The Horseshoe's Implicature

Speranza

Or robbing Peter (Wason) to pay Paul (Grice).

It all started, for Newstead, when he submitted for the Journal of  Memory
and Language that brilliant essay,
"Griceian implicatures and syllogistic reasoning" and it was  accepted!

Grice used to say, "Grice saves; but, mind, there's no free lunch".

The implicature was that if you, like Peter Wason and Paul Grice  himself
did (Robbing Peter To Pay Paul) take "⊃" to stand for the 'if' in  the 'if'
utterance of Wason's AD47 card selection task, you may SAVE (problems  that
may arise if you take a different sort of 'formal device' to represent 
something apparently as
naïf as "if" sounds like.

But that was precisely Strawson (also called Peter, like Wason, 
coincidentally, "Peter without a lesson," as I call him -- since at least Peter  Wason
is taking the horseshoe properly) and his mistake.

Now, in the case of Wason's 'if' utterance involved in the AD47  card
selection task, Leda Cosmides called it "abstract" -- she's no philosopher  but
we can get her drift.

Seriously, it's not so much about the 'subtle grammatical shift' (or nuance
or conversational implicature) but about how  you turn that 'grammar' into logic. Grice, echoing Russell, used to say that  'grammar' is a 'pretty good guide' to 'logical form', and I think in
THIS case  he did mean 'pretty pretty' (as opposed to the pretty in pretty complex).

For when we see the abstract 'if' utterance as

p⊃q

--surely there are dismissing 'formal devices' that Grice didn't in his 
"Logic and Conversation" lectures,
where he lists "∀" to stand for "all", "∃" to stand for "some (at least 
one)" and "ix" to stand for "the".

The Wason subject will first approach the task syntactically -- and he may 
fail at that level.

He will then approach the task 'semantically'.

And he will test the task pragmatically. He may wonder: "Is it really 
believable that Wason is in doubt about the truth value of the 'if'  utterance?
Don't think so. Is he teasing me, or trying to tease me? Isn’t it  more
likely that Wason just typed down a true "if" statement — the  more so since the
background "if" utterance (‘letters on one  side, numbers on the other side’
) must also be taken to be true, if one follows  Grice's cooperative
principle ("be truthful" -- his conversational category of  Qualitas).

But back to Step 1 and 2 (syntax-cum-pragmatics), the Wason subject may 
feel that the 'if' utterance needs some sort of predicate, or 
quantifiational, logic, and that its logical form requires the use of predicates  and
quantifiers, notably because of the occurrence of expressions like "one  side"
and "other side".

If, like Grice, you are enamoured with System Q (that Myro created in 
Grice's honour), the way to provide the correct logical form (the middle and 
second Witters hated that phrase!) formalise Wason's 'iffy'  utterance in
predicate or quantificational logic should use the  following predicates:

V for visible
I for invisible
O for vowel
E for even number

We now need propositions:

i. Ox

will read as "x is a vowel".

ii. Ex

will read as "x is an even number".

Next we need a 'relational logic' alla Russell, for the  introduction of
the predicate 'visible'. For it's not just 'visible' we want,  but the ability
to formalise, 'x is on the visible side of card y'. Let

iii. V(x, y)

represent that, where V stands for a dyadic relation).

And finally we need again 'relational logic' to represent, 'x is on the 
INvisible side of card y'. Let

iv. I(x, y) -- x is on the invisible side of card y.

symbolise that, where again "I" stands for a dyadic relation.

Now, Wason's iffy utterance can be given the proper logical form:

∀c(∃x(V(x, c) ∧ O(x))  ⊃ ∃y(I(y, c) ∧ E(y))) ∀c(∃x(I(x, c) ∧ O(x)) ⊃  ∃
y(V(y, c) ∧ E(y)))

After teaching Paul Strawson for a few terms at St. John's, Grice was 
appalled to see that his tutee still committed a few mistakes. It is these 
mistakes (in Strawson's "Introduction to Logical Theory") that moved Grice to 
accept the invitation at Harvard to give the "Logic and Conversation"
lectures.  In the "Prolegomena" Grice explicitly quotes verbatim from Strawson's
essay on  'if' as a mistake by 'ordinary language philosophers' who dismiss
pragmatic  phenomena -- and that can't stand the horseshoe's implicatures!

Similarly, and a few years later, Peter (same first name as Strawson) Wason
ends up finding that some of his subjects go astray at this point,
replacing the  second statement containing ⊃ by a bi-conditional (a different
formal device  altogether), or worse, by a reversed conditional.

Pinker knew this when he says that 'logical' words in language are 
'ambiguous'. He uses that ambiguous 'ambiguous' when he means is 'Griceian in 
nature': utterances containing them have a precise logical form and trigger 
implicatures that can go over Wason's subjects's heads (metaphorically) 
(although Pinker uses 'mind' LITERALLY!)

Pinker's list of 'logical words' in the paragraph preceding his précis of 
Wason's test, one may think he quotes just to reference the PhD by Cosmides
for  the Department of Psychology and Social Relations (Her advisor was
Sheldon H.  White, not Pinker, granted. From 1982 until 2003, Pinker taught at
the  Department of Brain and Cognitive Sciences at MIT, and eventually became

the  director of the Center for Cognitive Neuroscience, taking a one-year
sabbatical  at the University of California, Santa Barbara, in 1995–96 (to
which Cosmides is  associated). In 2003, Pinker became the Johnstone Family
Professor of  Psychology at Harvard).

In any case, Pinker's list goes:

"not", "and", "if", "same", "equivalent", and "opposite".

So, he wouldn't be surprised by a richer logical form to the Wason task 
than the mere

p⊃q

-- because, while "⊃", note that the expanded logical form in predicate 
logic makes use of "and" ("∧") and one can think of shorter expansions that
use  "not" (~) if not "opposite", "equivalent" and "opposite".

To sum up, a certain prejudice (or 'bias') against a logically-based 
account of reasoning (as Grice offers in "Aspects of reason", borrowing some 
ideas from his own "Logic and Conversation") cuts off the insights of
psychology  from a few applications. Those psychologists like Wason who explore
reasoning --  or worse some of his critics and some of his followers, although
hardly the  philosophical bunch of them -- seem allergic to formal semantics
as are  psychologists of reasoning. This 'bias' may be largely due to past
simplistic  attempts to apply formal theories in a monolithic way.

But since, after Grice, logic has become less monolithic (and  open the
gate to implicature -- as he said, "Implicature  happens"), psychologists of
reasoning cannot afford to continue  avoiding an enriched approach to logical
form like the rats!

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