R. B. Jones has interestingly written on "Scope and Truth value gaps".
Jones writes:
"Here is a bit of clarification of what goes on in relation to scoping of definite descriptions in [Whitehead's and Russell's] Principia [Mathematica] and in Grice's formal work, and where truth value gaps do or don't appear."
Thanks!
"Three philosophers appear here, Russell, Grice and Church, and three logical systems, with an aside to Strawson on truth value gaps."
Good -- Let us not forget Whitehead. I love Whitehead, and recall that Principia Mathematica is co-authored by Whitehead and Russell (rather than by Russell and Whitehead! Is 'and' commutative or not?)
Jones goes on:
"Let's get the latter out of the way first."
That is: Sir Peter.
Jones:
"Strawson was not so fond of Russell's theory of descriptions."
No, he wasn't. I sometimes wonder. In "Introduction to Logical Theory", Strawson writes that he is acknowledged to Grice "from whom [he's] never ceased to learn about logic". That is a broad thing to say. One learns truths. Can one learn mistakes? In other words: Strawson could never have learned about truth-value gaps from Grice because Grice disbelieved in them. On top of that, it was a coinage by Quine!
Jones goes on:
"[Strawson, or Sir Peter (as he then wasn't)] thought that statements [expressions] involving definite descriptions were meaningless if the definite description was not uniquely satisfied, and hence that they lacked a truth value."
I'm not sure he meant 'meaningless' as "Caesar is a prime number", "The Absolute is lazy", or "The nothing noths". I think he was happy with truth-value gaps. Perhaps he (Sir Peter) thought that
"The king of France is bald"
uttered when there is no king of France is bald has no USE. Recall that Grice's Prolegomena is meant to correct the Anti-Wittgensteinians. For Wittgenstein:
meaning = use
and therefore
meaninglessness would be non-use
Grice is intended to attack the new dogma that meaning is NOT use, which yields further paradoxes. In other words, Grice had to COIN (as it were) 'implicate', qua verb, because Strawson had doubts about it.
In "On referring", Sir Peter (as he then wasn't) says that to utter
"The king of France is bald"
AND
"The king of France is NOT bald" (for that matter)
both "imply"
that there is a king of France.
He meant: "implicate", but he wouldn't use the word because Grice had yet to invent it. And I doubt Sir Peter (who never studied classics, would have cared to check with the Lewis and Short "Latin Dictionary" (Oxford) that does have an entry for 'implicatura', unfortunately defined as a mere 'entanglement'. It is used only by Sidonius, which doesn't help.
Later, Sir Peter thought he could use another verb, and famously declared that uttering
"The king of France is bald"
and
"The king of France is NOT bald" (for that matter)
PRESUPPOSE
that there is a king of France. This W. C. Kneale cared to quote as a development of the old theory of 'suppositio' elaborated by Occam (or Ockham if you are from Surrey) and others.
Jones goes on:
"To realise that maxim in a formal logic you would typically use a "non-classical" logic with more than two truth values (and you translate "no truth value" as "neither true nor false")."
Exactly. It would also mean to avoid being "in the tradition of Kantotle". For the second part of Kantotle (Aristotle, not Kant) has the 'tertium non datur'.
Quine had no problem with deviant logics, nor did Susan Haack ("Deviant Logics", and "Philosophy of logics", citing Grice) but Grice did! (He was too much of a Kantotelian to allow for a proliferation of truth values like THAT).
Jones goes on:
"The three systems discussed here (Russell's PM, Church's STT, and Grice's system Q) are all "classical", i.e. two valued, and hence there can be no truth-value gap in definite descriptions in any of these systems, every sentence is either true or false (under every interpretation), so if there is a problem (as Strawson might insist) it is just the lack of a truth-value gap or a third truth value."
Exactly. The problem is that there is a problem, indeed, for Sir Peter. It is here when I find that that colloquial phrase, "robbing Peter to pay Paul" (whose origin apparently has to do with robbing the parishioners of Westminster, where Peter's Abbey is, to pay the new rich of the cathedral of St. Paul) cannot be better applied.
For Peter Strawson's mistake gives Paul Grice to rob Peter to pay Paul, i.e. himself.
