense pragmatically.
Form a truth-conditional perspective, the sentence "Colorless green ideas sleep furiously" is a syntactically well-formed sentence (not only a formula, but a sentence), and well-formed sentences by definition have a truth value as their extension.

Before we proceed, one should be careful about conflating the extension of an expression with its meaning. The extension of a sentence is its truth value in a particular situation; but few speakers would intuitively agree the the meaning of "The cat is on the mat" is just "1" (or "0"). The intension of a sentence is, under the classical formal semantic treatment, an abstraction over the sentence's possible truth values under different possible worlds, or equivalently, the set of all those possible scenarios in which the sentence becomes true. This idea comes closer to what is commonly understood as "meaning", but since the intension of an expression builds up on its extensions, it ist still useful to continue the discussion about truth value extensions here.

Truth-conditionally seen, the sentence is not meaning-less, because it does a truth condition: namely that it is true if and only if the situation it gets evaluated in fulfills the conditions that there is some x to which the predicate "idea" applies, and which is part of some sleeping event which takes place furiously, and so on.

The point is just that these theoretical truth conditions will never actually yield a true sentence, because, for example, the condition that our x is located in the intersection between "colorless" and "green" - which is obviously an empty set - will never be able to get satisfied.
Since there is no situation in which the sentence can become true, it would be dubbed semantically contradictory - but this still implies that it has a semantic value, namely "false" in all situations.

Strictly truth-conditionally, a sentence would only be nonsensical if it was syntactically not well-formed so the truth conditions which arise from a functional combination of the parts it is composed of cannot be formulated (like in your first example sentence, where you have a determiner + determiner + verb construction, for which there is no rule how to combine these types into a logical assertion under which the involved elements make a true sentence).
At the level of formulating such truth conditions from the individual elements and their syntactic structure, the logic is "blind" for issues like semantic incompatibility; as long as "ideas" is a noun that predicates some individuals which form the subject, "sleep" is a verb that applies to these individuals and so on, it is truth-conditionally meaningful in that we can formulate conditions (namely "= 1 iff there is some x such that Idea(x) and ...") under which the sentence becomes true.

---

Truth-conditional semantics has a rather narrow world view, in which everything - as long as it be a grammatical sentence - can either be true or false.

For example, the famous "King of France" problem:

Assume that there is no present king of France.
Is the sentence

The present king of France is bald  

true, false, or nonsensical?

would by most truth-conditional semanticists be answered by "false", because the assumption is that the sentence can simply be transformed into the predicate logic formula

 ∃x((KoF(x) ∧ ∀y(KoF(y) → (y=x))) ∧ Bald(x))

"There is something which is the king of France (and for all other things that are also the king of France, this other thing is equivalent to the first individual, such that there really only one king of France), and this individual is bald"

which can simply be negated by negating the whole sentence

 ¬∃x((KoF(x) ∧ ∀y(KoF(y) → (y=x))) ∧ Bald(x))  

"There is nothing which is the king of france and ..."

thereby negating the existence of such a king, which makes it a perfectly well-formed sentence with a truth value.

This would be the Russelian point of view - Frege (and I) would disagree and say that if the presupposition of a sentence (like that there exists a present king of France) is false, this sentence can neither be true nor false, because negating the sentence ("The king of France is not bald") would still exhibit the presupposition (in fact, that the logical entailment follows from both the affirmative and the negative sentence is the very definition of a presupposition! So under the account presented above, the sentence would have no presupposition at all, which I find simply implausible). But enough semanticists would say that the sentence is meaningful, truth-conditionally.

---

Long story short: Truth-conditionally seen, the sentence has a truth value due to being syntactically well-formed, which, however, will turn out to be false in all situations, because the truth conditions that there is an x which is both colorless and green etc. are inherently combined in such a way that they can never succeed to make the sentence true - but then the answer is just "false" rather than "nonsensical".

In order to get a more decent answer, you'd have to tune your semantics up quite a bit, for example with features. Then you can say that, for example, "sleep" is a predicate that can only apply to living entities (something like [+ LIVING]), a feature which ideas (being abstract entities) don't have, so the elements' types within the sentences don't match and then you can indeed say that the sentence is nonsensical rather than simply false.
But that requires a lot of complicated set-up and ontology and is not what is usually meant when talking about truth-conditional semantics.

