Validity and Anaphora

The following argument appears valid:

Some deity is called 'Zeus.'
Zeus is wise.
Therefore, some deity called 'Zeus' is wise. (D. E. Buckner, Reference and Identity, 118)

Now if an argument is valid, it is valid in virtue of its logical form.  What is the logical form of the above argument? The following argument-form, Buckner correctly states, is invalid:

Ex Fx
Ga
Ex (Fx & Gx)

So if the form just depicted is the only available form of the original argument, then the validity of the argument cannot be simply a matter of logical form. And this is what Buckner concludes: "It is clearly the anaphoric connection between the premisses that makes the argument valid, but no such connection exists in the formalized version of the argument. "(119)

Buckner seems to be arguing as follows:

a) The original argument is valid. 

b) The only form it could possibly have is the one depicted above.

c) The argument-form depicted is plainly invalid.

Therefore

d) The validity of the original argument cannot be due to its logical form, but must be due to the anaphoric connection between its premises.

I do not find this argument rationally compelling. (b) is rejectable.  I suggest that the original argument is an enthymeme the logical form of which is the following:

1) For some x, x is called 'a.'
2) For any x, if x is called 'a,' then x =a.
3) a is G.
Therefore
4) For some x, (x is called 'a' and x is G).

Comments

[oz]

premiss (2) is false. more than one thing could be called 'a'. try replacing 'a' with 'john smith'.

[David Brightly]

Ed proposes a form for the following example argument:

1. Some deity is called 'Zeus.'
2. Zeus is wise.
3. Therefore, some deity called 'Zeus' is wise.

It seems to me that the argument is a straightforward application of the rules of inference Existential Elimination and Existential Introduction, back to back. There is an implicit inference,

[1a] Zeus is a deity,

by EE from (1). Then (3) follows by EI from (1a) and (2). In English the surface grammar of EE is a little different from the language of PC, but I think it's clear that (1) simultaneously makes an existential claim and introduces a name for a witness to that claim. In PC the argument would go

1. ∃x. Deity(x)
2. Deity(Zeus)
3. Wise(Zeus)
3. Therefore, ∃x. Deity(x) ∧ Wise(x)

Perhaps the question here is whether we should see the English predicate --is called-- as a logical predicate.

[oz]

Numbering correctly:

1. ∃x. Deity(x)
2. Deity(Zeus)
3. Wise(Zeus)
4. Therefore, ∃x. Deity(x) ∧ Wise(x)

How do you get (2) from (1)? In any case, 4 follows directly from 2 and 3 so you don't need (1). And (1) follows from (2), not the other way round.

>Perhaps the question here is whether we should see the English predicate --is called-- as a logical predicate.<

Well you can make up any predicate you like, the problem is the quote marks. You have

for some x is_called(x, 'Zeus')

but you cannot infer

is_called(Zeus, 'Zeus')

from it.

[David Brightly]

(4) follows from (2) and (3) by Existential Introduction, agreed. Likewise from (2) we may infer (1) by EI. But (2) follows from (1) by Existential Elimination. If there is some x such that Deity(x) then we may give it a name, ie, introduce a new logical constant, viz, Zeus, by which we can refer to this thing that witnesses that ∃x. Deity(x). For example, as in (3), state that it is wise. EE is the only mechanism in PC for introducing a new name beyond those that might be given initially.

[oz]

If EE, how do we derive premiss (3)? It does not follow from (1).

[oz]

EE is:

∃x Fx implies Fa

Are you saying that

∃x Fx implies Fa and Ga ?

[BV]

The first comment, @ 1:04 PM, misses the point. I am merely displaying the logical form of the original argument. Note that more than one thing can be called 'Zeus.'

[BV]

Ed,

You need to tell me whether you agree with my OP up to the point where I write, "I do not find this argument rationally compelling."

[BV]

David writes, >>Likewise from (2) we may infer (1) by EI. But (2) follows from (1) by Existential Elimination.<<

Surely from 'Zeus is a deity' we may validly infer 'Something is a deity.' I call this move Particular Generalization. Others call it Existential Generalization. Brightly calls it Existential Introduction (EI).

