February 27, 2007
JC's Column
I spent some time in the departure lounge of Calgary airport on Friday, with Agustin "the Mexican Multiplier" Rayo and JC Beall, and JC mentioned how annoying he found it that some philosophers used the expressions "philosophical logic" and "philosophy of logic" interchangably. In fact, he thought he might write something up about it and try to get people to take notice. Not being one to stomp on a worthy cause, I asked him whether he'd let me post such a thing to a blog. He agreed, and so I give you JC's Column (an occasional series?) Reform or perish...
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Terminological Theme: Philosophical Logic, Philosophy of Logic, and Formal Philosophy.
There's reason to think that confusion exists over the terminology of "philosophical logic" and "philosophy of logic". It would do the profession -- and, perhaps, aspiring graduate students -- well to have uniform terminology. While terminological differences certainly exist across the English-speaking countries (e.g., in parts of the UK, "philosophical logic" is often synonymous with "philosophy of logic", though not so in Oz), here is a fairly standard -- though admittedly (perhaps perforce) vague -- classification, one that, if broadly adopted, would at least diminish some of the confusion.
A. Formal Philosophy: formal (mathematical) methods used in the service of philosophy.
(This comprises a lot, including philosophical logic, some areas of mathematical logic, decision theory, what Branden calls "formal epistemology", some areas of foundations of mathematics, some incarnations of philosophy of logic, some incarnations of philosophy of language, and much more. Similarly, some work in metaphysics -- particularly, formal ontology, formal mereology, etc. -- would certainly fall under this banner. So, this category is perhaps the broadest category, but it's worth including here. What is crucial is that formal, mathematical methods -- as opposed to just using symbols as abbreviations, etc. (!) -- is essential.)
B. Philosophical Logic: formal logic (usually, applied maths) in the service of philosophy; in particular, a formal account of *consequence* for some philosophically interesting fragment of discourse.
[If we take Logic to be concerned with *consequence*, then philosophical logic aims to specify -- in a formal, precise way -- the consequence relation over some philosophically significant fragment of our language. (Usually, this is done by constructing a formal "model language", and proposing that the logic of the target "real language" is relevantly like *that*.) Usually, philosophical logic overlaps a lot with formal semantics, but may often be motivated more by philosophical concerns than by linguistic data. Work on formal truth theories -- i.e., specifying the logic of truth -- is a familiar example of work in philosophical logic, as are the familiar modal and many-valued accounts of various expressions, and similarly concerns about 'absolute generality' and the *consequence* relation governing such quantification, and much, much else. What is essential, as above, is a specification of a given *consequence* relation for the target, philosophically interesting phenomenon. Whether the consequence relation is specified "semantically", via models, or proof-theoretically is not critical -- although the former might often prove to be heuristically better in philosophy.]
C. Philosophy of Logic: philosophy motivated by Logic; philosophical issues arising out of a given, specified logic (or family of logics).
[While competence in (formal) logic is often a prerequisite of good philosophy of logic, no formal logic or, for that matter, formal methods need be involved in doing philosophy of logic. Of course, philosophy of logic often overlaps with philosophy of language -- as with many areas of philosophy. The point is that philosophy of logic, while its target *may* be mathematical or formal, needn't be an instance of either philosophical logic -- which essentially involves formal methods -- or, more broadly, formal methods. A lot of work on "nature of truth" might be classified as philosophy of logic (though much of it probably isn't motivated by logic, and so shouldn't be so classified), and similarly for "nature of worlds" etc. Whether the classification is appropriate depends, in part, on the given project -- e.g., whether, as with Quine and Lewis, one is directly examining the commitments of a particular logical theory, as opposed to merely reflecting on "intuitions" concerning notions that are often thought to be logically significant. The point, again, is just that philosophy of logic is a distinct enterprise from philosophical logic, each requiring very different areas of competence, and each targeted at different aims.]
It would be useful if the profession, in general, but especially *practitioners* adopted terminology along the above lines. Of course, there's still room for confusion, and the foregoing hardly cuts precise joints. It might be useful to discuss refinements to the above terminological constraints.
