This post examines the division between those word strings that are false and those that are strictly meaningless by looking at borderline cases.
Some general principles I will try to adhere to, yet which may prove tricky in navigating some of the following considerations are:
- Contradictions of the form P & ~P are meaningful and false (even false tautologically as a fundamental axiom of some formal systems, e.g. first-order logic). Note: P stands for a proposition–i.e. something capable of being either true or false, and not both (i.e. capable of following the Principle of Bivalence).
- Word strings that specify (or constitute) no P are meaningless, for not being something that could be either true or false.
And so, the guiding question for all of the following word-strings will be, is the sentence capable of being either true or false (and not both)? Consider some claims (or purported claims) about a number, the number 3:
a) “The number 3 has weight.”
b) “The number 3 weighs 50lbs.”
c) “The number 3 is heavy.”
a) specifies a property (having weight) and that the number 3 has this property. The property and the subject of the property are irreconcilable, it may readily be thought (and correctly so, I’d say). This is because the subject and the property of the subject are opposed. Such are contradictions, we might say, between the subject and property themselves. Since a tie is to be made between a subject and property in such atomic propositions P (i.e. propositions that aren’t composed of other propositions), such an opposition would be internal to the proposition P itself. However, a tie is specified between the subject and property; it is for this that a) is meaningful yet false. More on this shortly.
It strikes me that a case of a meaningful proposition where an opposition between the specified property and subject it is tied to may be sometimes what is meant by an a priori judgement, or of a judgement about a proposition where its truth is evident prior to an empirical investigation (in contrast with an a posteriori judgement). if a) is meaningful, then a) is false, and priori-ly or necessarily so, since the property and the subject it’s attributed to are at odds. For the same reasons and with the opposite result, perhaps claiming a cow has weight would be a priori true, since weight applies for all entities of the domain, cows. However, such an a priori truth is slightly more problematic than the a priori falsity that I just mentioned. To illustrate, we might think that in contrast to the a priori, when we say a particular cow weighs 500lbs, reflecting that the property and the subject of the property are not at odds with one another will not be sufficient to determine the truth value of the proposition; such propositions are a posteriori. However, perhaps we just don’t know enough about the particular subject we’re dealing with. That is, if we just knew more about the subject, this particular cow, we’d immediately know it is at odds with any other weight than that of 500lbs. With this in mind, my final consideration of the distinction between the a priori and the a posteriori is that perhaps the distinction is between those specified fixed properties that range over objects that are a) compatible with said property, and b) that admit of a variation of the expressed property for such things (a posteriori), and those that do not for going against either a) or b) (a priori). Back to meaning!
How is a) meaningful? That is, how can a) be considered to be something that could be either true or false (before our reflecting that if so, it’s false)?
I defend that a) is meaningful, since it satisfies three sufficient conditions for meaning: i) it specifies a subject (a thing, loosely speaking), ii) it specifies a property, iii) it applies or attributes the property to the subject. Since the subject, the number 3, doesn’t have weight, the property ascribed to it, a) is false.
Such conditions, I strongly believe, are also necessary for meaning, but I’d need to say more about where existential quantification and scope fit in. It will suffice for me to say that it is either that a proposition, in order to be one, ties a property to a subject, or else that it gives an existential scope to a subject. Denying that these conditions are necessary calls for alternative conditions for meaning. And it is clear enough that such conditions are sufficient for truth-functional meaning in every commonplace example.
Similarly with a), b) specifies a property (having 50lbs) and that the subject, number 3, has this property. Since the number 3 doesn’t weight 50lbs, b) is false.
Finally, c) specifies a property (being heavy) and that the subject, the number 3, has this property. Since the number 3 isn’t heavy, c) is false.
My argument that b), and c) are false and meaningful is simple enough: a) is meaningful and false. If a) is meaningful, b) merely specifies a subset of the property expressed in a), and the very same subject in a); thus b) is meaningful (and false). Ditto for c).
Note that c)’s being false doesn’t make a proposition that replaces c)’s property with its opposite property (being light) true. It would be just as readily discovered that it is false that the number 3 is light. From our previous discussion, indeed it may be a priori true that the number 3 is neither heavy nor light.