(Recall that in "Presupposition and Conversational Implicature", in Grice's WoW, the first page, is all about how hateful the idea of a truth-value gap as a metaphysical concoction Grice finds).
Jones goes on:
"Passing onto curiosities of scope, the main issue arises in PM, persists in Q but is addressed in a different manner, but is absent from STT."
Good to know.
"It arises from Russell's Theory of Descriptions, which treats descriptions as "incomplete symbols" involving contextual definitions."
-- indeed, and I would add Whitehead, since at least a segment from "Principia Mathematica" Ostertag found relevant enough to include in "Definite descriptions: a reader". But it is true that Russell's "On denoting" owes little to my dear A. N. Whitehead.
Jones goes on:
"For the full low-down on this see the Stanford Encyclopaedia of Philosophy entry on the notation of Principia Mathematica, this is the shortened bowdlerized version."
And a good entry it is too!
What I love about the Principia Mathematica (Grice calls everyone who follows it a 'heir of modernism') is that it kept changing with the editions. When Scheffer invented the stroke, Whitehead and Russell, in an 'improved' edition, say they regret having introduced "and" for ".", and "or" for "v" and "if" for ")", etc. --"not" for "-" --, when using the Scheffer stroke would have done.
(Grice agrees, but realizes that 'negation' is psychologically PRIOR to any stroke that Scheffer may have come up with).
Jones goes on:
"Russell's incomplete symbols are defined only in specific contexts and the elimination is sensitive to the context chosen. This gives rise to potential ambiguities of scope, to resolve which Russell adopts a rather cumbersome notation, for which Grice offers an alternative in his System Q (for Quine) in the form of subscripted quantifiers."
Indeed. Along with a square-bracket device, which some find more intuitive. Harnish does. Harnish example is:
A: I didn't know you were pregnant.
B: You still don't.
Here the implicature is 'ambiguous'. On the idea that
'to know'
means
'to believe that p'
'to be justified in believing that p'
'p is true'.
one seem to think that "You still don't" (i.e. know that I am pregnant) IMPLICATES that "B is pregnant" is FALSE. But in principle, the negation could apply to A not being JUSTIFIED so, or not believing so.
In other words, if we have an analysis of 'knowledge'
K
as
JTB
justified true belief
The order of clauses would be
JBT
justified belief about what is true.
If we take the implicature of "You still don't" to be "B is not pregnant", Harnish argues (in "Implicature and Logical form"), is because we take as common ground that A is justified in saying what he says and in believing it?
In the square-bracket notation:
[JB]T
The negation becomes
~[JT]T
But whatever is included within the square brackets is, Grice says, immune to negation ('assertoric inertia' has been used for this, but it applies to non-assertoric utterances, too).
Grice does use the square bracket devices TWICE in "WOW" (Way of Words). He mentions them in passing in the "Retrospective Epilogue", and he develops them in some detail in "Presupposition and Conversational Implicature" (which should read as "Presupposition AS Conversational Implicature"). Curiously, but knowing him, one can see, he doesn't care to mention the subscript device introduced in "Vacuous Names" (perhaps Quine, in his "Reply to Grice" in the volume where Grice's "Vacuous Names" appeared, called it 'superfluous').
Jones goes on:
"The simple example offered by Speranza involves the infamous King of France"
Apparently, the first to introduce royalty in these debates (Russell would know about it, even if he was from Cambridge) was Oxford philosopher F. H. Bradley. His example, granted, mentions "Utopia" (a reference to Gilbert & Sullivan? Don't think so).
The king of Utopia died last Tuesday.
Bradley rights, conceding that this is 'ambiguous'.
Jones goes on:
"and the ambiguity which arises when a negation is added to the allegation of baldness."
-- because Grice preferred to stick with Russell (or Lord Russell, if you mustn't -- he was born in what is now part of England, but was then part of Wales; hence the lord), and Russell used 'bald' -- He (Russell) concludes his "On Denoting" by remarking that if you are feeling Hegelian and in need of a 'synthesis' you might just as well grant that the king of France is wearing a wig.
All that humour was lost on Sir Peter, who preferred to call the present king of France, of all things, 'wise'.
---- (Perhaps in opposition to the last real king of France who acted more or less unwisely).
