Clear Language, Clear Mind

In an earlier essay I mentioned that that meaninglessness is contagious with respect to sentences. One can pretty easily formulate the principle in normal english – if a sentence is meaningless, then so is any more complex sentence of which it is a part of. To get a proper, formal formulation of this we may simply think of the rules in logic systems used to form well-formed formulas (=wff’s) and then formulate some similar principles for the meaninglessness of sentences. Here’s what I have in mind:

Negation. For all sentences, iff it is not the case that a sentence is meaningful, then it is not the case that the negation of that sentence is meaningful.

(∀S)(¬M(S)↔¬M(¬S)

Conjunction part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the conjunction of that sentence with another sentence is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S∧Z)¹

Disjunction part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the disjunction of that sentence with another sentence is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S∨Z)

Implication/conditional part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the implication of the first sentence to the second is meaningful, and it is not the case that the implication of the second sentence to the first is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S→Z)∧¬M(Z→S))

Bi-implication/bi-conditional part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the bi-implication of the first sentence to the second is meaningful, and it is not the case that the bi-implication of the second sentence to the first is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S↔Z)∧¬M(Z↔S))

This should cover propositional logic. It is left to the reader can invent the relevant principles for modal logics and predicate logic.

Notes

In a sentence theory of truth bearers, what it means to say that a sentence is cognitively meaningful is that it is true or false. To say that it is not cognitively meaningful (i.e. cognitively meaningless) means that it is not true or false.

In contrast, in a proposition theory of truth bearers, what it means to say that a sentence is cognitively meaningful is that it expresses a proposition. To say that a sentence is cognitively meaningless means that it does not express a proposition.¹

I wonder if there is some problem with cognitively meaningfulness, logical implication and a sentence theory of truth bearers. Consider:

P. Colorless green ideas sleep furiously.

Q. The Earth is spherical.

Now consider a sentence that is about a logical implication from (P) to (Q):

S. That colorless green ideas sleep furiously logically implies that the Earth is spherical.

Now, it seems to me that if (P) is cognitively meaningless, then any sentence of which (P) is an antecedent or consequent, is cognitively meaningless too.

But now recall how a logical implication is defined. P implies Q iff there is no possible world in which Q is false and P true. But this is the case above. (P) is not true in any possible world at all² and so any logical implication in which it is the antecedent is true. (Also where it is a consequent.) Thus, (S) is true. Contradiction. Something is terribly amiss.

Maybe some other definition of logical implication is needed. Suppose we stop defining it in terms of truth and falsity, and use the “is the case” phrase instead. Logical implication can then be defined as this: P logically implies Q iff there is no possible world in which P is the case and Q is not the case. Presumably all cognitive meaningless sentences are not the case. They are not false either because the semantic truth relations only hold for cognitively meaningful sentences. Now given the definition of logical implication all logical implications with a cognitively meaningless sentence as the antecedent are true. Again contradiction.

Notes

It sometimes happens that one is analyzing some theory and one discovers that the theory in some sense implies something meaningless.

But that doesn’t make sense when we think about it. Meaningfulness/meaninglessness is only applicable to sentences, and not to propositions. Implication is only defined in relation to propositions. So when we encounter the scenario that a theory “implies something meaningless” we should say that the theory implies that some sentence is meaningful but it isn’t. In that way we can use an inference (MT) to conclude that the theory is false. We can’t do that with something meaningless.

When I write ‘applicable’ I mean only in meaning, not some possibility of some sort.