Consider cases when the specified property of some supposed proposition is problematic. For instance, when something contradictory is specified:
a1) The cow is weightily weightless.
b1) The cow weighs -50lbs.
c1) The cow weighs (simultaneously) 50lbs and 500lbs.
Taken literally, a1) specifies, if it specifies anything at all, one property, and that property is one of having weight and having no weight simultaneously. This is a contradictory property and thus doesn’t exist (part of what it is to be a contradiction is to be incapable of unification under one event of an identical place and time; since if something exists, it exists under an event of identical place and time, contradictions don’t exist). Since a1) specifies no coherent property, we don’t know what property might even be meant. Because of this, a1) fails to specify any property at all. That is, a1) fails at the level of specification itself , rather than at the level of the specification’s failing to be true. Since specifying a property is required prior to determining whether or not the proposition is true or false, it follows that such a “sentence” is neither true or false, i.e. not a P. a1 is not false (or true)–it is strictly meaningless.
b1) is tricky. It specifies a contradictory property, but in a different way than that of a1). b1 specifies a negative value of weight, which is a value (a property) that is at odds with the property of weight. b1) is not making a claim that weight has such a value (which would be a meaningful claim that is false), it is rather attempting to specify a single property of such a weight of such a value. Since it is this property that such a specification is aiming for, and since such a property is impossible, b1) never specifies a property, and for the same reasons as a1) for specifying no property, b1) is meaningless. I think the difference between a1) and b1), which is slight in the end, is that a1) tries to specify a property composed of a property and its opposite, while b) tries to specify a property composed of a property and its annihilating value.
c1) is a contradictory specification of a property, since weight admits of only one value, lest it be a contradictory property. Since c1) specifies no coherent property, it will suffer the same fate as a1) and b1), being meaningless.
Consider a case when the specified subject is problematic. For instance, when it fails to refer to anything:
a2) The man composed entirely of dark energy and a dozen black holes went swimming.
Note: a2) is not claiming that such a man exists (which would be a meaningful claim). Rather, a certain subject is claimed to go swimming (to have the swimming property). However, the subject is an impossibility straightaway and thus fails to refer to anything. Is such a failure of reference nonetheless potentially meaningful (e.g. for successfully specifying a property)? I think it’s clear that not: since there is no application of a property on a subject by the sentence (for there being no specified subject), there is no possibility that the sentence is either true or false (and not both). This is just because falsity is at the level of checking a specified property on a specified subject. And without a subject, this cannot be done.
Can’t some of the claims determined to be meaningless become meaningful if we give a more charitable interpretation of them, for instance, if in a2) we just embed the relevant existential claim? I like this idea. Let’s try!
a2.1) There exists a man composed entirely of dark energy and a dozen black holes, and this man went swimming.
Clearly, a2.1) is not an atomic proposition, since it is composed of a proposition, namely the proposition that there exists a man composed entirely of dark energy and a dozen black holes; this proposition is affixed to a larger string of words via the AND connective of first order logic. Such constructions (called conjunctions) are true or false, if and only if the propositions affixed by AND are either true or false. Otherwise, the truth value of the conjunction cannot be resolved to be either true or false. Since this is the same as saying the conjunction in such cases cannot be either true or false, such conjunctions are meaningless.
Unfortunately the first conjunct is meaningful and false, while the second conjunct suffers the same problem as a2), and is thus meaningless. Thus, a2.1) is meaningless.
The same would occur with the relevant changes for any other existential supposedly embedded. I cannot see any other way of giving charity to such meaningless sentences to possibly make them meaningful, without going against the sufficient conditions for meaning I mentioned near the start. And if we were to do this for some reason, we would need alternative conditions, which I presume would not be forthcoming (hence why I have been considering them to also be necessary conditions).
From this look-see into sentences at the borderline between the meaningful and false, and the strictly meaningless, I conclude that meaning fails when either a subject or a property fails to be specified, and that specifying a subject and a property is enough for the specification as a whole to be meaningful, for being either true or false.