My new semantic argument (abridged version)

In this post, I propose a deductive argument for the claim that positive universally held properties are necessarily universally held. I take properties to be whatever can be attributed by a predicate that uses no or one leading noun qualified by zero or more adjectives, such as ‘being Aristotle’, ‘being red’, ‘being a table’, ‘being a red table’ and ‘being a large red table’. What I say about positive properties is compatible with various available accounts of positive properties in the literature. My argument consists of three premises. The first premise is the Frege-Russell-Quine view of existence. On this view there are no things that do not exist. I take the first premise to state a necessary truth.

The second premise is a statement of the Fregean theory of meaning. I call the meanings of the meaningful subexpressions of an expression the meaning elements of that expression. If a meaning lacks meaning elements, it is called elementary. A complex meaning is a meaning that is not elementary. Since everything is a ‘being’, the meaning expressed by ‘being’ does not add anything to the way a reference is presented or thought about. But then the meaning expressed by ‘being’ is not an additional meaning element of a meaning.

For the third premise I need the notion of a reference set for a meaning. First, the reference set of an elementary meaning is defined as the reference of that meaning. So, the reference set of the meaning expressed by ‘red’ is the set of all red things. Similarly, the reference set of the meaning expressed by ‘being’ is everything that exists. Second, the reference set of a complex meaning M is defined as the union of the reference sets of the meaning elements of M. Note that a meaning element can itself be a complex meaning and thus have meaning elements. The definition of reference set is therefore recursive.

Consider the following example. The reference set of the meaning expressed by ‘unicorn’ is the union of the reference sets of its meaning elements. The meaning elements of the meaning expressed by ‘unicorn’ are the meanings expressed by ‘horn’, ‘forehead’, ‘tail’, ‘hoof’, etc. So, the reference set of ‘unicorn’ is the set comprised of all horns, all foreheads, all tails, all hooves, etc. Unless, of course, these elements are themselves complex. In that case, the reference set of the meaning of ‘unicorn’ is the union of the reference sets of the meaning elements of our original meaning elements. And so on.

I now state the third premise. Let M1 and M2 be two meanings expressed by positive predicable generic expressions, then M1 = M2 if and only if RefSet(M1) = RefSet(M2). If M1 and M2 are elementary, the criterion trivially reduces to M1 = M2 if and only if Reference(M1) = Reference(M2).

Why should we accept the criterion? The ‘only if’ part follows straightforwardly. I shall offer two reasons to accept the ‘if’ part of the criterion. First, if the reference sets of two meanings coincide, then these meanings are entirely indistinguishable in what their meaning elements refer to, all the way down to their most elementary parts. Both meanings are thus entirely similar in how they map onto the world. But then, given the close connection between meaning and reference, it is plausible that these meanings are themselves identical.
The second reason is constituted by an induction over the collection of meanings of positive predicable generic expressions. It is very easy to come up with different meanings that have different reference sets. Just pick any two meanings that are so different from each other that it is obvious that their reference sets do not coincide, e.g. {‘motorcycle’, ‘bicycle’}, {‘red’, ‘blue’} and {‘house’, ‘rock’}.

It becomes more interesting once we consider pairs of different meanings with the same reference. Take the meanings expressed by ‘three-sided’ and ‘three-angled’. Clearly, they differ. This provides us with another confirmation of the ‘if’ part of the criterion, since the reference set of the meaning of ‘three-sided’ is not identical to the reference set of the meaning of ‘three-angled’. The former, but not the latter, includes all sides.

Consider also the meanings expressed by ‘water’ and ‘H2O’. On Fregean semantics the mode of presentation of both expressions differ. The meaning of ‘water’ is thus not the same as the meaning of ‘H2O’. But what about their reference sets? The meaning expressed by ‘water’ has as meaning elements the meanings expressed by ‘transparent’, ‘potable’, etc., whereas the meaning expressed by ‘H2O’ has as meaning elements at least the meanings of ‘hydrogen’ and ‘oxygen’. But then the reference set of the meaning expressed by ‘water’ is not the same as the reference set of the meaning expressed by ‘H2O’. We thus obtain further confirmation of the ‘if’ part of the criterion.