Jones goes on:
""The King of France is not bald" can be translated in two different ways depending on whether the negation is considered to be in the context for elimination of the description or outside it."
Indeed. But surely, by default, external scope is operative; hence the need for the square-bracket device to point out that we are not dealing with a default reading here -- and that we need a rather extravant notation -- the square brackets -- which are made, as if by magic, immune to negation (Paul Elbourne is not at all happy with Grice's way out here! -- "Definite descriptions" (Oxford University Press).
Jones goes on:
"There are two options for representing this sentence according to Russell's theory:
[(
x)(Kx)] . ~B(x)(Kx) and
~[(
x)(Kx)] . B(x)(Kx)where the placement of the notation "[(
x)(Kx)]" serves to identify the relevant context for the elimination of the definite description, and we can see that in the first case the negation is inside that context and in the second case it is not."I see. There seems to be some repetitive stuff here:
[(
x)(Kx)] . ~B(x)(Kx) ~[(
x)(Kx)] . B(x)(Kx)notably a repetition of "
". I mean, one thing is to come to love ""; another thing to overuse it. But then Elbourne notes a typo by Grice: when defining 'the' -- as in "the king of France" -- we cannot go on and use 'the' in the definiendum. Therefore Elbourne replaces Grice's "nothing which is not the king of France" for "nothing which is not king of France is not bald". Since Grice is already working with the 'reduction' of the iota operator to the tripartite analysis, he does not need to use the iota operator at all when he comes to use his own version of the square-bracket devices (for 'common ground assignment' -- here's the keyword Grice would use):
The affirmative version being -- using the three conjoints --:
∃x[∀y(Ky ≡ y=x) and ~Bx]
The two negative readings being:
∃x[∀y(Ky ≡ y=x) and ~Bx] -- INTERNAL SCOPE for negation
~∃x[∀y(Ky ≡ y=x) and Bx] -- EXTERNAL SCOPE for negation
Schwarz calls the first the 'preferred reading' and I agree, even if preference is in the eye of the logician ("Notes from the pragmatic wastebasket: on a Grice[I]an explanation of the preferred interpretation of negative sentences") (Schwarz credits both Grice and Myro).
So we must indeed distinguish between Whitehead's and Russell's use of the square-brackets and Grice's idiosyncratic one, for a mere common-ground status assignment, which is always cancellable because is part of a mere conversational implicature, however 'generalised'.
Jones goes on: "
"When these are eliminated we get the two sentences previously quoted by Speranza: ∃x[∀y(PKFy ≡ y=x) and ~Bx] and ~∃x[∀y(PKFy ≡ y=x) and Bx] (the SEP article has slipped from using "K" for "King of France" to using "PKF"!)"
Indeed. If you think of this, one wonders. And even if you don't?
"Present" seems like a different predicate!
and "of France" does not necessarily mean "French" (I know a famous queen of France was Italian, and a Medici, too!)
Jones goes on:
"So long as there is a King of France these two sentences will have the same truth value, according as he is or is not bald, but if there is no King of France then the first sentence will be false and the second true."
Indeed. Or, as Grice prefers, using the neologism he invented and the one Moore did ("Some like Witters but Moore's MY man"), the affirmative "The king of France is bald" ENTAILS the existence of the monarch; the negative merely IMPLICATES it.
Jones goes on:
"The method used by Church is quite different"
Perhaps Grice should have spent more time at the logic library when visiting Harvard! --where I'm sure Church's influence -- at the time when Grice was delivering the William James lectures -- was very strong. Just the list of his doctoral students -- (not necessarily Harvard) is impressive -- C. Anthony Anderson, Peter Andrews, George Alfred Barnard, Martin Davis, Leon Henkin, David Kaplan, John George Kemeny, Stephen Kleene, Gary R. Mar, Michael O. Rabin, Hartley Rogers, Jr, J. Barkley Rosser, Dana Scott, Raymond Smullyan, and Alan Turing!) -- and I'm sure Church is over-quoted in publications by logicians associated with Harvard and that Grice got to know while at that august university.
