---- By J. L. Speranza,
---------- For the Grice Club
-------------------- THIS IS MY ELABORATION on Kramer's interesting proposal in his comment to my longish excerpt from Strawson -- just only on the '.' and 'and' interface. Kramer rightly proposes a formal system (for which it would be quite a BIT of a stretch to say it has a 'grammar'!) where there is NO NEED, and indeed it would be PRETTY UNCALLED FOR, TO hypothesise a cross-reference between any ol' 'formal device' and its 'vernacular counterpart'.
Grice SHOULD HAVE stressed the thing more clearly. After all, good ol Nel D. Belnap, Jr. had cared to publish his "Tonk, Plonk, and Plink" in Analysis for 1961/2 (vol. 22, pp. 130-134), one of the funniest essays -- and shortest -- ever written by a philsopher of his formal ilk. And the fun side of this is that Strawson cared to reprint it in his influential Philosophical Logic for 1968 ("Oxford Readings in Philosophy", ed. by G. J. Warnock). This Strawson collection -- which I used along with Haack, as basis for ch. ii of my PhD dissertation -- is a bit of a gem for at least two reasons:
-- It relies extensively on Grice, but in a cavalier fashion, with Strawson saying things like "Unfortunately, most of what Grice has said on this remains bitterly unpublished".
-- He has a specific section, Strawson does, on 'meaning' versus 'use' as it applies to, say, the connectives: the tonk, the plonk, and the plink. Strawson takes up the polemic from an early textbook by Quine, Mathematical Logic, and goes on to challenge Quine's hardly subtle points about the meaning or use of such things as 'tonk', 'plonk' and 'plink'. This is important vis a vis Grice, because his Prolegomena to the William James Lectures, which Strawson cites, and which made a trademark of Grice in Anglo-American philosophy, explicitly displays that his concern was to debunk some of the myths of the time-honoured distinction -- ('meaning' vs. 'use').
-- It is slightly ironic, but knowing Grice we understand it, that Strawson had to publish, instead of Grice's very specific things on this, his rather general "Meaning" in _Philosophical Logic_! As such, though, Grice is given credit and a "list-of-contributors" note has him as "William James Lecturer for last year" or something.
----
Belnap's point seems to be essentially Kramer's. Indeed, there is NO NEED to suggest that '.', 'v' and '>' signify 'and', 'or' and 'if', because they don't. And I don't care! As a Griceian I am only concerned with what _utterers_, on occasion, mean. And it is a bother to have to provide cross-references between the Russell, and the Strawson, and the Grice! But not impossible, and if you are in the right mode for it, could turn out to be quite fun, too!
By using, 'tonk', 'plonk', and 'plink', Belnap is exactly wanting -- isn't that obvious!? :) -- to detach this 'syncategoremata' from any vernacular associate!
In the same vein, I have found Gazdar's treatment, in his PhD for Reading, "Formal Pragmatics" (nothing formal about it!) on
p * q
where "*" means "any connective" you bother/please. Gazdar is more systematic. As he notes: in a bivalent system, the number of connectives is finite. I will retrieve the truth-tables in a comment I hope. Thus
p . q
1 1 1
1 0 0
0 0 1
0 0 0
is just ONE of the 'truth-functions' or 1-functions. For there is no need to really cross-reference '1' with 'true' (vide Grice on Strawson on "Truth" -- one of the few essays Grice cares to quote from in WoW -- section 'Truth' in WoW:iii).
Now, Gazdar, in something that has NOTHING to do what he purports to call "formal" pragmatics -- since it's so empirical it hurts! -- goes on to propose things. Since his thing was submitted to the Department of Linguistics at Reading (under Palmer), he is sufficiently concerned with the natural-language interface, and from what I recall he cares to quote from some 'furrin' lingo -- and not just "English".
But when it comes to English, he makes some outrageously funny Gricean predictions. Why is it, for example, that ... speakers of English are so Gricean?
The Gazdar point is however more clear when it comes to the monadic operator. The 'not'.