One more -- just for those who might be wondering:
D. Mathematical Logic: formal logic in the service of (usually classical!) mathematics, as well various subfields of mathematics. (E.g., standard limitative theorems and classical metatheory is mathematical logic, as is reverse mathematics, many aspects of category theory, many aspects of set theory, areas of abstract algebra, areas of recursion theory, and so on. Mathematicians need have no interest in philosophy to engage in such areas, in contrast with the philosophical logician who is driven to use "mathematical methods" in an effort to clarify the consequence relation of some philosophically interesting "discourse". There's more to be said here, but this is chiefly a post about A, B, and C.)
** One note: it may well be that anyone talented in B is interested in C, but it hardly follows that one who is talented in B is talented in C. Similarly, one who is talented in C may well have little talent or interest in B. My hunch is that, on the whole, those who do B (or do it well) are usually talented in C. It's unclear whether those with a talent in D are naturals for B or C -- or A, for that matter -- but one can think of excellent philosophers who also engaged directly in D. (The obvious such folks were also good at A, B, and C, as well as D. Russell comes to mind, as does Kripke, but there are others.)
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Posted by logican at 03:16 PM | Comments (3) | TrackBack
March 03, 2006
Logic, Logic and Logic
I've just noticed that Greg and JC's book on logical pluralism is now out in paperback.
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December 13, 2005
Logic and Belief Revision
My copy of the newest issue of Phi-News arrived today. You can also get it online here, but sometimes it's nice to have a little bound booklet that you can read over breakfast (if you finish the cereal boxes, of course.)
Starting on page 4 there's an interview with Patrick Blackburn that has lots of interesting stuff about the relation between logic and natural language, but there's also a passage that will have Frege turning in his grave:
I mean, if you go back and look at the logic books that were written about 1870 and 1880 and so on, for instance, John Stuart Mill's text or other texts that were written about that stage, two things strike you. First of all, in on sense they are surpisingly modern. There is actually explicit reference to psychology, to cognitive themes, in some sense, and above all, to language. [p.7, my emphasis]
There's an except from the forthcoming Formal Philosophy
In my graduate quantum theory course, the professor labored to produce a clumsy theorem (I don't remember what) of the form "if p then q." The contrapostive was more intutive and I said so, only to hear the professor adamantly deny the truth of the contrapostive formulation, and when presented with the general logical principle, deny that too. I learned, as had better minds before me, Boole's and Frege's included, that logic is not about people. [p. 20]
Go check out what he has to say about his Heidegger prof. Phi-news is brilliant. They'll be selling it by the checkout in your local supermarket before the end of the year...
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May 22, 2005
Via Richard Zach
John MacFarlane on logical constants.
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May 12, 2005
Woolley Logic
Jim Hacker: Are you telling me the Foreign Office is keeping something from me?
Bernard Woolley: Yes.
Jim: Well, what?
Bernard: Well I don't know, they're keeping it from me too.
Jim: How do you know?
Bernard: I don't know.
Jim: You just said that the Foreign Office is keeping something from me. How do you know if you don't know?
Bernard: I don't know specifically what, Prime Minister, but I do know that the Foreign Office keep everything from everybody. It's normal practise.
- A Victory for Democracy, Yes, Prime Minister
On Monday Richard Zach posted a quote from Buffy as an example of someone asserting a disjunction without being prepared to commit to either of the disjuncts. In the same spirit, the above is a quote from Yes, Prime Minister in which Bernard Wooley believes himself justified in accepting an existential claim, though he didn't infer it from any instances of that claim.
Unfortunately, it is hard to get these kinds of examples to tell us anything interesting about logic (that intuitionist logic is unnecessarily restrictive, for example) once we have distinguished inference and implication, since the example is clearly one about what it is reasonable to believe or infer, given the evidence, and we are looking for a conclusion about implication relations between sentences. Similarly, it is reasonable for me to believe that sun will rise tomorrow, but that does not show that sentences expressing the data on which I base that belief deductively imply the sentence "the sun will rise tomorrow."
I don't see why that should stand in the way of a quote from Yes, Prime Minister though. There are some more below the fold, and some of them have a tenuous link to language, though nothing I won't even attempt the kind of sophisticated analysis that you'll see here. If you want to see a few clips, the BBC has some posted here.
