Adhominem

I’ve been thinking (for a while) about the mismatch between different ‘bits’ of logic. Take a standard example, which can be found throughout the literature on defeasible logics:

Bird(x) -> Flies (x)
Penguin(x) -> not Flies(x)

Bird(Tweety)
Penguin(Tweety)
?Flies(Tweety)

The point is to highlight the deafeasible nature of the knowledge, develop arguments, etc.

There are, basically, two bits to the formulae: the predicates (e.g. Bird(x)) and the inference rules (what to do when you encounter ->). Now, if you turn to any textbook of logic, which gets more coverage? - the inference rules. Indeed, in the two I’ve been using (Schum’s Outline of Logic and Kelly’s Esssentials of Logic), much more weight is given to the inference procedures than to the predicates; in fact, the only time the predicates are really discussed is to explain the prefix syntax.

I think this has some (really) quite odd effects. Many of the defeasible systems make a distinction between ’strict rules’ which deduce ‘facts’ and defeasible rules that deduce ‘assumptions’ or ‘defeasible facts’. The problem is, these ‘fact’ predicates are rarely properly defined; in fact, they are rarely defined AT ALL. Indeed, given how tight everything else is in the descriptions of logic languages, the abandon with which one can introduce a new predicate is, frankly, shocking. It is this distinction that I am trying to point out when I talk about Ontology vs. Inference. The conventional treatment of problems in logic concentrates too much on one, and not enough on the other. The end result is that we harness a wonderfully powerful inference mechanism to some really shoddy foundations: and then we wonder why it goes wrong.

Consider the following example:
Man(x) -> Mortal(x)
Man(Socrates)
?Mortal(Socrates)

The answer (according to every textbook I have ever read) is yes: Socrates is Mortal. That’s fine, but in that case:

Woman(Sappho)
?Mortal(Sappho)

Should be “No”. Now, Sappho was most certainly mortal, as her suicide proves. So where did we go wrong? Well, we committed the fallacy of ‘equivocation’, in that we have confused ‘Man’ as ‘Man as separate from Woman’ with ‘Man as representing all of humanity’. But the reason we have been able to do so is because there is no definition of Man. No one told us what it meant, and so either interpretation is reasonable. This fuzziness is because there is no underlying ontology defined.

Unfortunately, it gets worse; even if we try and define an ontology, we run into problems. We might try and divide people into Men and Women; such a distinction seems natural enough, until you reach the edge cases: children, pseudo-hermaphrodites, true intersex and those with testicular feminisation. All of these cause different problems for any classification criteria you might want to choose.

The real problem is that an ontology is a model; and like all models, it fails sometimes. John Sowa has been trenchant in his criticism of ontologies that purport to offer the ‘truth’ about the world, and points out that the attempt to represent a continuous, non-discrete world in discrete lumps is bound to fail.

And so we come back to the problem with logical inference. The rules of inference may be sound, but the formulae they apply to are built on treacherous sands and, try what you might, the sands cannot be made solid. We could certainly solve the problems with the intial models by committing to an ontology, but at some point our ontology will fail. And even where it doesn’t fail, there are still exceptions (like Tweety).