There are four monadic operators which are logically possible in combos:
Np -- which inverts the truth-value
So that if p is 1, Np is 0, and vice versa.
But there's also
Tp -- which keeps the value.
So that if p is 1, Tp is 1; if p is 0, Tp is 0
There's also the
Kp -- which yields 1 regardless
So that if p is either 1 or 0, Kp is always 1
And finally, there's
Qp -- which yields 0 regardless
So that if p is eiter 1 or 0, Qp yields always 0.
The good thing here is to keep a Lukasiwicz sort of notation -- one slot for an operator, be it monadic or not. Note that truth-functions can be triadic too (alla "... between ... and ...")
Now, Gazdar makes the Gricean question: why is it that English only 'realises' (good linguistic jargon there! to avoid unless you mean it!) 'not'?
Gazdar's Gricean (or more strictly, Cook-Wilsonian) answer: the three other operators have no METIER: the truth-operator, that yields only truths, would totally breach the maxim of quality; and so would the falsity-operator: imagine if all we said were _false_, and knowing it. Finally, the redundancy operator, that repeats the value of the 'radix' is so otiose it hurts. This leaves us only with "not" or Np, as the ONLY 'reasonable' operator to have when it comes to a 'metier'.
So Grice would like to say that, qua philosophers, or even philosophical logicians, we SHOULD be interested in things (vague) as "English". So he proposes a few 'operations' which we seem to be able to abide by:
-- conjunction
-- disjunction
-- implication (material)
and the rest is legend!
For then he wants to say that perhaps Russell was _onto_ something when he spoke of, with his fellow mathematician Whitehead, of '.', and 'v' and '>'.
Grice -- the locus classicus here is William-James No. iii -- which he reprinted in WoW:iii under the misleading title of "Indicative Conditionals" but it surely covers more ground -- goes on to combine this with his thoughts on '-' and 'not' AND, interestingly, what he calls "one of the strokes".
By "one of the strokes" he means at least Scheffer.
--- From wiki, Sheffer stroke
"The stroke is named after Henry M. Sheffer, who proved (Sheffer 1913) that all the usual operators of propositional logic (not, and, or, implies, and so on), could be expressed in terms of it. Charles Sanders Peirce (1880) had discovered this fact more than 30 years earlier, but never published his finding. Peirce also observed that all boolean operators could be defined in terms of the NOR operator, the dual of NAND."
----
Grice is adamant _not_ to give much credit to Sheffer. He finds 'incompatible' pretty incompatible with our ordinary ways of talking and thinking. Plus, Grice's very first paper was on 'negation' and he was well aware of Plato's problem with the circular attempts to get rid of 'diaphoron' in terms of 'other than'.
---
So with Sheffer _kaput_, Grice is left with ... Cock Robin.
He spends the rest of the lecture on variants of the nursery rhyme.
Who killed Cock Robin?
Jenny the Wren?
No.
For he found that Cock Robin, as the wiki notes is "aa murder archetype in world culture."
Who killed Cock Robin?
I, said the Sparrow,
with my bow and arrow,
I killed Cock Robin.
Who saw him die?
I, said the Fly,
with my little eye,
I saw him die.
Who caught his blood?
I, said the Fish,
with my little dish,
I caught his blood.
Who'll make the shroud?
I, said the Beetle,
with my thread and needle,
I'll make the shroud.
Who'll dig his grave?
I, said the Owl,
with my pick and shovel,
I'll dig his grave.
Who'll be the parson?
I, said the Rook,
with my little book,
I'll be the parson.
Who'll be the clerk?
I, said the Lark,
if it's not in the dark,
I'll be the clerk.
Who'll carry the link?
I, said the Linnet,
I'll fetch it in a minute,
I'll carry the link.
Who'll be chief mourner?
I, said the Dove,
I mourn for my love,
I'll be chief mourner.
Who'll carry the coffin?
I, said the Kite,
if it's not through the night,
I'll carry the coffin.
Who'll bear the pall?
We, said the Wren,
both the cock and the hen,
We'll bear the pall.
Who'll sing a psalm?
