Aside from the Stanford Encyclopedia paper "Paul Grice", first published Tue Dec 13 2005 with substantive revision Mon May 8 2006 (which I downloaded without recording the name of the author/reviewer, although I will try to remedy that soon), Editor Zalta, I have not read much of Grice, but I have some comments already on his views about probability.
Grice appears to prefer conditional probability of B given A rather than material implication of B by A (A --> B or even more usually in some works A ==> B). There are supposedly 2 mainstream literature concepts:
1) P(BA), the conditional probability of B given A, defined as P(AB) divided by P(A) if P(A) is not 0 where P(AB) in mathematical probability/statistics is the probability of the intersection of set/events A and B, while P(A) is the probability of set/event A. This has an analog P(A ^ B)/P(A) for P(A) not 0 in logic, because A ^ B, the conjunction of what are now propositions A, B, is a logical analog of AB (also written A upside-down U B) if A or now let us write a is the proposition that A occurs and so on. If P(A) = 0, the conditional probability is undefined.
2) P(A-->B) or P(A==>B) depending on whether one regards (A-->B) or its propositional analog (a-->b) as not requiring "truth" separately from the probability, as in the first symbol P(A-->B), or as requiring "truth" separately from the probability as in P(A==>B), although even these distinctions are sometimes ignored. The first symbol P(A-->B) = 1 + P(AB) - P(A), which can either be taken as a definition or can actually be PROVEN (!) by noticing that (a-->b) = ~(a ^ ~b) = ~a V b and analogously (A-->B) = (AB ' ) ' = A ' U B, from which it follows that we have P(A-->B) = P(A ' U B) = P(A ' ) + P(B) - P(A ' B) = 1 - P(A) + P(BA) = 1 + P(AB) - P(A) from the ordinary laws of probability. P(A==>B) is not usually calculated differently from P(A-->B), and even in the rather disorganized theories of quantum logic the ideas of P(A-->B) are used but with modifications due to certain oddities alleged for the quantum domain.
Readers who compare P(BA) and P(A-->B) will observe that the former involves division, while the latter involves subtraction as well as adding 1 to the result. So the distinction is essentially between division and subtraction of the same quantities, a distinction which has no authoritative mainstream literature exploration in the 30 years that I have searched for it, perhaps because as Grice might say, the former seemed to work and therefore it exists. However, it can be proven (!) that P(BA) and P(A-->B) can differ by 0 in some contexts and 1 in other contexts and any intermediate value in other contexts, and since probability itself only takes values between 0 and 1 inclusive, with 1 being the probability of the entire universe, any result obtained by using one in preference to the other can be as "wrong" as the entire universe! At the very minimum, both should be studied. In my studies (I refer to them as non-mainstream), I refer to P(A-->B) as "the probability that A causes/influences B", although I formerly named it "the probability of material implication of B by A" in the early 1980s. It could with equal "legitimacy" be described as "the probability that if A then B", which sometimes is used for the conditionl probability P(BA).
Grice (see Section 3, Conversational Implicature, of the Stanford paper), claims that P(BA) and not (A-->B) (he doesn't even get to P(A-->B) ) is "appropriately assertible" by arguments regarding the intentions or purposes or goals or directions of conversational speakers. Engineers and Artificial Intelligence people sometimes give similar claims based on some "standard examples" that allegedly involve incomplete information, with a tendency to use "birds flying" as the scenario of their standard example. Remarkably, there is no mathematical or logical proof or even mathematical or logical symbols leading from their example(s) to their conclusions. This would be thrown out of physics and mathematical probability as pure speculation or at most "conjecture", not because it is "new" but because it is "missing" key parts of the argument!
Look again at P(A-->B) = 1 + P(AB) - P(A) and notice that if P(A) is 0, then P(A-->B) is still defined, whereas in P(BA) = P(AB) divided by P(A) (symbolically P(AB)/P(A)), if P(AB) is 0 the division is UNDEFINED! Philosophers sometimes think that events of probability 0 are unimportant or impossible. It is true that the "null set" or "empty event" has probability 0, but so does the top of a table in 3-dimensional space, the tip of a sword in 3-dimensional space, the edge of a table in 3-dimensional space when the probability of continuous random variables is calculated as increasing with volume analogously to Lebesgue Measure (a concept in "post-graduate calculus/analysis", which the reader hopefully will not worry about just now). In fact, for the above types of probability, the probability that a particular person will be at a particular place at a particular time (place specified exactly, such as the exact center of a room or city) is 0 and yet it happens "all the time"! For example, the probability that the Kaiser would be assassinated at exactly the place and time that he was, under assumptions of continuous random variables, is 0, and yet it happened!