Answer 2

Theorems are wffs, but not all wffs are theorems. Theorems are demonstrable, but other wffs are not. The central notion in logic is implication, not truth. It is possible to answer questions about implication without making any appeal to truth -- a logic constructed that way is a "logical syntax".

If there is appeal to truth, in a "logical semantics", it is always through implication. If sentences you take to be true imply "Colorless green ideas sleep furiously", then the latter is true; if its negation is implied, then it's false. That's all a truth functional logic can tell you about the truth of this example.

Answer 3

Following my comment

I read between the lines of this [@GregLee's] answer, that there is no reason to construct such a sentence (2) and therefore no reason to estimate its semantic truth value condition thingamabob. That's a good answer, except that the phrase exists for some reason (even if, say automatically produced by a nonsense generating markov chain) and on the metalevel it's become symbolic for a certain statement, so effectively it proves itself

I answer thus:

Of course phrases can be paradox. This is known from various examples, and however Truth Conditional Semantics treat Russel's Paradox (There is a set that contains all sets that don't contain themselves) and similar head nuts. Many set theoris thus simply don't allow a set to include itself. Just as we are predisposed to allow no color to be colorless.

But, if wordplay is allowable, then the phrase could mean any number of things. It's just not tractable, because of lack of information.

But in this case we do have the essential information: The sentence is supposed to be nonsense. Well, who could disagree with the elaboration? No amount of logic can.

Whereas, if somebody would coincidentally see meaning in it, they will be referencing a different context.

Truth conditional semantics then has to decide which contexts to allow, to infere the conditions from. They cannot allow contexts that they don't know.

I don't even know what "cat" is. I do know what "mat" is: Either the guy from the bodega next door, family name "the" (his cultural affiliation dictates family names first, and minuscles for names, I guess), or "An alloy of copper, tin, iron, etc.; white metal." [en.wiktionary]. Either makes sense, but the phrase is undecidable, because cat is not a concept in my calculus of constructions. Even if cat should be an odd short form from Latin cattus, I'm affraid I know neither which cat, nor precisely how it is on.

That is to say: If they rely on uniquely identifiable objects to analyze natural language, they are going to have a bad time. But this is the goal.

---

After reading @LemonTree's answer I'll add my interpretation of it.

1. The pretense that colorless green had to be an empty set is untennable, if we have walking dead, and other viable oxymorons quite frequently. Indeed, yellow-green or even red-green could be rejected the same way, but they do make sense (color + color); being observably colorless, but intrinsically green also works. Green has a dozent different definitions, anyhow (color + adj).

That is not even the point. Rather, the opposite sentence supposedly wouldn't make sense either. But it is a multifaceted statement and so has not one unique opposite, but many axes of symmetry. If we reject that ideas could be green for any intent or purpose, than saying they were simply not green would be true, and the opposite false; and so on.

2. The fact that the sentence would be overall unacceptable anyhow must mean that it is ungrammatical. This could be upheld with a stringent type theory to some extend, to say the sentence is not well formed. This is a generally applicable notion that should be workable with the subject under study.

3. Point in case, saying "how does truth conditional semantic treat ..." should be deemed ungrammatical. Guns don't kill people, I do. Your wording allows you to denigrate any answer, because nobody individually embodies the truth conditional semantic theory; which is just your interpretation (and you are welcome to have it, I won't argue; maintaining a kind of objectivity is exactly the point of the wording). It is acceptable however, even more so than theorists or any more elaborate circumscription.

Which gives us a fourthe type, syntactically not well formed sentences with semantic meaning that can be reconstructed (mock up examle exercise reader left to is). This completes the picture and shows that wffs without semantic interpretation musf belong to (1), for sake of the argument, if wffswos can be said to exist at all.

Chomsky didn't show that 2d shapes don't have volume. Rather, some shapes don't have corners. (I didn't think this through, my statement might belong with (2)). I mean, he picked an example with too many dimensions, that confuses people. But it has become a one-dimensional point instead, and thus entails a lot of meaningful verbiage in our shared context. I would prefer to just ignore it, though.