I, said the Thrush,
as she sat on a bush,
I'll sing a psalm.
Who'll toll the bell?
I said the bull,
because I can pull,
I'll toll the bell.
All the birds of the air
fell a-sighing and a-sobbing,
when they heard the bell toll
for poor Cock Robin.
--- The exercise of course is to find 'and', 'or' and 'if' as used in the rhyme and see if they may correspond or not with '.', 'v' and '>'. Recall Grice's 'conversational metiers' for each: 'and' allows to denegate a complex claim ("Jenny and the Sparrow killed him"), 'or' is for contingency planning ("Either Jenny or the Sparrow killed him" and 'if' is for interrogative sequencing in x-patterns ('If Sparrow did not kill Cock Robin, Jenny did")
At this point a ref. to Latin may prove interesting, since it has been much discussed and it tells a little moral. The Romans distinguished types of 'disjunction'. Since everybody inevitably did end in the colisseum with the lions, one shouldn't bother -- but in English, 'or' is SUCH a trick, that eventually logicians won the day, and managed to have TWO operators for this venerable particle. In my PhD I DENIED the logical status of 'w' (alla Gazar) but the fact that it has a symbol of its own is proof of the pudding that tonk, plonk and plink win the day in St. Giles! (* St Giles is where this Oxford don teaches "Mathematical Logic" today, and also the address for Grice's St. John's -- hence a veritably interesting ambiguity). Etc.
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You might find my page on Boolean Operators amusing:
ReplyDeletehttp://www.rbjones.com/rbjpub/logic/log048.htm
It's lovely!
ReplyDeletehttp://www.rbjones.com/rbjpub/logic/log048.htm
---
It's lovely and I'll consult it in more detail. Loved the nimp and the iff along with the nand. And the iff.
Horn has written on 'iff' recently: his first piece for the Journal of Pragmatics. He may credit me there. I recall discussing with him this bit by Pears, in "Ifs and Cans" (1971) to the effect that 'if' conversationally implies "iff" -- what Horn calls, with other linguists, or pragmaticists, 'strengthening'.
Incidentally, I would often get into discussion, friendly of course, with Horn on this or that. e.g. the "nand" and the 'neall'. In "History of Negation" he does mention the Boole "nand" but argues that it cannot 'realise' in natural language. I say, what was Boole, a Martian? But he wouldn't listen!
I was amused to let Horn have the editorial of the Performadillo that he had somehow forgotten about. This was the proceeding of the 1973 talk which Horn shared with Grice. Horn was fresh with his PhD (UCLA, 1972) and so presented his ch. ii as a talk -- the grammar realisation of logical operators. Grice presented his convoluted symbolisms on reason. Neither paper was published in the proceedings "for technical reasons", the editors go.
Horn was amused to be paired with Grice on that. In any case, I asked him, "I suppose Grice was very pleased by what you said." He said, in his typical ironic, terse mode. "Well, at least he was not too displeased." Meaning, "he did not utter one word," from what we recall. Ah well.
The OED entry quotes extensively from Horn 1972, which is Horn's PhD as circulated by I forget what company in I forget what state of the United States. The thing is UCLA deposited and he was much influenced by Montague and the formalists in writing that: the logical operators, it is called -- but the phrasing of "English" in title makes it more of a linguistics piece than a philosophy one. Especially for those who don't speak it! Etc.
Gazdar's Gricean (or more strictly, Cook-Wilsonian) answer: the three other operators have no METIER: the truth-operator, that yields only truths, would totally breach the maxim of quality; and so would the falsity-operator: imagine if all we said were _false_, and knowing it. Finally, the redundancy operator, that repeats the value of the 'radix' is so otiose it hurts. This leaves us only with "not" or Np, as the ONLY 'reasonable' operator to have when it comes to a 'metier'.
ReplyDeleteI'm not clear on what a metier is, but I'm not convinced that the three "other operators" are not "realized" in English.
First, I would argue that to state the effect of an operator in English is to "realize it" in English. I think JLS made this point parenthetically in one of his comments or posts, but I cannot find it.
