Tuesday, August 30, 2011

More on predicate logic and direct reference

David Brightly asks here whether modern predicate logic (MPL) in fact rests on ‘weak’ (relativistic) reference, not strong (direct) reference. He appeals to the way that singular sentences in MPL seem to have a meaning even when the singular term is empty.

There are a number of connected reasons why this won’t work. Here are the main characteristics of the relativistic logic (RL) I have defended here.

  • In RL there are two forms of negation for singular sentences. There is a narrow form ‘nonF(a)’, which is true when a exists and it is not the case that Fa, i.e. a is a nonF; and a wide form, which is true when either nonF(a), or when a does not exist. Excluded middle applies only to the wide form, naturally. By contrast, MPL has no such feature.
  • In RL, a sentence of the form ‘An S is P’ has a narrow existential sense: it is convertible with ‘An S-P exists’. There is no distinction between a wide sense conveyed by ‘some’, and the narrow sense conveyed by ‘exist’. RL is not Meinongian. By contrast, it is open to MPL to invent an existence predicate ‘E()’, which may satisfied by some members of the domain, but not by others.
  • As a direct consequence, in RL, the wide negation ‘it is not the case that Fa’ never implies the existential form ‘some x is non-F’. By contrast, in MPL a singular sentence is existential, at least in the wide sense: ‘~Fa’ implies ‘Ex ~Fx’.
  • A further difference, though probably not relevant here, is that some relational statements in RL are not existential. ‘aRb’ does not always imply ‘Ex aRx’, namely in the case where ‘R’ is not logically transitive. This is how RL avoids the problem of intentionality without invoking Meinongian non-existent objects.
The net result is that if a singular sentence in MPL cannot be empty in the sense required by RL. Let ‘v’ be a term that purports to denote Vulcan (a non-existent planet). Then either Vulcan lies within the orbit of Mercury - OM(v), or not - ~OM(v). The problem is the negation. If ~OM(v), standard predicate logic implies Ex ~Fx. But that leads to Meinongianism, i.e. positing objects that do not exist, and this, as I have argued, is a form of direct reference. Direct reference is the thesis that if ‘Fa’ is meaningful, then ‘a’ refers to something - possibly a non-existent something.

Another argument: how can we even say in MPL that Vulcan does not exist? The sentence ‘~Ex x=v’ will not do, for it asserts of something in the domain that it is not in the domain; ‘~E(v)’ will do, where ‘E’ means ‘exists’, but this comes at the price of Meinong’s junkyard. For ‘~E(v)’ implies ‘Ex ~E(x)’, which is precisely Meinongianism. Or it can be denied that ‘OM(v)’ means anything at all, which is strong Direct Reference of the familiar variety. There is no escape. Either we adopt a radically different semantics and inference schemata, on the lines of RL above, or we are left with Direct Reference.

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2 Comments:

Blogger David Brightly said...

1. You seem to be suggesting that if the relative reference theory is adopted as our understanding of how reference 'works', ie, as an account of the semantics of proper names, then we are obliged to adopt RL. I'm still not convinced. Can we not express, say, the Aeneas story in MPL *together with some existence assumptions*? A useful way of assessing the truth of a story is to check its consistency. An inconsistent story cannot be true. But with a relative theory of reference consistency is independent of existence considerations. And this can be done in MPL.

2. >> how can we even say in MPL that Vulcan does not exist?
We can't. But we can say that nothing exemplifies the salient properties of the putative Vulcan.

3. >> If ~OM(v), standard predicate logic implies ∃x ~Fx. But that leads to Meinongianism.
Does it? In the informal MPL argument I give, ∃v.~OM(v) is not inconsistent with ∃v.Planet(v) previously *assumed*. Note, though, that my 'v' is a *variable* where yours is a *constant*. Can their semantics differ so much?

4. My Vulcan argument makes informal use of MPL language, in particular the existential quantifier, just as my earlier root 2 argument did. My worry about these arguments is that for a strict application of MPL we need to specify the domain we are quantifying over, and that's not clear at all. However, we can tighten up the Vulcan argument by quantifying over 'ways of arranging matter within the orbit of Mercury'. We would show that there is no matter distribution that counts as a planet, is too small to be visible, yet is large enough to affect Mercury in the required way. But I find that the informal argument comes quite naturally---it's the way I was taught to think! My question is, Can this informal style of argument be formalised without resorting either to the somewhat unnatural rendition I allude to above or to a formulation in RL? A second question is How would the Vulcan argument go in RL? I'd very much like to see how you do this.

5. Thanks for the clarification regarding DR and Meinongianism. I understand now what you mean by "if ‘Fa’ is meaningful, then ‘a’ refers to something - possibly a non-existent something." But in my informal Vulcan argument, 'Planet(v)' is meaningful and 'v' refers (relatively) to the same planet as the one whose existence is assumed in an earlier premise. I don't see this as Meinongian. Do you?

6. I think that we'd agree that in 'the Greeks worshipped Zeus' the name 'Zeus' does not refer. Just out of interest, what about 'Aston Villa' in 'Nigel supports Aston Villa'?

4:17 pm  
Blogger Edward Ockham said...

>>Can we not express, say, the Aeneas story in MPL *together with some existence assumptions*?

You would need to clarify what 'existence assumptions' these are. According to the relativity theory, most statement in fiction are false (except for true statements in historical fiction, of course). That includes apparently contradictory statements. On the assumption that Aeneas never existed, 'Aeneas founded Rome' is false, and so is 'Aeneas did not found Rome'. That is not because RL denies excluded middle. Rather, the second statement is not a genuine negation of the first. And the point of doing that is to avoid the existential implication of 'Aeneas did not found Rome'.

>>we can say that nothing exemplifies the salient properties of the putative Vulcan.

Which would be the approach of the theory of descriptions. But as I have said, there are strong arguments against names being descriptions, all of which RL concede. But RL does not concede the dichotomy "either names are descriptive or names directly refer". Names are not descriptive. Nor do they directly refer.

>>Note, though, that my 'v' is a *variable* where yours is a *constant*. Can their semantics differ so much?

A variable cannot occur in a separate proposition. I say

There is a unicorn called 'Frank' in the garden.
Frank is white.

The second proposition is distinct from the first. But we cannot say

Ex [unicorn x & calledFrank x & in my garden x ]
x is white

This is not well-formed.


>>How would the Vulcan argument go in RL? I'd very much like to see how you do this.

I haven't really thought about formalising it. In MPL there is a step that involves 'interpreting' the logical constants. In RL, you wouldn't bother. You would just introduce constants as you liked (much as in fiction you introduce characters). The default assumption would be that there is an object corresponding to the name, i.e. that Vulcan exists. But if you can derive a contradiction, you can challenge this assumption, thus showing that it does not exist.

>>what about 'Aston Villa' in 'Nigel supports Aston Villa'?

Difficult. If the truth conditions of this involve physical facts like going to matches etc, then it's logically transitive. Otherwise, I am not sure.

8:21 am  

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