The lines quoted above are immediately followed by:
Jim: Who does know?
Bernard: May I just clarify the question? You are asking who would know what it is that I don't know and you don't know, but the Foreign Office know that they know, that they are keeping from you so that you don't know but they do know, and all we know is that there is something we don't know but we want to know but we don't know what because we don't know. Is that it?
Jim: May I clarify the question? Who knows Foreign Office secrets apart from the Foreign Office?
Bernard: Oh that's easy, only the Kremlin.
- A Victory for Democracy
OK, made it down here? For the uninitiated, the main characters are Jim Hacker, a self-centered cabinet minister (later Prime Minister) Sir Humphrey Appelby, the brilliant and devious head of the civil service for Hacker's department and Bernard Woolley, who is Hacker's Private Secretary (a kind of civil servant.) Bernard is often torn between loyalty to Hacker, and loyalty to the civil service.
British democracy recognises that you need a system to protect the important things of life, and keep them out of the hands of the barbarians. Things like the Opera, Radio Three, the countryside, the law, the universities ... both of them.
- Sir Humphrey, Power to the People
Sir Humphrey: If local authorities don't send us the statistics that we ask for, then government figures will be a nonsense.
Jim: Why?
Sir Humphrey: They'll be incomplete.
Jim: But government figures are a nonsense anyway.
Bernard: I think Sir Humphrey wants to ensure they're a complete nonsense.
- The Skeleton in the Cupboard
Sir Humphrey: It is so difficult for me you see, as I am wearing two hats.
Jim: Yes, isn't that rather awkward for you.
Sir Humphrey: Not if one is in two minds.
Bernard: Or has two faces.
- A Real Partnership
A clarification is not to make oneself clear, it is to put oneself in the clear.
- Sir Humphrey, The Tangled Web
Sir Humphrey: East Yemen, isn't that a democracy?
Sir Richard: Its' full name is the Peoples' Democratic Republic of East Yemen.
Sir Humphrey: Ah I see, so it's a communist dictatorship.
- A Victory for Democracy
Jim: I am going to do something about the number of women in the Civil Service.
Sir Humphrey: Surely there aren't all that many.
- Equal Opportunities
Geoffrey: Personally I find it hard enough to believe that one of us was one of them, but if two of us were one of them ... two of them, all of us could be ... um could be ... um ...
Jim: All of them.
- One of Us?
Bernard: You remember that letter you wrote Round Objects on?
Jim: Oh yes.
Bernard: It's come back from Sir Humphrey's office, he's commented on it.
Jim: What does he say?
Bernard: Who is Round and to what does he object?
- Equal Opportunities
Well, I suppose we could put some sort of government health warning on the rifle butts, this gun can seriously damage your health.
- Sir Humphrey, The Whisky Priest
It [conscription] will give our young people a comprehensive education, to make up for their Comprehensive Education.
- Jim, The Grand Design
Bishops tend to have long lives, apparently the Lord isn't all that keen for them to join him.
- Sir Humphrey, The Bishops Gambit
Peter: Soames has been waiting for a bishopric for years.
Sir Humphrey: Long time no See
- The Bishops Gambit
It is necessary to get behind somebody before you stab them in the back.
- Sir Humphrey, A Conflict of Interest
Sir Humphrey: Unfortunately, although the answer was indeed clear, simple and straightforward there is some difficulty in justifiably assigning to it the fourth of the epithets you applied to the statement, inasmuch as the precise correlation between the information you communicated, and the facts insofar as they can be determined and demonstrated is such as to cause epistemological problems, of sufficient magnitude as to lay upon the logical and semantic resources of the English language a heavier burden than they can reasonably be expected to bare.
Jim: Epistemological, what are you talking about?
Sir Humphrey: You told a lie.
Jim: A lie?
Sir Humphrey: A lie.
Jim: What do you mean, a lie.
Sir Humphrey: I mean you ... lied. Ah yes, I know this is a difficult concept to get across to a politician um ... you ah ... ah sorry ... ah yes, you did not tell the truth.