The choice of one alternative of measurement, such as P(BA), instead of another such as P(A-->B), because of human intentions in conversation, needs to be "balanced out" or "traded off" by the loss of information or knowledge that can occur if this is decided!
Best Regards,
Osher Doctorow
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I have just noticed that the symbol which refers to conditional probability, which should be P(B given A) where "given" is typed ordinarily by a single short vertical line, did not print out in this post. Wherever the reader sees P(BA) rather than P(AB) in the paper, he/she should read "P(B given A) rather than P(BA) as the meaning of the symbol, because of this.
ReplyDeleteOsher Doctorow
Excellent to have your intriguing views, Osher. I will read and re-read your points and consult with the Stanford entries for "Grice" (by Grandy) and on implicature (by Bach).
ReplyDeleteWhen I think of probability and implicature I think:
First -- of Frank Jackson. His book on "Conditionals" takes up some argument by D. K. Lewis on the application of Grice's conversational implicature theory. Along with Appiah, and others, Jackson argues for a 'conventional' implicature view. I was lucky to study all that when I attended a seminar with Dorothy Edgington on 'if' claims having NO truth-values.
--- Grice has at least one important paper with "Probability" in the title. I first run across it in Levinson, Pragmatics (Cambridge UP, 1983), where it is quoted as being,
"Probability, defeasibility and mood operators".
It turns out that was a typo, and that it should read:
"Probability, DESIRABILITY, and mood operators". I know because when I had to justify my PhD syllabus, I had to quote the source -- along with Allwood, "Logic in Linguistics" for the use of 'boulemaic'.
This paper, by Grice, is dated, exactly, 1973, since it was given in Texas. I went as far as to order the proceedings of the Texas conference ("Implicature", ed. Murphy and Wall). I was slightly disappointed and later told this to Horn and he was amused by this. The preface read, "These are the proceedings of the conference on implicature. Two papers, alas, have not been reprinted, due to technical reasons. One is Horn's -- [it was ch. ii of his PhD] --, the other is Grice's."
Ah well. The thing IS safely deposited at U. C./Berkeley -- I forget what folder.
Grice did quote from this essay in his published thing on Davidson on akrasia (that Grice co-wrote with Baker).
Finally, Grice may be seen to have expanded on that 1973 "Probability" paper in his book for 2001, Aspects of Reason. I have elsewhere, maybe this blog -- "Precis" blog post -- summarised that book section by section. It is a pretty well organised book, and has some comments, then, on "Pr" as the symbol for probability.
Most likely, Grice was influenced on this by work by Davidson. Davidson had worked and researched on probability back since his Chicago days, and he possibly convinced Grice of the importance of the probability calculus and how it relates to basic and fundamental issues in philosophy.
Grice also quotes from Kneale, "Probability and Induction", p. 103. This is a passing reference, but it shows Grice as interested in the ontological commitment of probability theorists. In this case, -- cited in his "Reply to Richards" -- where Richards is his multiple personality to avoid to have to refer to Richard Grandy and Richard Warner every time -- the reference to Kneale is to suggest that Kneale may have had to accept 'properties' as basic ontological items just to justify "secondary" induction -- i.e. the principles at work in 'primary' induction.
ReplyDeleteThe "complete" citation of the Stanford Encyclopedia paper is: Grandy, Richard E., Warner, Richard, "Paul Grice," The Stanford Encyclopedia of Philosophy (summer 2009 Edition). Edward N. Zalta (ed.), URL = < http://plato.stanford.edu/archives/sum 2009/entries/grice/ >. Grandy has email address rgrandy at rice.edu (using the "at" symbol), and Warner has the email address rwarner at kentlaw.edu ("at" symbol).
ReplyDeleteMy wife Marleen and I first explored the probability of material implication in 1980, and I presented a paper on it at a U.C. Berkeley philosophy symposium in 1981, which was published in 1983 in the Far West Philosophy of Education Society (FWPES), although I have very few copies remaining and one of the internet sites that claims to contain a list of internet publications of FWPES and "Philosophy of Education of Society" does not list it! Other publications of mine that discuss the topic are my paper in B.N. Kursunuglu et al (Editors), "Quantum Gravity, Generalized Theory of Gravitation, and Superstring Theory-Based Unification," Kluwer: N.Y., 2000, pages 89-97. Also my paper in Electronic Journal of Sociology 2004 or 2005 online, which at that time was Edited by the Calgary University Sociology Department, Canada (only the second half of the paper is relevant to probability; the first half, on dimensional analysis, was based on printer proofs that somehow were interchanged!). Readers can also find detailed information in my Quantum Gravity thread in sci.physics (it requires a free subscription via Google Groups or some similar server), which is now in its 398.9th subsection, although the site is unmoderated and is so far above most papers in sci.physics that typically only "trolls" (attention-seeking Dunces) make comments on it if at all.