But does 'Zeus is a deity' follow from 'Something is a deity'? One thing is clear. If it is true that something is F, then, necessarily, there is at least one individual item a that makes it true that something is F. So if something is a deity, we may introduce by EE (or what I call Particular Instantiation) the term 'Zeus' to denote one of the items (perhaps the only item) that makes 'Something is a deity' true.

But that is not to say that 'Zeus is a deity' follows from 'Something is a deity.' For 'Zeus' was introduced as an arbitrary constant to denote whatever makes the general statement true. As such 'Zeus' functions like a free variable and not like a name. A sentence with a free variable in it such as 'x is a deity' does not have a truth value and so cannot figure as a constituent in an actual argument.

That is why (2) does not follow from (1).

[David Brightly]

Ed >> If EE, how do we derive premiss (3)? It does not follow from (1).

(3) is not an inference. It's a further claim about the witness claimed to exist in (1).

Ed >> Are you saying that ∃x Fx implies Fa and Ga ?

Certainly not! Just Fa where a is a new constant.

Bill >> For 'Zeus' was introduced as an arbitrary constant to denote whatever makes the general statement true. As such 'Zeus' functions like a free variable and not like a name.

My emphases. Isn't there a tension here? I'm pretty sure I have got EE (aka Existential Instantiation) right.

[BV]

Suppose that there are ten individuals each of which suffices to make '∃x Fx' true. EE sanctions the introduction of the arbitrary constant 'a.' To which of the ten individuals does 'a' refer? It refers to any of them. In this respect it is like a free variable, no? 'a' is not a name of any one of them.

I know that there are faithful husbands even if I can't name one. I also know that 'There are faithful husbands' cannot be true unless there is at least one nameable faithful husband. So I introduce 'Al' by EE as the term to denote one of these guys. Is 'Al' a constant or a variable or neither?

[oz]

Note also that (in English) the following is valid:

someone is called 'Zeus'
Zeus is a god
therefore Zeus is called 'Zeus' and is a god.

But that is not valid in pred calculus, because the rule is that "the constant c [here 'Zeus'] introduced by the rule must be a new term that has not occurred earlier in the proof, and it also must not occur in the conclusion of the proof.

[oz]

Bill objects “I am merely displaying the logical form of the original argument” and says “You need to tell me whether you agree with my OP up to the point where I write, "I do not find this argument rationally compelling.”

Replying. First, I do not agree that Bill’s argument (1)-(4) above displays the logical form of the ‘Zeus’ argument. Nor do I agree the original Zeus argument is an enthymeme.

Second, I do not agree with your OP up to the point where I write, “I do not find this argument rationally compelling” for I do not agree with your “b) The only form it could possibly have is the one depicted above.”

In my view, the logical form of the Zeus argument is given by the ordinary language of the argument itself. My point is that this ordinary language form cannot be captured by predicate calculus as it stands.

The issue of logical form is dealt with extensively in Chapter 2 ("Rules for Reference").

The question of what authors wish to convey through their work in general is an old and difficult question, but we cannot doubt their specific ability to successfully convey, which individual they are writing about. Our ability to comprehend a narrative involves keeping track of which character is which. There are about 31,000 verses in the whole Bible, and (from Aaron to Zurishaddai) about 2,000 characters. The biblical narrative would make little sense if we were unable to tell whether the same character was the subject of any two of those verses, or not. There are more than 700 occurrences of the proper name “Moses” in the Old Testament, and it is crucial to our ability to comprehend the work that we understand that these are not ambiguous names for 700 different people.” (p.20)

[oz]

Bill:

>For 'Zeus' was introduced as an arbitrary constant to denote whatever makes the general statement true. As such 'Zeus' functions like a free variable and not like a name. A sentence with a free variable in it such as 'x is a deity' does not have a truth value and so cannot figure as a constituent in an actual argument.<

Correct.