by JLS
for the GC
AS J REMARKED in commentary in this blog, Grice omits the well-known logical equivalence (pun there) between a statement to the effect that "iff p, q" and one to the effect that p ≡ q.
Why?
He lists seven 'formal devices' in the locus classicus: "Logic and Conversation":
1. ~
2. /\
3. \/
4. )
5. (x)
6. (Ex)
7. (ix)
---- Surely "≡" would fall immediately after 4.
As I note, Horn and others (including D. F. Pears in the Canadian Journal of Philosophy, early 1970s) have noted the implicature of 'if' to the effect of 'iff' -- but that does not say much about 'if'.
When reading Horn, I realised HOW literal, 'if and only if' is:
for
'if and only if'
is a conjunction:
a. if
b. if
----
what about the 'and only'?
--- Surely, this has to do with necessary and sufficient conditions. But why bring 'only'?
Atlas has devoted a good essay to this: The importance of being "only". But I do not think it deals with a reading of "≡" as "if AND ONLY if". But it should!
Etymologically, 'only' is of course a weird sort of adverb. The '-ly' qualifies an adjective, 'one'.
As when we say,
"One clown was funny".
"Only one clown was funny".
If 'beautiful' yields 'beautifully', surely 'one' can yield 'onely'.
For some reason, circa 1400 or something people started to spell 'onely' as 'only' -- to confuse _me_.
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Boolos includes <-> . Given that Sir <-> is a tautology and quite different, truth functionally speaking, than our old friend Mr. ->, the inclusion seems warranted. Doesn't The Wittster (as in Ludwig) include it in Tractatus-ness? Ich denke so.
ReplyDeleteYes, Grice SHOULD have included it! Recall this is a mimeo. He never titled ANY of his Wiliam James lectures. It justs starts, out of the blue, "It is a commonplace in philosophical logic that "if" is the horseshoe" and so on. As Martinich said, "It never takes up the topic no more" or words. But it does. Grice had discussed "if" in the ONLY quote direct from Strawson in WJ I. In WJ II he mentions "if" again and ), and in WJ III he goes to modified Occam's razor, and in IV he re-entitles it years later in 1987, as "Indicative conditionals".
ReplyDelete----- Nothing about 'if and only if', indeed. Of course it may have been Mr. Grice's most famous enemy. Between 1948 and 1968 his friends in Oxford would not leave HIM along but with bothering him, like flies, with counterxamples to what he later calld the 'prong of necessity' (my meaning too strong?) and the 'prong of sufficiency' (my meaning too weak). He dedicates WJ V then to just what we mean by 'iff' when it comes to one particular analysis of one elucidation in terms of the paradox of analysis.
Grice quotes from Boolos in his "Vacuous Names" -- along with other logicians he credits the acknowledgment note. Since Boolos was Cambridge, Mass., I like to think he attended the WJ. He died recently, of cancer, I believe. One of the most entertaining logicians of all time!
But again, 'if and ONLY if' is perhaps a 'vulgar' thing to say. In a way, 'if' is ALSO a pretty vulgar thing to say. As Alice says in the book, "I only said if". "Did NOT! You said a great deal more than THAT!" the wonderful wonderland critters reply!
It sounds as if you think the "ONLY" is otiose, so I'm going to explain why it is essential I and apologise if this is something with which you are thoroughly acquaitained, but are engaged in some badinage too subtle for my comprehension.
ReplyDelete"If and only if" and "≡"
Well we know that A ≡ B means the same as (has the same truth table as in most boolean logics):
A ) B /\ B ) A
So its definitely not quite the same as "if".
Note that "A if B" is "B ) A" its like horseshoe with the arguments the other way round.
The "only" comes in to make an "only if" which is exactly the same as the horseshoe:
"A only if B" means the same as "A ) B".
So we have "A if B" and "A only if B" means the same as "B ) A /\ A ) B" which is "A ≡ B".
So, in iff, the "if" is implication in one direction, and the "only if" is implication in the other direction. Both together give you BI-implication aka logical equivalence.
I apologise again, its one of my failings to give literal responses to things that were intended as jokes, often because I didn't get it, and sometimes because I did but no witty rejoinder comes to mind.
RBJ
No. Explanation welcome.
ReplyDeleteI suppose a more verbose way to say this would be:
"If p, q, and if q, p."
That version above does NOT feature "only" and of course does feature "and" (&), which is just the standard way to state this
p iff q
(p ) q) & (q ) p).
So, one wonders -- but then it IS late -- where the syntax for the "if" comes from, such that it inverts the order:
"p if q"
is then "p ) q".
And then we have:
"p if AND ONLY if q".
So I should revise Jones´s explanation.
In particular:
"The "only" comes in to make an "only if" which is exactly the same as the horseshoe: "q only if p" means the same as "p ) q"."
I suppose the reason eventually trades on the idea of necessary AND sufficient condition.
From wiki´s entry for "necessary and sufficient condition":
"Formally, a statement q is a necessary condition of a statement p if p implies q.
...
Formally, a statement p is a sufficient condition of a statement q if p implies q."
-- where of course "implies" is vague and it´s best to read it as ... HORSESHOE!
Or something!
Well, I guess it was late!
ReplyDeleteYou got the if and the only if the wrong way round (the opposite of what I said). You did say you were revising what I said, but why is a mystery to me.
On the other hand, what you say about necessary and sufficient conditions is right, subject to the caveat I just posted on a different thread about their modal status.
RBJ
I guess I was referring -- but mind, it is late as I write this, too! -- to Jones's:
ReplyDelete""A only if B" means the same as "A ) B"".
and a syntactic explanation (rather than semantic?) as to why the adverbial modification, 'only' produces the inversion of arguments like that?
"A only if B" still seems to have, as per the surface syntax form, B as the antecedent or protasis -- but this is no longer the case in the rewrite given above. Or something.
Reflecting on this I am struck despite my previous complaint about he modalities, by the explanatory power of the sufficient and necessary conditions talk, if one can manage to detach the modal connotations and think in terms of simple truth values
ReplyDelete"A if B" means "if B then A" i.e. The truth of B is a sufficient condition for the truth of A.
"A only if B" means "the truth of B is a necessary condition for the truth of A"
i.e. if A is true then B will is also be true.
Not sure whether this helps or not, probably not!
Part of the problem is that when we convert the conditional phrase "If A then B" into something that looks more like an infix operator in ordinary language, we could use implies which is the right way round but perhaps suggests relevance and if we use "if", then we find that we have changed the order of the arguments:
"A if B" means "If B then A"
and the one with the arguments the right way round is "only if":
"A only if B" means "if A then B"
so if you don't note carefully that "A if B" is the "wrong" way round (i.e. opposite to "if A then B"), then "A if and only if B" really looks like a pleonasm.
RBJ
Jones:
ReplyDelete"if you don't note carefully that "A if B" is the "wrong" way round (i.e. opposite to "if A [...] B"), [...] "A if and only if B" really looks like a pleonasm."
Good. It couldn't be just me! Of course you mean: it looks like a pleonasm but it ain't!
---- Still, this from wiki, then, while I re-read what you wrote --
"if and only if", wiki:
"It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its pre-existing meaning. Of course, there is nothing to stop us stipulating that we may read this connective as "only if and if", although this may lead to confusion."
Or pleonasm, but it shouldn't lead to any -- with R. B. Jones's help!