Osher Doctorow
Osher Doctorow
All this is VINTAGE Grice. Or rather, non-vintage Grice. The vintage Grice on probability is nonexistent, but it relates to the earliest work on this in Oxford.
ReplyDeletePerhaps the first was Toulmin. Not really Oxonian, but close enough. His "Uses of Argument" has a beautiful section on "Probability". In fact, that chapter originated from a separate essay which had been reprinted by Flew. In this connection, Toulmin qualifies as proto-Gricean in that he quotes from Urmson, who had worked hard to analyse the 'implicatures' of 'probability' notions.
In the case of Urmson, and the early Grice, the idea of probability is indeed subjective, Bayesian, psychologist, and epistemic. In this connection, I learned from Sir John Lyons.
Lyons notes that,
Probably, it will rain.
contrasts with
Possibly, it will rain.
Lyons claims that if p > .5, you use "probably". If p < .5, you use 'possibly'. That's Sir John forya! I'm never such a stickler, but see his point!
Finally, Noel Burton-Roberts has noted some odd implicatures of "Possibly" and "Probably".
"Possibly, 2 + 2 = 4" is NOT really 'false'. What is necessary IS possible. I have discussed elsehwere this with that brilliant theorist, Seth Sharpless.
Grice avoided possible-worlds semantics like the plague, but having read Myro, and having learned of his System G (which he deviced after Grice) I'm not sure it _is_ a plague: possible-world semantics can be pretty harmless if properly manipulated, and it helps to understand some of the basic algebra besides possibility and probability utterances.
See also Pavel Hajek, "Metamathematics of Fuzzy Logics," Kluwer: Dordrecht, 1998. Hajek is with the Czech Republic in computers in their Academy of Sciences. It actually should be titled "Metamatematics of Multivalued Logics", but at that time Fuzzy Logics were more recognizable by the public.
ReplyDeleteMy wife Marleen and I proved that Lukaciewicz/Rational Pavelka Multivalued Logics are analogs of P(A-->B), while Product/Goguen Multivalued Logics are analogs of P(B given A), and Godel Multivalued Logic is the analog of Independent Probability P(AB) = P(A) times P(B). It turns out that "Generalized Boolean (Multivalued) Logics" are generated by any 2 of the above 3 Multivalued Logics. Note that Hajek's chapter on Probability is in my opinion uninspired - he does not relate it to the 3 types of Multivalued Logics.
Osher Doctorow
Yes, J. L. Speranza, possible-world semantics in the form of Multivalued Logics (that take truth values continuously, usually between 0 and 1 as fractions or decimals and so on) are somewhat remarkable in their applications.
ReplyDeleteHajek explains the 3 types of Multivalued "Continuous" Logics very well, but for the reader who is not a professional mathematician I recommend focusing not on "t-norms" as they are referred to in his book but on "logical conditions" of form a-->b. Once the equations of the latter are understand, then one can always go back to the more complicated t-norms if one wishes to.
The most remarkable thing of all that I have found in my Quantum Gravity thread on sci.physics is that Continuous Multivalued Logics and their Probability analogs can be very useful in analysing our own Universe without even going "outside" to alternative Universes. To be more precise, it appears that OUR Universe follows P(A-->B) rather than P(B given A), that is to say it follows Lukaciewicz/Rational Pavelka rather than Product/Goguen Multivalued Logics, although it also follows Godel Multivalued Logic. Any similarities with P(B given A) are because in certain scenarios they are close in value.
Osher Doctorow
I seem to have just replied to a post by J. L. Speranza which I cannot locate now. I will try to summarize that post briefly.
ReplyDeleteI think that Grice was mostly right about intentions, meanings, communication, psychology, belief, procedures, "ought", "should", and so on, and in fact I think that we can develop specific rules for translating physics and mathematics research into ordinary relatively simple English so that physics and mathematics can be seen as branches of philosophy.
