Predicates in search of nouns


According to standard categorical logic, statements of the form All S are P must have some sort of noun-phrase in the predicate.  Thus ‘All dogs are mammals’ is fine, but ‘All dogs are four-legged’ is not.  Why is this a requirement?

Standard categorical logic is based on Aristotelian logic and inherits some of Aristotle’s assumptions. So one reason for the noun-requirement is that given Aristotelian metaphysics predicates without nouns are incomplete. To understand why, we first need to keep in mind that predicates(parts of language) correspond to properties(parts of the extra-linguistic—that is, non-linguistic—world). 

Predicateswithout nouns are incomplete because, according to Aristotle, properties must inhere in objects in order to exist.  There are no free-floating properties—properties without some subject that they attach to.  Thus, statements like ‘All dogs are four-legged’ while grammatically adequate are semantically or logically incomplete.  We can see this by thinking in terms of class logic: all dogs are four-legged may be rendered in class logic as follows: every member of the dog class is a member of the four-legged class.  The problem is that there is no such thing, according to Aristotle, as the four-legged class.  If there were, then there would be properties that float free of there subjects, and Aristotle is clear in saying that there are no such things.  Thus, the class logic rendering is semantic non-sense, given an Aristotelian metaphysics.  The property four-legged must be attached to something to exist, and hence some noun, namely, ‘animal’, must complete the predicate ‘four-legged’.  So, the proper semantics of ‘all dogs are four-legged’ is ‘all dogs are four-legged animals’.  

We can approach this in another way (this is not quite equivalent to the above).  Standard categorical statements are meant to make the reasoning process clearer.  At a minimum, the basic operations of conversion, obversion, and contraposition should not turn a categorical statement into non-sense.  If they do, then we may conclude that the original was incomplete. Consider:

1. No dogs are nine-legged

This is an E statement and conversion is valid for E statements.  The converse of 1 is

2. No nine-legged are dogs

But 2 is ungrammatical. So we went from a seemingly grammatical statement to an ungrammatical one via the valid operation of conversion (valid for E statements).  Something went wrong.  The solution is, it seems, to say that 1, while appearing to be grammatical and semantically complete, turns out, upon analysis, to be either ungrammatical or semantically incomplete (or both).  

We can approach this in still another way. If it were permissible to completely detach properties from their subjects (and thus predicates from their nouns) then a bunch of obviously bad arguments would turn out to be rationally permissible. For example, consider the following argument

X is a big ant
All ants are animals
Hence, x is a big animal

Yikes! What went wrong? One plausible analysis is that we were led to an obviously false conclusion because we regularly say things like ‘that’s big’ or ‘that’s small’ or more generally ‘x is big’ or ‘x is small.’ In such statements, the modified noun is left unspoken (or unwritten). But, the context has to provide it or else we won’t be able to make sense of the statement. We will go hunting for the modified noun. Thus, properties like bigness and smallness cannot exist apart from the object (actually objects here since these properties are comparatives) they modify and neither can predicates. 

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