--- by JLS
----- for the GC.
FURTHER TO DOCTOROW on "Probably Grice" some cutting and pasting from Grice, WoW:
Grice notes that the contraposition of
p ) q
is
~q ) ~p
In his special game of chess -- wiki: Grice's paradox: this yields:
8/9 if Yog has white, Zog wins.
1/2 if Yog loses, Yog has black.
2/10 Yog doesn't have white or he wins.
----
Grice writes:
"Dummett and Kripke"
in discussion at Harvard -- no reference given -- this is 1967 --.
"suggest that we distinguish between I and II."
I. the notion of the probability
of a conditional relative to certain evidence "h" -- a notion
which is NOT altered if, for that conditional, we substitute
its standard contrapositive, or (for that matter) its
standard disjunctive counterpart; ofr
(if p, q)/h
is equivalent to
(if ~q, ~p)/h
and also to
(~p v q)/h
---
II. the notion of conditional probability as it is
exemplified in the probability of p, relative to both
q and h, a notion which cannot be treated as
identical with the probability of the negation of q,
relative to the conjunction of the negation of p
and h."
----
"Dummett and Kripe further suggest that the puzzle
about Yog and Zog should be taken to relate to
CONDITIONAL PROBABILITIES and not to the probability
of conditionals."
---
Grice goes on:
"If we are aksed the PROBABILITY that if
Yog did not win, he did not have hite, and if we take
this question to "look forward" the possession
of information that Yog did not win, we consider
only what was the case when Yog did not win (as
regards his having white or black) and ignore
cases in which Yog won."
----
"If we are aked the PROBABILITY that Yog either
did not have white or won, there is no direction
(pointeering) to modus ponendo ponens, so we
consider the whole series of games which were either
ones which Yog had black or ones in which Yog won."
---
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ReplyDelete(edit). It's a bit confusing as stated (and I misread it at first). However---if you don't mind un petite dissent, or a challenge to Guru Grice--I'm not sure it's an actual paradox as much as a exercise in producing accurate statements. Here's the wording of original:
ReplyDelete(i) 8/9 times, if Yog was white, Yog won.
(ii) 1/2 of the time, if Yog lost, Yog was black.
(iii) 9/10 times, either Yog wasn't white or he won.
It seems a bit weird that one would reverse the order of the conditional in ii, and that anyone would give the situation as it is stated in iii (Yog won 8/10 times as white, and lost 2/10 half as white, half as black) . So why state it otherwise, except for some obscure point?? Gricester sort of betrays his own maxims of, oh, being accurate or something)
It could just be factual statements..."Yog won 80 out of 90 games as white, AND Yog lost 10 out of 10 games as black. So he won a total of 80 out of a 100, and when he lost he was black 50% of the time."
Literal and boring as like a Chilton's manual, but I don't see any profound paradox, but perhaps a warning not to misuse conditionals...(and the contraposition logically equivalent to conditional and to material implication per truth tables anyway)
Yes. There is something obscure about Grice's stating of the paradox. Kramer is our expert here. He specialises in mathematical problems. What makes a problem a mathematical problem? As Kramer notes: it's the Griceian maxims. You have to give away just as much, but not more than you need to give away to make the point trivial. (There's this lady who specialises in these things, that Kramer quotes).
ReplyDeleteI think Grice is sticking of that sort of quiz scenario, where the thing is totally artificial. He is isolating a scenario where one should NOT be driven by the implicature of 'if':
"If the butler did not kill her, and it is not the case that either the maid or the gardener was a lesbian, then ..." sort of Miss Marple quiz.
So, yes, the point may be about some odd things connected by 'if'.
Note that 'protasis' and 'apodosis' make no sense when 'if' is understood truth-functionally. It's just a combo of truth values for p and q. So 'reversing' the conditional should best be reread as 'invert' the order of 'p' and 'q' in the original formula. I dislike the use of 'conditional' for that reason; for, in the truth-functional 'sense', 'if' is LESS than a conditional.
The Germans have 'ob' which relates to 'if'. The Latins had 'si' (modern Italian 'se'). Hardly the companion for /\ and \/ which are the two truth-functors that Grice mentions before 'if' -- i.e. 'and' and 'or' --.
In any case, the use of conditional probability and probability of conditional is tangential here. As the Richards mentions in the thing cited by Doctorow, the Griceian point is that, sometimes, the implicature tracks the conditional probability of the 'apodosis' (NEVER THE WHOLE CONDITIONAL) given the protasis.
What Lewis and Jackson on the other hand seem to be expanding is on the probability of the 'if p, q' as such -- and in SOME way of looking at the Yog-Zog 'paradox' -- a reference to the probabilities seems relevant.