Jones goes on:
"No special scoping problems arise in [Church's] system. Church does not use incomplete symbols and instead may be understood as using an explicit non-contextual definition of the definite description operator (though in fact the symbol is a primitive and so is defined by an axiom, an implicit definition in Hilbert's terminology). Church here is more interested a solution which is technically satisfactory than one which correctly renders definite description as it is used in natural languages."
I see. Where by 'natural language' we should mean English -- for perhaps there is a natural language that lacks 'the'. The Romans apparently never used the definite article!
For some reasons, the Italians thought they should! But found that since the Romans never used it, they did not know where to derive it from. Eventually, they reached a compromising solution.
They chose a DEMONSTRATIVE expression used by the Roman ('illus') and turned it into the definite article. Since for the Italians it's the SOUND of the expression that matters, from the very same old Roman demonstrative, they get forms of some sort of 'definite article' in Italian which quite differ when it comes to the masculine and the feminine!
Il re di Francia è calvo.
La regina d'Inghilterra è calva.
(The second reads: "The [fem.] queen of England is bald" -- an example owed to Dummett -- he is discussing the unprovability of statements regarding the past and gives the example that Queen Elizabeth I was bald -- daring anyone to CHALLENGE that -- but I suppose he is trying to be funny -- Dummettian humour).
----- INTERLUDE, Short/Lewis, "Latin dictionary" on the source for the Italian definite article.
*********************
ille (old orthog.,
I gen. illī̆us (usu. illĭus in epic and lyric poets;
sol me ille admonuit, that sun, in illa tranquillitate atque otio jucundissime vivere, cum omnis arrogantia odiosa est, tum illa ingenii atque eloquentiae multo molestissima, in illa vita, illum Aurora nitentem Luciferum portet, ergo ille, cives qui id cogit, etc., tum ille, Non sum, inquit, nescius, etc., illum ab Alexandrea discessisse nemo nuntiat, de illius Alexandrea discessu nihil adhuc rumoris, ne illi sanguinem nostrum largiantur, Galba erat negligentior, quam conveniret principi electo atque illud aetatis, illud horae, itaque cum primum audivi, ego ille ipse factus sum: scis quem dicam, qui cum illis una ipsum illum Carneadem diligenter audierat, ille quoque ipse confessus est, huic illi legato, hunc illum fatis Portendi generum, hic est enim ille vultus semper idem quem, etc., hic est ille status quantitatis, est idem ille tyrannus deterrimum genus, eandem illam (sphaeram), cum et idem qui consuerunt et idem illud alii desiderent, illum reliquit alterum apud matrem domi, sic oculos, sic ille manus, sic ora ferebat, non ille timidus perire, etc., Parmenides, Xenophanes, minus bonis quamquam versibus, sed tamen illi versibus increpant, etc., Xenophon, Socraticus ille, auditor Panaetii illius, a qua (gratia) te flecti non magis potuisse demonstras, quam Herculem Xenophontium illum a voluptate, ut ex eodem Ponto Medea illa quondam profugisse dicitur, magno illi Alexandro simillimus, honestum illud Solonis est, illa verba, velocitas, hoc dico, nullum te Aspendi signum, Verres, reliquisse, quaesisse, num ille aut ille defensurus esset, commendo vobis illum et illum, philosophi quidam, minime mali illi quidem, sed, etc., ludo autem et joco uti illo quidem licet, sed, etc., Q. Mucius enucleate ille quidem et polite, ut solebat, nequaquam autem, etc., alter bellum comparat, non injustum ille quidem, suis tamen civibus exitiabile, sequi illud quidem, verum, etc., ex illo fluere et retro sublapsa referri Spes Danaūm, tempore jam ex illo casus mihi cognitus urbis Trojanae, solis ex illo vivit in antris, scilicet ex illo Junonia permanet ira, nunc ego me illa per posticum ad congerrones conferam, hac vel illa cadit, ac ne pervium illa Germanicis exercitibus foret, obsaepserat, ipsum quin etiam Oceanum illa tentavimus, forte revertebar festis vestalibus illa, qua, etc., principio ut illo advenimus, ubi primum terram tetigimus, neque enim temere praeter mercatores illo adit quisquam, nam illo non saxum, non materies advecta est, positiones huc aut illo versae, haec omnia Caesar eodem illo pertinere arbitrabatur, ut, etc., to that very purpose, spectat, illim equidem Gnaeum profectum puto, quid illim afferatur, omnem se amorem abjecisse illim atque in hanc transfudisse, i. e. from her, ----- END OF INTERLUDE.
