On the Enforcing and Permitting of Coreferentiality by Argument-Forms

This argument is invalid:

Cicero was a Roman
Tully was a philosopher
-----
Some Roman was a philosopher.

Quite simply, there is no middle term. The example is an instance of the dreaded quaternio terminorum. But of course we learned at Uncle Willard's knee that Cicero = Tully. Add that fact as a premise and the above argument becomes valid. As a general rule, any invalid argument can be rendered valid by adding one or more premises.

So sameness of reference is not sufficient for sameness of name. 'Cicero' and 'Tully' have the same reference, but they are different names. They are both token- and type-different. Since they are different names, that fact must be accommodated in the form diagram, which looks like this:

Fa
Gb
---
(Ex)(Fx & Gx).

This form is clearly invalid. The most one can squeeze out of these premises using Existential Generalization is '(Ex)Fx & (Ex)Gx.'

It is worth pointing out that the use of the different signs 'a' and 'b' does not entail that a is not identical to b; it leaves open both the possibility that a = b and the possibility that ~(a = b). It is because of the second of these possibilities that the argument-form is invalid.

Commenter Edward  Ockham in a comment on the old blog wanted to know  why, given that we had to add a premise to make the Cicero argument valid, we don't have to add a premise to make the Alexander argument valid. That argument, from the days when men were men and went around 'seizing' women, proceeds thusly:

Alexander seized Helen
Alexander did not seize Helen
-----
Someone seized and did not seize Helen.

Ockham wants to know why we don't have to add an identity premise to secure the validity of this argument. But what premise would he have us add? It can't be 'Alexander is Alexander' for that is necessarily true and therefore true whether or not both occurrences of 'Alexander' in the original argument are coreferential. Presumably, Ockham wants us to supply '"Alexander" is coreferential in both of its occurrences.' But this goes without saying. There in no need to affirm this in a separate premise since it is implied by the fact that 'Alexander' in both occurrences is a token of the same word-type. We needn't say what is plainly shown. (He said with a sidelong glance in old Ludwig's direction.)

Ockham is bothered by the possibility of equivocation. Well, either there is an equivocation on 'Alexander' or there isn't. If there is an equivocation, then the argument instantiates an invalid form, and Ockham's contention collapses. If there is no equivocation, then the argument instantiates a valid form but it is not the case that both premises are true; so again Ockham's contention collapses. Either way, his contention collapses.

In a further comment, Ockham writes:

Either we capture the reference [of a name] in the form, and my objection collapses. Or you concede that the form covers only the visible or audible outward form of the word. In which case, my specious Alexander argument really does have the right form, and we have to add on the condition about reference, and my point stands.

I grasp something like the first horn. If 'a' occurs two or more times in a form diagram, then no argument of that form has an equivocation on a term whose place is held by 'a.' This is to say that the form diagram enforces coreferentiality on any terms whose place is held by 'a' in the form schema. Otherwise, the argument would not be of the form in question.

Ockham wants to have it both ways at once. He wants his argument A to be of valid form F without F enforcing coreferentiality on the occurrences of 'a' in A. This is just impossible. If there is an equivocation on 'a,' then A does not instantiate F. But if A does instantiate F, then there cannot be any equivocation of 'a.' Why? Because the form does not permit it. The form enforces coreferentiality.

Now look back at the Cicero argument. It is invalid because its form (depicted above) is invalid and the argument has no valid form. But I don't say that the invalid form enforces lack of coreferentiality on the singular terms whose place is held in the diagram by 'a' and 'b.' I say instead that the invalid form permits coreferentiality of these terms. Thus there is an asymmetry between the Alexander and Cicero cases.

Related articles

Abbreviations, Place-Holders, and Logical Form

Three More Putative Instances of Valid Is-Ought Inferences

The Stromboli Puzzle Revisited

Transitivity of Predication?

Comments

[london ed]

Goodness we were arguing about that even then. Bill, please note that the Powerblogs link gives a 500 (internal server) error.

>>If 'a' occurs two or more times in a form diagram, then no argument of that form has an equivocation on a term whose place is held by 'a.'

So according to you, the following argument

Alexander seized Helen
Alexander did not seize Helen
:. Someone seized and did not seize Helen.

is not of the form

a is F
a is G
some F is G

when 'Alexander' is equivocal. At that time I was reading Lukasiewicz, who is very strong on 'form' being the outward and visible manifestation of a sign. As I understand him, he would say that the 'Alexander' argument does have the same form as the placeholder argument below. This is because the first token of 'Alexander' is spelled in exactly the same way, i.e. has the same letters in the same order as the second token. Indeed, we couldn't even talk about the 'first' and the 'second' token of that name type unless that were the case.

Or do you hold that, when the names are equivocal, i.e. have different meanings, then they are not tokens of the same type. I.e. the first occurrence of 'Alexander' is not a token of 'Alexander', nor is the second. But then how can I even talk about 'the first occurrence' of that name? It's very confusing.