- The Tangled Web
Master: It's such an awful country, they cut peoples' hands off. And women get stoned when they commit adultery.
Sir Humphrey: Unlike Britain where women commit adultery when they get stoned.
- The Bishops Gambit
Jim: Now this happens and they charge in like a herd of vultures.
Bernard: Not herd, Prime Minister.
Jim: Charge in, like a herd of vultures.
Bernard: No, I mean vultures don't herd, they flock. And they don't charge they ... um ...
Jim: Yes, what do they do Bernard.
Bernard: They ... er ... (does imitation of vulture)
Jim: Sit down Bernard.
- Official Secrets
Many thanks to Shawn Stanley and this very helpful site for the quotations.
Posted by logican at 12:12 AM | Comments (1) | TrackBack
April 19, 2005
Searching for Logic
Adam Morton - holder of the Canada research chair in Epistemology and Decision Theory - spoke here on Friday. His talk was a flurry of impressive and confusing ideas concerning the relation of something he calls searching to three other topics - logic, the teaching of logic, and human reasoning.
Adam's main example of searching is using a computer in a library to do a Boolean search of a database for records. We can get the library computer to search for all the English books by putting "English" in as our search string, and then it will return the records for each of the English books in the library. We also can use the three Boolean operators to form more complex search strings. "English AND Non-fiction" will return the records for each of the English non-fiction books, and "German AND ¬Non-fiction" will return German fiction and "Play OR biography" will return the records of books that are either plays or biographies.
Adam maintains - on the basis of an experimental logic class which he taught last year (sounds cool, right?) - that if you teach this, and then explain deduction in terms of it, it helps to even out the heartbreaking teacup of a grade-graph familiar to many intro logic teachers, which records the fact that half the class found logic easy, and half the class were failing from the get-go. He found that students learned searching readily, and that it was simple to justify some of the less intuitive argument forms in terms of searching. Here is (what I remember of) Adam's dramatised two ways of teaching the disjunction rules:
Disjunction 1a
Adam: George skis in spring OR George likes spinach, therefore, George likes spinach is not a valid argument.
Students: Yes it is. George loves spinach. He eats it all the time. Even in class. Look! There he goes again!
Adam: But suppose the last sentence wasn't 'George likes spinach' but 'George likes apples'...
Students: But the last sentence IS 'George likes spinach.' THIS is the argument we're talking about, stop changing the subject!
Disjunction 1b
Adam: roughly, an argument is valid iff any record which is a search result for each of the premises is also a search result for the conclusion. For example, suppose we have as our premise 'George skis in the spring OR George likes spinach' and as our conclusion 'George likes spinach.' This is not a valid argument because 'George skis in spring' is a search result for the premises, but not for the conclusion. See?
Students: Indeed, Professor Morton. You are very wise.
Disjunction 2a
Adam: George skis in the winter, therefore George skis in the winter OR George likes spinach.
Students: What? You are crazy, sir! Where did the spinach come from?
Adam: Erm...
Disjunction 2b
Adam: Any search result for "George skis in the winter" is also a search result for "George skis in the winter OR George likes spinach." So the argument is valid, see?
Students: But of course, Professor Morton, even the smallest child knows that.
I think he is right that the second approach to explaining the rule is likely to be the more convincing. But it should not really be surprising that we can explain these arguments in terms of searching, because being a result for a search-string is very closely related to being a model which satisfies a statement, and we can already explain validity in terms of satisfaction by models: an argument is valid just in case any model which satisfies the premises satisfies the conclusion. Similarly, the argument is valid just in case any record which is a result for each of the premises is also a result for the conclusion.
In my experience it is usually the case that justifications of logical laws proceed best through semantics. It is common, for instance, for students to balk at the idea that explosion
is valid. And one good way to explain its validity is to say: look, by definition an argument is valid if and only if every model which satisfies the premises satisfies the conclusion. No model can satisfy contradictory premises, so trivially, every model which satisfies contradictory premises satisfies the conclusion. Hence the argument is valid. (Then say reassuring things about it being a degenerate case, just a consequence of our definition, 'valid' is a technical term etc.) This works for disjunction introduction too (once one has explained about inclusive and exclusive disjunction, but I think you have to do that for searching as well.)