Second, I wonder if the "other three" monadic operators really have no more traditional English realizations.
Np - Not
Tp - Indeed. If p is true, then "indeed p" is true. If p is false. Then "indeed p" is false. I assume this is the 1-ary operator that RBJ says "does nothing." Perhaps silence is the English "realization" of that operator. If we want to negate p, we say "not p"; if we want not to negate p, we NOT say "not p." We could call that the "strange incident of the dog in the night" operator.
Kp - Either...or not. If p is true or if p is false, "Either p or not [p]" is always true.
Qp - and Queen Anne lives. Just tack that (or any other fib) on to any p, and the resulting sentence is false. QP has no non-ironic use because it is uncooperative, but it does have ironic uses:
I just squared the circle!
Yeah, and I invented anti-gravity.
Perhaps Gazdar just didn't realize...
--------------
I also wanted to comment on your quote of Strawson in the Strawson Polemic thread:
And the identification of '.' with 'and', or with a full stop, is not a simple mistake. There is a great deal of point in comparing them. The interpretation of, and rules for, '.' define a minimal linguistic operation, which we might call 'simple conjunction' and roughly describe as the joining together of two (or more) statements in the process of asserting them both (or all).
The problem with this approach, for me, is that it seems to treat English as a superset of Logiclandian, when it makes more sense, to me, to see Logiclandian as a subset of English, extracted for convenience. The . operator actually seems to me more closely identifiable with the mathematical dot representing multiplication. I assume it is no coincidence that where a and b are the binary truth values (0,1) of p and q, then
a · b
will give the binary truth value of
p.q.
As a reductionist, I think there is something to be learned about anything we don't understand from just about anything we do understand. So if we understand Logiclandian, that may help us investigate English. After all, logic has more applications than just logic.
Kramer's excellent points all taken! Indeed, the parenthetical Kramer is alluding may, but then it may not, be my reaction to Horn's claim, "'NAND' is not realised in English'. "And so -- Boole was from Mars?". We also discussed 'neall'. This, the OED noted, and I showed that to him, I think he knew because I think he quotes that in his "Hitory of Negation". This "neall" is very pertinent, because it expands beyond the truth-functional gambit (or gamut). Horn's big thing, in his PhD indeed is the Square of Opposition, i.e.
ReplyDeleteA-E
I-O
(where indeed, as Grice has it, there is a correspondence between 'all' and multiplication or better addition, and conjunction and &, and 'O' and disjunction or 'or' /\x -- "all x" \/x, "some x". /\, and, \/, or. So Horn is, naturally, as Grice was, concerned with realisation of operators. Now, the Anglo-Saxons did use 'neall' to mean Not-All, but not as negation of "A" yielding "E" but as something different (I would have to revise the logical form. Maybe as O). Anyway, it got dropped in modern English, alas. Horn has referred to the sad story of this in his "The Story of O". (Can't he _be_ witty? Love him!). Yes, I do think Gazdar may say that those operators are not as silly as I painted them. There are actually strict letters to use, which I should revise. I think Tp is the truth one, but I forget the letters for the "and Queen Anne is alive" and "either ... or" ones. Very clever examples by Kramer, thank you. I'll have a look at possible conversational métier for them. Matter of fact, while Grice cites Cook Wilson, I never was able to find the passage in that dry, old, otiosely posthumous book, Statement and Inference. The thing makes for such dry reading! Wilson's comments on 'negation' are easy enough, but what Grice may be meaning by "I am hereby taking up a suggestion by Cook Wilson" is a bit of a red-herring or blue-herring, if you must. Cook Wilson, however, was the Wykeham Professor of Logic, so that's something. As Ayer said to this famous American boxer when he was picking up a girl. "You may be a famous American boxer, but I am the former Wykeham professor of logic". Or something. Ah well. Gazdar's thing was distributed in 1976, but I have, somewhere in my Swimming-Pool Library, his 1979 Academic-Press reprint. This press was so academic it hurts, and it's a signal of our times it went bankrupt! I like the idea of 'silence' or the 'strange incident of the dog in the night' operator. And of course, Strawson was wrong and Grice right -- see my "Robbing Peter to Pay Paul" (this refers to the ugly St. Paul's Cathedral so-called, 'ugly', in comparison with the beautiful St. Peter's cathedral, i.e. Westminster, originally). The idea that English is a superset is just what one would expect from him. When in Hungary he was challenged: "All you say strikes me as petit-burgeoise". "Well, I am one. I am a petit-burgeoise", he said, and with a straight face! But he always loved Grice in his ways, so that's charming of him, too. I love him, too! (And his son, Galen!). Etc.