Another class of examples are cases where both meanings have an empty reference, such as those expressed by ‘round square’ and ‘married bachelor’. Clearly these meanings differ. The reference set of the meaning expressed by ‘round square’ includes all round things and all square things. The reference set of the meaning of ‘married bachelor’ contains all married people and all bachelors. So both reference sets differ as well, confirming the ‘if’ part of the criterion. In the absence of counterexamples, the above examples – and many more like them – provide strong inductive support for the ‘if’ part of the criterion.

I shall now derive the conclusion that all positive universally held properties are necessarily universally held from the premises. Suppose for reductio that there is a positive universally held property that is not necessarily universally held. Let P be such a property. Since a property is whatever can be attributed to something by a predicate, it follows that P can be attributed by the predicate ‘being P’. Since property P is positive, the predicate ‘being P’ is positive. That is to say, ‘P’ is a positive predicable expression.

Furthermore, since P is universally held, ‘P’ is in fact a positive predicable generic expression. Let M be the meaning expressed by ‘P’. Since ‘P’, when used as a predicate, attributes a universally held property, the reference of M is everything that exists.

M is either elementary or complex. Suppose M is complex. If we recursively unfold M’s meaning elements, we find at some stage at least one elementary positive meaning element M*. But then the reference of M* is also everything that exists. And since M* is elementary, the reference set of M* is the reference of M* and thus everything that exists.

Hence, RefSet(M*) is everything that exists. Now, the reference set of the meaning expressed by the positive predicable generic expression ‘being’ is everything that exists as well. So, it follows that RefSet(M*) = RefSet(meaning of ‘being’). According to the identity criterion for meanings of positive predicable generic expressions, if follows that M* = meaning of ‘being’. This contradicts the fact that M* is a meaning element. For, as mentioned above, the meaning of ‘being’ cannot be a meaning element. There are no meanings that have the meaning of ‘being’ as one of their elements.

The only remaining option is that M is elementary. Recall that the reference of M is everything that exists. Since M is elementary, the reference set of M is the reference of M and thus also everything that exists. It follows that RefSet(M) = RefSet(meaning of ‘being’).

So, according to the identity criterion, M = meaning of ‘being’. Now, M is the meaning of a predicable expression that, when used as a predicate, attributes a property that is not necessarily universally held. But then the meaning of ‘being’ is the meaning of a predicable expression that, when used as a predicate, attributes a property that is not necessarily universally held. From this it follows that it is possible that there are things of which it cannot truly be said that they exist. In other words, it is possible that there are things that do not exist. But this contradicts the first premise of the argument – the Frege-Russell-Quine view of existence – according to which it is impossible that there are things that do not exist. We thus have to reject our reductio assumption. Hence the conclusion follows: All positive universally held properties are indeed necessarily universally held.

I’ll now point at a number of interesting ontological corollaries. Consider the positive property of being material. This property is positive and plausibly not necessarily universally held. For the existence of a thing that is not material seems at least possible. But then the property of being material is not universally held either. After all, according to the conclusion of the argument, if everything would be material, everything would be necessarily material. It thus follows that there are immaterial things, which is to say that materialism – the thesis according to which everything that exists, is material – fails.

Analogously physicalism and naturalism fail as well. There are non-physical and non-natural things. We can go on: The property of being contingent is positive and plausibly not necessarily universally held either. For a necessarily existing thing seems at least possible. But then it follows that this property is not universally held. So there is at least one non-contingent and thus necessarily existing thing. It can furthermore be shown that there is at least one contingent thing, which refutes fatalism – understood as the thesis that everything exists necessarily.

Take the positive property of being caused. It certainly seems possible that not everything is caused. So this property is not necessarily universally held. But then it follows that it is not universally held. That is, not everything is caused. There must be at least one uncaused thing. By now the recipe for deriving further interesting consequences will be clear enough.