Explaining these inferences by teaching searching first, and then explaining them in terms of searching, might have the following pedagogical advantages over the traditional semantic explanation:
One. Even students who never really understand the explanations might learn the useful practical skill of Boolean searching. I don't think we should scoff at this. One of the nice things about undergraduate teaching is that even when we fail to get a student to do good philosophy, we can still help them to improve their writing skills, close-reading skills, and maybe research skills like Boolean searching. This is very cool and likely to be of use to them in later life. Similarly, A basic logic course which gives half the class a grounding in logic, and half the class C+s and the ability to do Boolean searches, is surely better than a basic logic course which gives half the class a grounding in logic, and leaves the other half in tears with C-s, and no new skills.
Two. Students can see the point of learning to search, and everyone learns better with a motivation.
But here are some worries you might have about teaching logic this way, the first two are just developments of things that Adam touched on himself:
One. Maybe the success of teaching logic through searching is a product of the extra excitement and interest involved in teaching such an experimental class. If that were true we might expect results to tail off as the method became standardised. (Well, that might be the explanation, but we should not assume that. We could say the same thing about any new, successful teaching method, and presumably some of those would be better than the methods we use now.)
Two. Searching isn't so helpful when it comes to thinking about conditionals. Students are puzzled when asked to search for "if it is English then it is non-fiction". This matters because the collection of arguments known as "the paradoxes of material implication", for example, these guys:
are traditional sticking points with students, just like disjunction introduction. (You might think this is fine though, since teaching conditionals is a bit tricky anyway. I might have more to say about this in a latter post, since I thought Goldfarb's explanation of the truth-table for the conditional in his recent Deductive Logic was one of the best I have seen.)
Three. New logic students already have difficulty separating syntax and semantics and often find it difficult to understand the point of a completeness proof. I worry that this method risks confusing them further by mixing up the semantic notions with the more syntactic looking database records. Maybe this worry is unfounded; after all, when we give Tarski-models for first order logic, we represent them using linguistic items - numerals, brackets etc. Maybe the database entries are just like that. But in that case shouldn't we be saying that what we're searching for is not itself a database entry, but whatever that entry represents? (like a book?)
Four. I'm a bit worried that searching encourages sloppiness with respect to the objects of certain properties. In taking about searching it's natural to end up talking about searching for "things that are in English" and then end up saying "being in English entails either being in English or being non-fiction since everything that is either in English or non-fiction is English." Though I am sure this makes perfect informal sense, it isn't the way we normally talk about entailment: entailment is a relation between interpreted sentences (or rather a set of interpreted sentences and a sentence. Or multisets of interpreted sentences, or...(sigh)), not between properties, (or even predicates.) Learning to be sensitive to such things is one of the tasks that is difficult for some students, and encouraging insensitivity to it early on might not be a kindness. (Though, thinking about it, one standard exercise is to ask students to demonstrate that "intransitivity entails irreflexivity", i.e. to show ∀x ∀y ∀z ((Rxy &Ryz) → ¬Rxz) entails ∀x¬Rxx. Perhaps by the time we start talking like this their understanding of validity is already safely entrenched.)
Finally, I have worries about introducing all this complicated build-up to the stuff we actually want to teach. There is potential for introducing all kinds of confusion. We'll be multiplying the definitions of validity, the kinds of objects students have to think about when thinking about logic, and with that the potential for confusing those objects and definitions. So a database record is kind of, but not entirely, like a model, but more familiar to non-logicians. This makes it helpful, but treacherous.
Lots of these worries are things that might be allayed by the details. Surely we can be as strict in the way that we talk about searching as we are when we talk about satisfaction. Perhaps this class is best aimed at high school students? Perhaps the class might only be aimed at students who are never going to need to follow a completeness proof. Perhaps there is a way of teaching the conditional through searching. If someday a version of this idea could help to overturn those teacups, then I'm all for it.
Adam also had some interesting things to say about the relation between searching and human reasoning, so I might talk about that in the near future.