I thought a bit about multiplication vs. addition, and decided that multiplication was the better choice because the arithmetic matched the truth tables. I think the key is to understand "multiplication by 1 or 0" as sui generis. These bits of "multiplication" aren't really multiplication at all, just extrapolations of mathematical models of multiplication. The correspondence, therefore, is between the . operator and multiplication by 1 or 0.
ReplyDeleteIf I understand correctly, and there's lots of reason to think I don't, p.q is not like "ham and eggs." It's like "The chicken makes a contribution and the pig makes a commitment." What we care about is not p+q, per se but the truth of p.q. The probability of two things being true is the product of their respective probabilities. The probability that p and q are true, where p and q are true, is 1 x 1 = 1. It is not 1+1. If either is false, the probability that both are true has a multiplicative factor of 0, and so is 0. The operator does not join p and q; it operates on the conjunction of p and q. It's just that the "is true" is "understood" in Logiclandian. Holy Cow, an implicature in Logicland. Now what?
Is it not correct to say that
if p then q
is equivalent to
if p is true, then q is true
If so, then shouldn't we understand
p.q as
p.q is true (multiplicative)
and not as
p and q are true (additive)
?
God knows! And, what, with Dawkins saying that the population in South America grows geometrically while the food grows arithmetically I'm never hungry!
ReplyDeleteYes, I think it must be right -- what _Kramer_ says, about the 'multiplication'.
It's an odd word, anyway, 'multiply'. "Do not multiply", Occam said. God, on the other hand, said, "Do Multiply!". Borges said: copulation and mirrors irritate me: they multiply _men_!
What strikes one is that while addition and multiplication _are_ commutative: I.e. five apples six oranges is thirty grapefruits; six oranges five apples is thirty grapefruits. In symbols
5 X 6 = 6 X 5; or ab = ba.
Addition is _also_ commutative: one ham, two eggs: one triple; two eggs, one ham: one triple. In symbols:
1 + 2 = 2 + 1; or a+b=b+a
Their inversions are NOT. Notably
5:6 ≠ 6:5; in symbols a:b≠b:a
and
1 - 2 (which is -1) ≠ 2 - 1 (which is 1). In symbols: a-b≠b-a.
What Kramer says about the horror (perish the thought) of an implicature in Logiclandian is very pertinent, and will elaborate on that. Horn has! He has grown so tired of people calling him a neo-Griceian that he is now calling himself the neo-Fregean implicature guy instead. He is re-reading all his Frege, and has thoughts to express on 'neben', Koloratur, Farbung, Implikatur, Forz, Assertung Zignum, and the rest of them. Apparently, Frege, the author of the Begriffschrift was the greatest Griceian of them all! (Neale considers this in "Heritage of Grice", too, and Harnish. Frege tried and tried and tried to make Leibniz's dream come true (to use R. B. Jones's apt metaphor -- vide his site --) yet he was more and more and more led to grow sceptical. The fact that he spoke German did not help. Grice once commented to Austin, (or words) "Perhaps Frege was not as otiose" (implicating, "as all that"). The next thing, we have Austin spending every Sunday translating one of the most boring books ever written in German Literature, "The Number" by Frege -- for Basil Blackwell, Ltd. Oxford. Or, for long: Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl(1884). Austin seems to have concluded that numbers _do_ exist. I have elsewhere (notably with Henninge) discussed Austin's failing to grasp some of Frege's implicatures and "tones". Etc.