Jones goes on:
"The result [in Church's system] is clearer and easier to work with, and though definitely not the same as ordinary language, it is probably no worse than Russell's rendition."
Some should revise it!
Jones notes:
"A few years back I did a bit of formal analysis of Grice's system Q using ProofPower, a proof tool for HOL a language based on Church's Simple Type Theory. Because the treatment of descriptions in Simple Type Theory does not follow Russell's Theory of Descriptions, the aspects of [Grice's] system Q providing an alternative scoping mechanism (which were not the main thrust of Grice's work) were not treated, as no scoping issues arise when system Q is rendered in Church's system."
I see. Someone should revise that too!
It seems that once Grice turns from 'names' to descriptions, he forgets, as it were, the subscript symbolism, and starts exploring these identificatory versus non-identificatory uses of 'the'. But the closing remarks of "Vacuous Names" -- in the very closing section -- seem particularly interesting in that he applies the distinction to scope issues, not so much having to do with pure scope per se, but to the occurrence of a description within what he would call a 'psychological predicate'.
Which brings us back to DIOGENES.
He became notorious for his philosophical stunts such as carrying a lamp in the daytime, claiming to be looking for an honest man, as per the painting by Tischbein:
(Other painters that depicted the episode include: Jordaens, van Everdingen, van der Werff, Pannini, Corinth, de Ribera, Castiglione, Petrini, Gérôme, Bastien-Lepage and Waterhouse.)
where -- 'an honest man' is an indefinite description', HM in the SEP symbolic notation.
Indeed, Diogenes is credited with using to stroll about in full daylight with a lamp; when asked what he was doing, he would answer, "I am just looking for an honest man."
The SEP symbol perhaps should not be HM (for 'honest man').
Diogenes Laertius, Lives of Eminent Philosophers Ⅵ:41 is the locus classicus.
Modern sources often say that Diogenes was looking for an "honest man" (in symbols "HM"), but in ancient sources he is simply looking for a "human" (anthrôpos) -- in SEP terminology, "M". Diogenes's implicature, according to the modern sources, seems to be that the unreasoning behaviour of most people around him meant-n to him that these people did not qualify as "human". The use of the lamp is hyperbolic.
Note that he was never looking for "THE" honest man.
But if we now turn to "Mary", to use one of Grice's examples in the concluding section of "Vacuous Names".
She may believe that Tom is the honest man in the gang.
Mary believes that Tom is the honest man in the gang.
She also believes that he wears very elegant clothes.
Mary believes that the honest man in the gang wears very elegant clothes.
Grice plays with scope considerations regarding these type of 'psychological predicates'.
-- Mary believes that the honest man in the gang wears elegant clothes, but she is wrong, since there is no honest man in the gang, never mind wearing elegant clothes.
Grice then turns to 'desires' and wants.
Mary may believe that there the honest man in the gang who wears elegant clothes is non-existant (as per today's gang's membership) but she may still DESIRE to meet him.
Mary wants ('want' is the verb used by Grice) to mary the honest man in the gang that wears elegant clothes.
Grice thinks his approach will surely simplify all this -- and bring in the good conversational implicature for good measure!
Posts
I am very bad in attributing PM to Russell alone.
ReplyDeleteUsually I am talking about Russell's Theory of Types, or related doctrines of Russell's such as his theory of descriptions, which is my excuse for thinking just of Russell.
But nevertheless, if I refer to PM rather than specifically to Russell's work I should credit both authors.
I'm not convinced by the rejection of the implication from simple statements about definite descriptions to the existence of something satisfying the description, since this does seem to hold in PM and I would think in system Q.
ReplyDeleteCan you tell us how Grice comes to the conclusion that the implication does not hold?
The reason for including the definite description in the square brackets in indicating the scope is that there may be more than one definite description in that scope, and you need some way of telling which definite description is being assigned that particular scope.
ReplyDeleteThere may by another which occurs in that scope but whose scope is broader.
You could allow that detail to be omitted and write only "[].s" if only one definite description appears in "s", but I don't think Russell does.