Posted by logican at 04:28 PM | Comments (6) | TrackBack
April 07, 2005
Even More Harmony
Lest you think that I am the only person to come down with tonk-fever, Charles Stewart has started a harmony page over on Greg Restall's Wiki, and Matt Weiner (newly of Texas Tech - congratulations Matt!) has an interesting post on truth and local reducation.
Posted by logican at 02:35 AM | TrackBack
April 04, 2005
Dummett on Harmony, Conservative Extensions and Local Reduction/Normalisation
This post will be a brief discussion of a family of related concepts - local reduction/normalisation, conservative extension and harmony - in the light of Dummett's "Circularity, Consistency and Harmony" in The Logical Basis of Metaphysics.
One thing that emerged in the comments here and here, it was that it is easy to get confused about whether Dummett's concept of harmony is identical with the satisfaction of local reduction. The idea of local reduction is reasonably straight forward: a logical constant has the local reduction property just in case, whenever its introduction rule is used to derive some formula A, and then its elimination rule is immediately used, taking A as the major premise, then the proof can be rewritten without that pair of steps. (See this post for more detail, with the caveat that the post ignores the complicated literature on the subject of local reduction with respect to negation.)
Harmony and Meaning
Harmony is slippery because it is something which Dummett has an intuitive grip on and is trying to make more precise. The idea is introduced through talk of meaning. Dummett thinks that verificationists and pragmatists have been investigating different parts of the meaning-elephant: verificationists identify the meaning of a statement with what is needed to establish its truth (verify it), pragmatists identify the meaning of a sentence with its consequences, but Dummett thinks neither sufficient:
Someone would not be said to understand the phrase 'valid argument', for instance, if he knew only how to establish (in a large range of cases) that an argument was valid but had no idea that, by accepting an argument as valid, he has committed himself to accepting the conclusion if he accepts the premises. The analogue holds good for a great many expressions...(213)
Rather, the meaning of an expression encompasses both principles governing how to verify it, and principles governing what follows from it. He wonders if these rules might somehow be in tension. Could they be inconsistent? Or might it be that one rule could be too lax or too restrictive given the other? Such a situation would be a failure of harmony. Since it is difficult to isolate the principles governing a particular expression in a natural language, Dumment takes the logical constants as a case study, since their verification and consequence principles are clearly established in their introduction and elimination rules.
Harmony and Logic
Dummett has other reasons for being interested in the logical constants as well. His interest in harmony stems from his interest in rejecting or accepting change in logic. He thinks that fear of lack of harmony makes us wary of changing logics. He gives two examples, first quantum logic's proposal to weaken the classical [VE] rule, so that this classically valid argument form becomes invalid:
And second the (fictional) proposal to strengthen counterfactual logics (John P. Burgess used to call these and modal logics "neo-classical logics") so that the following argument form is valid:
Wittgensteinian Conventionalisism
One possible view about such new proposals in logic - which Dummett attributes to Wittgenstein - is that any combination of introduction- and elimination- rules defines a legitimate logical constant. Some of these constants are not in use in any natural or formal language, but that is merely a matter of convention. In principle we could even adopt connectives like 'tonk' that allowed us to derive contradictions. (I suppose this might be the explanation of the Wittgenstein quotation at the beginning of Graham Priest's chapter on Paraconsistent Logic in the second edition of the Handbook of Philosophical Logic: "Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from 'consistency.'")
Dummett is looking to resist the Wittgensteinian conventionalist view by finding a way to criticise some combinations of rules. The concept of harmony appears to promise a foundation for this criticism. It looks as if what is wrong with 'tonk' is that the introduction and elimination rules are not in harmony; we can derive far too much from a claim of the form 'A tonk B', given what was required to establish it.
Harmony and two notions of Conservative Extension
Dummett suggests that the notion of a conservative extension can help us to make the notion of harmony more precise. Conservative extension is normally defined over theories, and a theory T2 is a conservative extension of T1 if i) it can be obtained from T1 by adding new expressions, along with axioms (or, according to Dummett) rules of inference which govern those expressions and ii) "if we can prove in it no statement expressed in the original restricted vocabulary that we could not already prove in the original theory." (218)
(Just a thought, but it seems to me that the ideas of language, theory and proof system are all mixed up together here, a la Tarski and Carnap.)
