by JLS
for the GC
Further to Jones´s comments in thread on "iff", this from
http://en.wiktionary.org/wiki/Talk:only_if
-- just quoting! Not analysing it!
From
G. Chartrand, A. Polimeni, and P. Zhang,
"Mathematical Proofs: A Guide to Understanding the
Basics of Abstract Mathematics and Constructing and Writing
Proofs of Your Own, as it stood at the end of 1999"
Chapter 2: Logic, pages 27–28.
"We have already mentioned that the
implication "p ) q" can be expressed as both
“If p, q” and “p implies q”. In fact, there are
several ways of expressing "p ) q" in words,
namely:
If p, q.
q if p.
p implies q.
p only if q.
p is sufficient for q.
q is necessary for p.
"It is probably not surprising that the first three of these say
the same thing, but perhaps not at all obvious that the
last three say the same thing as the first three. […]"
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On the whole the defects here are the defects of material implication, and, that defect understood, "only if" is just as good as the others.
ReplyDeleteThe last two fare badly however, because of the modal connotations of the terms "sufficient" and "necessary" which I would say rules out these locutions from contexts in which material implication is intended.
RBJ
Very good. I would rule them out TOO. I'm surprised logicians (or logic instructors, rather) kant be more careful about that.
ReplyDeleteThe ONLY way to represent 'necessity' is via a special operator to represent "must", to wit:
□
This, off (the) course, operates on p to yield
□p
And then we have the combos:
□p ) □p
and
□(p ) q)
---- and it's THERE -- as we get into deeper waters that some think a horseshoe won't do and bring in the fishhook.
Note that if we stick to "content" as Kramer uses "information" -- we may need to provide a predicate-calculus version:
□Fx (free-variable)
or
□Fa
--- which off the course is a bit of an essentialist thing for a few extensionalists to digest (vide Christopher Kirwan).
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ReplyDeletethe material implication makes sense--as does sufficiency and necessity-- when the conditionals are used as class membership, however primitive (ie felines -> mammals. isoceles triangle-> triangles). But when used with propositions, or quasi-causal statements things get fugly. And -p v Q also becomes -(p & -q) (demorgan ah believe). So in effect the conditional is also a denied conjunction (ie can't have a cat that's not a mammal)
ReplyDeleteMany logic teachers start with propositions (and "ordinary language" types probably part to blame) when they'd do better starting with Aristotle's Square of Opposition (which Frege himself updated with quantifiers).
Yes, I think 'ordinary language types' (read: Grice) are partly to blame for emphasis on 'proposition'. Or, as the first in greats in Lit. Hum. that Grice was would prefer, the 'that'-clause.
ReplyDeleteSurely you cannot get a 'that'-clause with cats and mammals!
The meeaning of a subsentential item (such as "if", "only", etc.) is the contribution to the meaning of the proposition to which it belongs.
Consider Joan Rivers:
"How old are you?"
"Only 74".
Strictly, "74" seems just as informative. So, there is a 'way' or sense in which 'only' has to have to be paid 'extra' to do the extra job. Note that K. Ryan wrote a song, "If only" -- NOT: "Only if".
But you'll argue that the order of things cannot alter the product:
Why is it that "if only" and "only if" trigger (or 'project') different imlpicatures?
Some people DO say,
"If only!"
but it's best to figure out their implicatures. Otherwise you get the reprimand as the one Alice gets from the White Queen:
"If I really am a Queen," Alice said, "I shall be able to manage it quite well in time.'
"What do you mean by "If you really are a Queen"?"
"What right have you to call yourself so?"
"I only said "if"", poor Alice pleaded in a piteous tone.
The Red Queen looked at the White Queen and remarked, with a little shudder:
"She says she only said "if"".
"But she said a great deal more than that' the White Queen moaned, wringing her hands. "Oh, ever so much more than that."
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ReplyDeleteYes -- but don't Mention "Necessity". Recall the Isle of Wight and Cowes Week: "to live is not necessary: to sail is necessary". Also Jones's point about 'necessary' being strictly modal, so not a matter of the horseshoe but the fishhook.
ReplyDeleteIt is used for emphasis. How it reverses the order of protasis and apodosis is a fact of English -- not Japanese.
(But hey, they don't have anything like 'if'. I tried to learn it once, the old method. So went to Japan (Tokyo) and said 'if'. Apparently it means 'cigarette').
Anyway, beware 'antec' and 'conseq'. If you are seriously into terms, and defining them in non circular ways, it is a bit otiose to say that the consequent is a consequence of what is not a consequent, but the antecedent. Try postcedent.
Decided to delete after savoring Carroll's (er, Dodgson's) bon mots on singular statements. Note that Alice is essentially correct, and uses the conditional as hypothetical--but the Reginas react as if she offered, like, a fact-statement (and as with most Carroll, there are political implications ..)
ReplyDeleteOK. For propositions, or ord.lang. and whimsy-logic I agree the antecedent/consequent seems a bit...strict. Yet...in a formal setting...i.e. first order logic.., the ordering does seem critical (and perhaps an issue for the material implication..IM a bit exhausted to think about it now). Mammalness is a necessary condition of...felineness; lungs are a necessary condition of breathing. That seems to get lost in the -P v Q translation, BUT the -> has the same truth value as mat.imp. Not really that profound, unless one takes the conditional to be profound (while not anti-logicist, at times I side with Toulmin-ists and late Wittgenstein in regards to the ultimate value of Formal Logic. It may help with some matters (like programming), and as a "heuristic", but ultimately it's rather limited and impractical...).
ReplyDeleteYou Kant be exhausted with the material implication. Have an orange juice, tea with scones, play a game of something, and be back! The material implication is INexhaustible!
ReplyDeleteYes, there is something idiotic in translating
"p ) q" with "-p v q". The whole point of uttering 'if' (even in the context of a proposition) is that you ARE uttering it. For Grice this has to do with Cook Wilson.
"&" is adding.
"~" is substracting.
then we get to
"v"
and
")"
--- Here Grice cites from Cock Robin. (Not Cook Wilson).
Who KILLED Cock Robin.
The wren OR the sparrow.
----- uninformative. Or, rather a yes/no question to the content will have Shannon turning.
What about:
Who killed Cock Robin?
If it wasn't the Sparrow, it was the Wren.
----
So, 'or' (which features in the definability of "p)q" as "~pvq") indicates a DIFFERENT, 'contingency-planning' Grice calls it, or pis aller, using his French, metier.
"if" introduces an interrogative subordination which is quite ANOTHER animal. So, in terms of metiers, even if 'p ) q' REDUCES to '~pvq', there are metiers -- in the conversational game -- that cannot be played by just '~' and 'v' unless you MUST.