logicandlanguage.net

Well, you might try making "essentially" a kind of modal operator (represented here by the box) and write:

Fine argues (quite convincingly) that this would be wrong, since this next sentence would be translated the same way and is false, where the above sentence is true:

It is an essential property of Socrates that he is a member of the set singleton Socrates.

(Having trouble getting the right intuitions about these sentences? Fine thinks of essential properties of an object as being those that are had by virtue of its nature. The thought is that it is part of the nature of singleton Socrates that it contain Socrates, though it is not part of Socrates' nature that he be a member of any set---it doesn't even follow from his nature that sets exist. Fine's project in this paper is that of "developing a logic of essence, not now as a fragment of a modal system, but as a system in its own right" (241))

Instead we have, for each predicate F, an operator:

and the predicate F picks out the subjects of the essentialist claim - that is, the objects (if any) whose natures underwrite the truth of the claim. We call predicates playing this role "delimiters."

To formalise "Singleton Socrates essentially has Socrates as a member", we need a predicate which applies only to singleton Socrates (singleton-Socratises? lx(x=Socrates)? (l is for "lambda"), say, F, and then we use the associated operator:

This can be read as "Rab is true in virtue of the nature of the objects that F". So where a refers to singleton Socrates, R expresses the membership relation, b refers to Socrates himself and F expresses the property of singleton-Socratising, this formula says that Socrates is a member of singleton-Socrates in virtue of the nature of singleton Socrates.

There is another piece of unusal notation that you need to understand the rigidity axiom. Fine doesn't only represent predicates using single letters. He also uses lambda abstraction and some abbreviations of expressions so formed. Most relevant here, he writes (y) for lx(x=y), where x is the first variable distinct from y (i.e. an predicate for the property of being identical to y) and in using such abbreviations in delimiters, he sometimes abbreviates them even further by removing the parentheses. Hence

can be read as "A is true in virtue of the nature of objects which are identical to y" (that is, of course, in virtue of the nature of y.)

Finally, it seems that if we have more than one predicate in the delimiter space, separated by a comma, that means that the inside formula is true in virtue of the objects which fall under the disjunction of the predicates. (I say "it seems" because I can't find a place in the text where this is stated, but I think it makes sense of what comes later.)

So now we have the resources to understand the rigidity axiom:

Roughly what this says is that if Px is true, then it is true in virtue of the natures of x and the objects which fall under P. Even more roughly, Px is true (if it is) in virtue of the natures of the things which it is about. Fine says that the axiom is "clearly correct" and gives a proof to demonstrate this correctness:

For if x is one of the objects x1, x2..., say xi, then it is true in virtue of the nature of x that it is xi, and hence true in virtue of the nature of x1, x2...that x is one of x1, x2....

What I'm puzzling over is this. I'm not sure that sentences are always true in virtue of the natures of the objects that they are "about," that is, the objects which they refer to or which fall under the predicates they contain. Take a sentence like "fred is a frog or it is not the case that fred is a frog." Is that true in virtue of the nature of Fred, or froggy things? One reason why you might think it isn't is that it would be true regardless of Fred's nature, or the nature of Froggy things. The expressions could even apply to things entirely unfroggy objects which are entirely unlike Fred, and the sentence would still be true.

So does that mean that there's something wrong with Fine's proof - er, no, though in the spirit of the intuition I've just drawn on I might deny that it is true in virtue of the nature of x that x is identical to xi. (After all, things are self-identical regardless of their natures, aren't they?)

But what I suspect is really going on here is that there are several senses of "in virtue of". Channeling the Quine of "Carnap and Logical Truth" for a second, we might say that there's a sense of "in virtue of" on which "Fred is a frog or it is not the case that Fred is a frog" is true in virtue of Fred's nature, and a sense in which it is not. It really doesn't make any difference which we choose to use (and so Fine is perfectly justified in using his sense.)

Here's another place where the assumption comes in - in the following rule of Fine's system:

This says that if you've proved A as a theorem, then it is also a theorem that A is true in virtue of the natures of the objects contained in the proposition expressed by A (Fine is working with a very Russellian conception of propositions) which seems like a fairly explicit committment exactly the claim I was puzzled about, namely that sentences are always true in virtue of the natures of the objects the refer to or which satisfy their predicates.

"in virtue of" is puzzling...

Two questions:

1. You wrote:
"Take a sentence like "fred is a frog or it is not the case that fred is a frog." Is that true in virtue of the nature of Fred, or froggy things?"
You then suggested that the answer is no, on the grounds that its truth-value is independent of the natures of Fred and of froghood. Could one reply that it is just very general features of Fred and froghood that are being appealed to here> For example, just because Fred shares the property of self-identicality (ugh) with every other object in the domain of discourse does not entail that that property is not essential to Fred. And if we wanted to stick to your excluded middle case, maybe thinking about cases of presupposition failure would make it clearer that the truth-value of this sentence does depend on features of Fred and froghood. "Santa Claus has a beard or it is not the case that Santa Claus has a beard," on negative and neutral free logics, does not come out true. Similarly, if we consider a predicate that is ambiguous (i.e., it is associated with more than one extension), then the law of excluded middle will not obviously hold. So there is something about Fred and froghood that makes the law of excluded middle come out true for them.

2. You paraphrase Fine as saying:
"if Px is true, then it is true in virtue of the natures of x and the objects which fall under P. Even more roughly, Px is true (if it is) in virtue of the natures of the things which it is about."
So, on Fine's view, are there really no accidental predications? I haven't read his original article-- have I misunderstood him and/or you?