Dummett then turns to natural languages, wondering whether there could be any disharmony in English, and says "if there is disharmony [between the rules governing an expression E,] it must manifest itself in consequences not themselves involving the expression E, but taken by us to follow from the acceptance of a statement S containing E."
And then: "A conservative extension in the logicians' sense is conservative with respect to formal provability. In adapting the concept to natural language, we must take conservatism or non-conservatism as relative to whatever means exist in the language for justifying an assertion or an action consequent upon the acceptance of an assertion. The concept thus adapted offers at least a provisional method of saying more precisely what we understand by 'harmony': namely that there is harmony between the two aspects of the use of any given expression if the language as a whole is, in this adapted sense, a conservative extension of what remains of the language when that expression is subtracted from it."(218-9)
I think this is wrong. It seems to me that it confuses lack of harmony (as he characterised it before: a lack of fit between introduction and elimination rules) with a particular KIND of lack of harmony. If the elimination rule of an expression licences too much, given what was needed to introduce the expression, then yes, we will be able to prove sentences not containing that expression which we were not able to prove before, and we will have a non-conservative extension. But if the elimination rule is too restrictive given the introduction rule, then it doesn't seem that we will be able to prove any more than before. Suppose, for an example, we start out with a classical propositional logic containing only '→' and '¬' and we add 'V', governed by the usual introduction rule, but taking as an elimination rule the weak, quantum version given in the picture above. Surely the result is a conservative extension of the original, on Dummett's definition. And if we had added the usual classical 'V' it would still be a conservative extension - all three systems are just classical proof systems, some of which are easier to use than others. Yet how could the introduction rule for 'V' be harmonious with two non-equivalent elimination rules?
Local Reducation
In chapter 11 of the same book, entitled "Proof Theoretic Justifications of the Logical Laws," we get another provisional definition of harmony, this time just for the logical constants, and it is here that local reducation/normalisation comes in:
"The analogue, within the restricted domain of logic, for an arbitrary logical constant c, is that it should not be possible, by first applying one of the introduction rules for c, and then immediately drawing a consequence from the conclusion of that introduction rule by means of an elimination rule of which it is the major premiss, to derive from the premisses of the introduction rule a consequence that we could not otherwise have drawn."
(I couldn't find an argument for this identification.)
Dummett has just claimed that susceptibility to local reduction (aka normalisation) (within some system) is the formal analogue of being a conservative extension of a theory (once that idea is adapted for natural language.) But since the notion of a conservative extension was originally a formal notion anyway, it looks as if he thinks that a logical constant will be susceptible to local reduction with respect to some proof system just in case adding it to that proof system results in a conservative extension. And, as noted earlier, he thinks that the rules governing a connective are in harmony with respect to a proof system just in case adding that connective to a proof system results in a conservative extension. So it seems that, for logical constants anyway, Dummett identifies all three of the notions we have touched on. (I'm thinking he is a lumper, rather than a splitter.)
Two Questions
There are two questions that I would be very interested in the answers to, if anyone out there knows them:
1) I have argued that Dummett is mistaken in identifying idea that the rules for a logical constant are in harmony (with respect to some proof system), with the idea that adding the constant to a proof system will result in a conservative extension. If the elimination rule allows too little, rather than too much, to be derived (given the introduction rule) then the addition will still result in a conservative extension, even though the rules are not in harmony. Am I right, or I am hopelessly confused on this and being unfair to Dummett?
2) Dummett says that adding a constant to a proof system will result in a conservative extension of that system if, and only if, all the "local peaks", (or non-normal pairs) can be removed - that is, if, and only if, the new connective has the local reduction property with respect to that proof system. I could not find the argument for this in Dummett, but he might be right anyway. Is he? What IS the relationship between local reduction (a.k.a. normalisation) and conservative extension?
Posted by logican at 05:32 PM | Comments (3) | TrackBack
April 03, 2005
Truth and Tonk
I've always been sympathetic to the Tarskian idea that languages can be inconsistent, and in particular to the idea that the truth-predicate is responsible for making natural languages inconsistent (yes, even when I'm being careful to distinguish proof systems from languages and even after reading Herzberger on the topic.) Sometimes, when I thought about this, I liked to think of the truth predicate as a kind of less obvious 'tonk'; introducing it to our language had made it possible to derive contradictions (such as the Liar paradox.)