1. Look at arXiv and Front for the Mathematics ArXiv online, which contain mathematics and physics research papers since 1991.
2. Usually the Abstracts at the beginning of those papers are in ordinary English, and often the Summaries or Conclusions at the end are also, because there are so many branches of these fields that "nobody is an expert" in most of them.
3. Papers that relate to Probability 0 (nil) and to differences of variables like u - v or w - z or 5 - 2 or u - 3 are of key importance in P(A-->B) and arguably for philosophy. These includes papers on G. 't Hooft's (Utrecht U. Netherlands) Holographic Principle, the Randall-Sundrum and Kaluza-Klein theories, and so on. For differences of variables, I will try to provide some guidelines later, but most papers provide a list or location of their "Main Results" or "Main Theorems" in the Introduction, and just scan those to see if they involve differences like u - v and so on. Look up the list of symbols, usually early in the paper, which hopefully are defined in English (if not, then they refer to symbols which ultimately have English definitions).
Osher Doctorow
Thanks for the clarifications. Glad to get your general perspective on the relevance of 'ordinary language', free of implicatures, perhaps, on this.
ReplyDeleteI like the idea of a continuum 0-1, and indeed, have often generalised Grice's more conservative use of "t" for true and "f" for "false" to read 1 and 0, respectively. Cargan, who contributes to this blog, has also taught me how the use of the continuum allows for a better taxonomic. E.g. I was onto classifying steps on things as per 1, 2, 3, ... and so on. Cargan argued: why not just start with 0 and proceed up to 1. In this way, my taxonomies became more Tractarian, alla Wittgenstein. But they cannot exceed the limit 1. In a way is like having a fruitful conversation between Achilles and the Tortoise, who were also into the continuum.
----
I have now copied and pasted Grice's references to 'conditional probability' and 'probability of conditional' as they apply to what he, rather presumptuously, but we love him, called "Grice's paradox". Notably his references to Kripke and Dummett, and Grice's own thoughts on the probability of "p ) q". I also commented briefly on the Richards's view that the 'appropriate assertability' of "if p, q" or "p ) q" "tracks the conditional probability of q given p". Very apt way of putting it.
I further made references to Jackson, Mackie, Lewis, Strawson, and Appiah -- not to mention Edgington -- who have all considered how 'if' relates to 'probability', etc.
A probability model greatly depends on the events/phenomena one is describing. Bayesians seem to forget that (as do comp-sci people who automatically apply Kolmogorov axioms...). In many if not most real world problems--ie, economics, medicine, even physics--researchers generally rely on frequentist models, however ...low-tech they seem to some (including logicians...). And what's nice about frequentism and the normal distribution--belief doesn't enter the picture (as it does with Bayesian....of course Bayes does have applications--say in a courtroom, or medicine-- but it's often misused...)
ReplyDeleteGood to have your input on this, J. I will revisit Bayes, etc. and we can expand on "probably Grice". One point I have been thinking is the role of probability (or, rather uncertainty, so this applies to what Urmson saw, wrongy, as the two 'senses' of 'probable') in intention.
ReplyDelete-- "if" . When it comes to "if p, q", while Grice does mention conditional probability and the probability of a conditional (exactly the title of Lewis's essay some 9 years after Grice delivered the William James lectures), the issue seems to be about the strict nature of the implicature involved. As the Richards have it in the Stanford entry on "Grice" 'the implicature tracks the conditional probability of q given p", or rather the implicature, conversational for Grice, of "if p, q" is that the conditional probabiliy of q, given p, is high, i.e > .5.
-- "intend". Now, to please Grice, Davidson argued that probability and uncertainty is similarly implicated when I say, "I intend to do A". Rather, certainty. Davidson claimed the implicature is that the intender thinks the conditional probability of the intended action given the circumstances is > .5. -- the point was repeated by Pears in "Intention and belief" in the Vermazen/Hintikka volume to which Grice/Baker also contributed. Insted, Grice, in an unpublication, "Davidson on intending" -- now at Bancroft -- thought the theory 'too social to be true'. He would rather have 'entailment' rather than 'implicature' here.
In the case of "if p, q", few have defended the entailment view of the conditional probability; in the case of "I intend to do A", Grice's point about 'entailment' should be enriched with his view on 'disimplicature' (as reported by Chapman in her _Grice_, which relate): some speakers use 'if' and 'intend' idiosyncratically: so what is an implicature for a few, may be an entailment for a fewer, and so on. "Implicature happens" but so does entailment, or for that matter, 'disimplicature' and loose talk!