"So there is something about Fred and froghood that makes the law of excluded middle come out true for them."

Doesn't it seem more likely that there is something about the terms (i.e. the fact they designate) and the semantics that makes it come out true (or false wrt 'Santa Claus')?

You may argue that the truth of "Fred is a frog or it is not the case that Fred is a frog" relies on features of Fred (and froghood) but how would explain the truth of "Unicorns are red or it is not the case that unicorns are red" since there is nothing that is a unicorn?

Hi Greg,

Thanks for your comments. You're right that my interpretation of the rigidity axiom makes it very strong - though of course that doesn't mean Fine doesn't endorse it. Your comment made me wonder again whether I've interpreted the commas correctly...

...so I went back through Fine's (enormous) list of notations and abbreviations for notations. He says that []x,yA abbreviates []lz((x)zV(y)z)A. (In these expressions '[]' is the box, 'l' is lambda, and everything except the box and the subformula 'A' is meant to be in subcript (as part of the delimiter.) '(y)' is, as already mentioned, an abbreviation for the predicate expression 'lx(x=y)' (i.e. it's interpreted as something like the property of being y. So it seems that '[]P,xA' must be an abbreviation of []lz((x)zVFz)A - that is, informally, A is true in virtue of the nature of x, or of the nature of the objects which F - as I thought originally.

That doesn't mean that the axiom of rigidity isn't compatible with the distinction between essential and accidental properties, of course. It's not so strong that it commits to Px->[]xPx (with the first 'x' subscripted to the box), but it does seem to have the surprising consequence that if 'Px' is true and yet not true in virtue of the nature of x (that is, it is merely an accidental property of x) then it is true in virtue of the nature of the objects that P (including x, of course.) At the moment this looks wrong to me.

Anyway, I'm still reading the article, so I'll add a comment here if I discover anything that would help.

There's one point in my original post that I wanted to retract though - at the end I wrote:

This says that if you've proved A as a theorem, then it is also a theorem that A is true in virtue of the natures of the objects contained in the proposition expressed by A (Fine is working with a very Russellian conception of propositions) which seems like a fairly explicit committment exactly the claim I was puzzled about, namely that sentences are always true in virtue of the natures of the objects the refer to or which satisfy their predicates.

From "which seems like" onwards is completely wrong; what the rule says is that if A is a theorem, then it's also a theorem that A is true in virtue of the nature of the objects contained in the proposition it expresses. That's not the same as saying that if A is true, it's true in virtue of the nature of the objects contained in the proposition it expresses.

I would like to pose two very simple questions:

1. If the box corresponds to essential and not necessary, what the diamond symbol ◊ corresponds to in the Fine system you are talking about?
2. Is the essence operator the primitive of the system or not? If not, how is it defined?

I am really curious about these two things, since I have not Kit Fine's paper.

Hi Tony,

The expression you get by writing a diamond, followed by a subcript predicate letter, followed by a sentence letter (e.g. "<>F A" (that's the best I can write it in these comments) is an abbreviation of "¬[]F¬A" and you can think of it as saying that it is not the case that in virtue of the nature of the Fs, not-A - that is, if you like, the nature of the Fs permits A.

And yes, the "essence" operator is a primitive in the sense that it isn't a notational abbreviation for any other expression, (though it does have formal and informal interpretations in a metalanguage.)

Thanks Gillian,

It is always good to talk to an important logician to learn more about those things, especially someone from an important family of logical ancestors.
If I am not mistaken, one standard idea is that modal systems are all consistent. João Marcos has proposed that, on the other hand, every modal system is paraconsistent. Now, talking about tautologies and ⊥, do we get any differences in terms of truth-tables and axioms if we take the box to be an essence operator rather than a necessity operator in a system like Fine's? Maybe it is just me, but I haven't been able to guess what is the practical advantage of talking in terms of essence rather than necessity or obligation.

Fine's E5 does have a distinct semantics and proof system from any modal system I've ever seen, though some of the axioms correspond to those of S5. You could check out his expositions in "The Logic of Essence" and "Semantics for the Logic of Essence" for the details.

Maybe it is just me, but I haven't been able to guess what is the practical advantage of talking in terms of essence rather than necessity or obligation.

Ah, but Fine isn't proposing that we give up talk of necessity for talk of essence. He merely argues that talk of essence is not reducible to talk of necessity. He thinks that there is more to being P essentially than simply being P necessarily.

It might help to realise that the motivation comes from trying to make sense of a metaphysical view that looks fantastically extravagant to some. Not everyone has a very good intuitive grip on what it is to be an essential property, and plenty of people last century (Quine especially) have thought that the very idea was confused. Fine is trying to do the hard work of developing the view against such opposition.

(p.s. I'm as fond of flattery as the next guy, but there ain't no important logicians in my family. I come from a respectable line of shale-miners and salmon poachers. You must be thinking of those upity sasenach Russells from south of Hadrian's wall.)

Oh! I honestly assumed you were Bertrand's grandson. It would make sense. Sorry, then.

I'd be seriously surprised to learn that Gill were Bertie's grandson. She doesn't look anything like him.

Sorry again. 'Gillian' in Brazil is a boy's name. (I had two male schoolmates who were 'Gillians'!) Oh, I am terribly ashamed! Thousands of apologies!