One might wonder how 'true' had managed to survive in our language, though it is hard to imagine that 'tonk' ever could. But I thought that since the problem with 'true' was a bit harder to discover than the problem with 'tonk', and since it didn't get in the way of our everyday thinking in the same way, and since a truth predicate governed by simple disquotational rules like these ('T' is our truth predicate):
was easy to use, the simple truth predicate had been allowed to remain, even though 'tonk' never would be.
But since the discussion of local reduction/normalisation in posts here and here, I've noticed something that does not sit very well with the view that 'true' is like 'tonk'. Though 'tonk' does not have the local reduction property, 'true' surely does and it is easy to show this, by showing that any proof containing an instance of [TI], followed immediately by an instance of [TE] can converted into a proof without those steps, like this:
It is tricky to know exactly what follows from this. Though it is always tempting to take technical ideas as obviously underwriting exciting philosophical conclusions, (just as it is tempting to take experimental results in psychology as having sweeping consequences in ethics) I suspect that hastiness in this area is more likely to lead to publication than to the right answer. Maybe there are such consequences to be had here, but there in no harm in going slow...
Following Michael Kremer's suggestion in the comments, I have been reading some of Michael Dummett's The Logical Basis of Metaphysics. In chapter 9, "Circularity, Consistency and Harmony," Dummett has a bit to say about the relationship between the local reduction property (though he doesn't call it that), conservativeness, harmony and the nature of deduction. So in the next post I'll have a go at reconstructing his position.
(Who says philosophy weblogs can't do dramatic tension? Logicandlanguage.net cares about those surfers who read this weblog for the plot.)
Posted by logican at 06:49 PM | Comments (9) | TrackBack
April 01, 2005
Proof in Print
Proof and Beauty in The Ecomomist.
Posted by logican at 03:25 PM | TrackBack
March 21, 2005
Inference vs Implication
When I was a graduate student at Princeton (many days ago), we used to joke that Gilbert Harman had only three kinds of question for visiting speakers:
- Aren't you ignoring < insert recent result in psychology >?
- Aren't you assuming that there is an analytic/synthetic distincton?
- So you say, < insert one of the speaker's claims >, but isn't that just conflating inference and implication?
These questions had a tendency to induce harmania in the subject and the inference/implication point, (which is what this post is really about) sometimes seems to me to have the following odd property: hardly anyone gets it if you explain it to them in conversation (I didn't get it that way), but everyone understands it if they read Chapter 1 of Change in View.
So naturally I want to try it out here. The following claims are ubiquitous and false:
- Logic is the study of the principles of reasoning.
- Logic tells you what you should infer from what you already believe.
Each overstates the responsibilities of logic, which is the study of what follows from what - implication relations between interpreted sentences; one can know the implication relations between sentences without knowing how to update one's beliefs.
Suppose, for example, that S believes the content of the sentences A and B, and comes to realise that they logically imply C. Does it follow that she should believe the content of C? No. Here are two counterexamples:
1. Suppose C is a contradiction. Then she should not accept it. What should she do instead? Perhaps give up belief in one of the premises, but which one? Logic does not answer the question - as we know from prolonged study of paradoxes - because logic only speaks of implication relations, not about belief revision.
2. Suppose she already believes not-C. Then she might make her beliefs consistent by giving up one of the premises, or by giving up not-C. Or she might suspend belief in all of the propositions and resolve to investigate the matter further at a later date.
Hence these questions about inference and belief revision - about what she should believe given i) what she already believes and ii) facts about implication - go beyond what logic will decide. That's not to say that logic is never relevant to reasoning or belief revision, but it isn't the science of reasoning and belief revision. It's the science of implication relations.
Convinced? Gil has a short and very clear discussion of this, and the pernicious consequences of ignoring it, in the second section of his new paper (co-authored with Sanjeev Kulkarni) for the Rutger's Epistemology conference.
Posted by logican at 09:05 PM | Comments (6) | TrackBack