Sentence negation, predicate negation and direct reference

There is a good post here by Vallicella, about a comment I made to an earlier post of his. My claim, in essence, is that if we cannot distinguish between sentence negation and predicate negation of a singular sentence, then we have to accept ‘direct reference’. Let me explain. Predicate negation – e.g. ‘a is non-F’ is such that its truth requires that the object denoted by ‘a’ exist, and that this object is not F. Sentence negation, by contrast, is when we simply deny whatever is asserted by ‘a is F’.

In modern predicate logic, we do not distinguish between these two forms of negation, at least for singular sentences. The negation ‘~Fa’ is true whenever some thing is referred to by the singular term ‘a’, and when that thing is not F. Thus it is predicate negation. But it is also the contradictory of ‘Fa’. If it is not true that ~Fa, then it is true that Fa, and conversely. Thus it is sentence negation.

By contrast, in traditional logic (or at least in standard forms of it) we do distinguish between these forms of negation. ‘Socrates is a non-runner’ is true when there is some such person as Socrates, and that person is not running. Thus ‘Socrates runs’ and ‘Socrates is a non-runner’ are contraries. Both can be false when Socrates does not exist.  This is predicate negation (or ‘indefinite negation’ as the scholastics called it). This was considered distinct from propositional or ‘extinctive’ or ‘destructive’ negation, where everything asserted by ‘Socrates runs’ is denied, and which can be true even when there is no such person as Socrates.  Thus ‘Socrates runs’ and the destructive negation ‘It is not the case that Socrates runs’ are contradictories. Both cannot be false, for one denies everything the other asserts.

Now I claim that in systems where there is no distinction between predicate and sentence negation, we have ‘direct reference’. This is easily shown. Direct reference in a singular sentence is when the sentence is meaningless when the singular subject fails to refer. Assume that ‘a is F’ is not meaningless. If it is true, then there is a referent for ‘a’. If it is not true, the sentential negation ‘It is not the case that a is F’ is true. If sentential negation is equivalent to predicate negation, it follows that ‘a is non-F’ is true, and so a exists, and so, there is a referent for ‘a’. But (by excluded middle) either ‘a is F’ is true, or its contradictory (the sentential negation) is true. In either case, ‘a’ has a referent. Thus if ‘a is F’ is not meaningless, ‘a’ has a referent. Conversely if ‘a’ does not have a referent, ‘a is F’ is meaningless. But that is Direct Reference, as I have defined it.

In systems where we can make a distinction between the two forms of negation, we do not have to accept Direct Reference. If ‘a is F’ is meaningful but false, this could either be because the predicate negation is true, or because there is no referent for ‘a’.

Both Peter Lupu and David Brightly challenged this idea in their comments to the post.

Peter argued that we do not have to accept Direct Reference if we accept the distinction between meaning and reference (or ‘sense’ and reference or whatever you call it). I reply: in accepting this distinction we have (tacitly) accepted the distinction between the two forms of negation. If we are happy that ‘a is F’ may not be true in the case that ‘a’ has a sense but not a reference, and assuming that (in this system) we can say that this is so, i.e. state that a does not exist, then we can state the disjunction ‘a does not exist or a exists but is non-F’. But that disjunction is equivalent to the sentential negation of ‘a is F’, and is therefore distinct from the predicate negation ‘a does exist and is non-F’.

David argued that we can suppose a singular term has a referent, and prove a contradiction in order to show that it does not have a referent. For example, let ‘a’ denote the largest ordinal number. That supposition leads to a contradiction, therefore ‘a’ cannot denote that. Yet (according to David) it is meaningful to make assertions using the singular term ‘a’. I reply: what is meant by ‘denotes’ here? If it means ‘signifies’ or ‘means’, then the supposition that it has a meaning leads to a contradiction, and so it does not have a meaning. This is Direct Reference. On the other hand, if it does not mean ‘signifies’, but rather that a exists, then my reply is the same as to Peter. To accept the possibility that ‘a’ is meaningful but fails to correspond to any existing number, is tacitly to accept the distinction between the two forms of negation.