29 September 2013

Peregrin on "Logic and Reasoning"

I just finished listening to this talk; I quite liked it. Peregrin defends the view that logic is "constitutive" of rationality, and not merely normative for it: the laws of logic make the game of giving and asking for reasons possible, not what tells you which moves within the game are good ones or bad ones to make. This isn't a new idea, but I very much liked seeing someone defend it without leaning on the fact that they are also trying to exposit Kant's views on general logic, or the views of the author of the Tractatus.

There is one bit that struck me as odd, though, which is why I'm writing this post: at one point Peregrin is concerned to show that his view can still claim that logic is "a supreme arbiter of rationality" despite not allowing the laws of logic to be violated (in general -- he allows occasional violations by single members of linguistic communities, but says very little about the details of how even this could happen given that the laws violated constitute the act which is supposed to defy them). His reply seems odd, though: He says that being rational may be "having implications, negations, etc., not using them in an appropriate way."

This seems wrong for at least two reasons. One is that minutes earlier he'd allowed that there might be linguistic communities which don't have implication (a group in Siberia studied by a Soviet scientist is discussed as a possible example), and so don't have modus ponens because of this. So, what he put on his slide seems to contradict his own commentary on it, unless he wants those linguistic communities to fail to be rational (which is obviously undesirable). The other is a generalization of the point Peregrin made when he allowed for a linguistic community to lack "implication": Why should the material conditional, classical negation, exclusive disjunction, etc. be so important for being rational as such? They seem to be of fairly recent invention, and sit uneasily next to the ordinary-language terms often used to characterize them (as anyone who has ever had to teach undergraduates well knows). How could having them in one's language be so important as to constitute the game of giving and asking for reasons?

The remedy to this, I think, is just what Sebastian Rödl talks about in the first chapter of "Categories of the Temporal": we should retain the idea that logic is constitutive of thought as such, but not identify this logic with a calculus (and so in particular not with classical logic and its material conditional etc.): the possession of any particular logical calculus is an optional tool in reasoning, not constitutive of it. The kind of logic which is constitutive of thought as such is transcendental logic, not general logic: it essentially involves reference to thought's relationship to its objects (and so refers to inquiry, the process by which those objects are known). Being rational cannot plausibly be "having implications, negations, etc." but it is (at least plausibly) having distinctions of truth and falsity, of oneself as an inquirer who can err and be corrected by others or correct oneself, of the objects of inquiry as being capable (at least in many cases) of settling questions about them when some are in error or ignorant of them, etc. It is hard to see how one can be a rational subject without such notions; it is the task of a transcendental logic to outline them and give their laws, without which thought as such is impossible.

I think this sort of view also helps to make sense of so-called disagreements in logic (as between classical and intuitionist logicians, or dialetheists and everyone else): they are not expressing conflicting views on "the" logic constitutive of thought as such (which would have to be a transcendental logic), but expressing views which disagree with one another on which calculus captures the laws of thought as such (or which are normatively correct in describing how one should proceed if one wishes to think rationally, as they tend to think of it). Accepting or rejecting the law of excluded middle, or rejecting (and not merely accepting) the law of noncontradiction can then be seen not as doing the impossible (if laws of logic are constitutive of thought as such), but as advancing rival views on a distinct question -- one which the transcendental logician may reject as relying on a false assumption, that the logic of thought as such is the logic of a calculus, is general and not transcendental logic. So long as it is not among the laws of transcendental logic that one has a particular view of transcendental logic (which would be a surprising result), the disagreements in logic do not need to be seen as violating laws which are constitutive of thought as such, and can be regarded as genuine disagreements without overturning the view of logical laws as constitutive.

It might be thought that the existence of dialetheists still posed a problem: Didn't I say that a distinction between truth and falsity was (plausibly) involved in the laws of transcendental logic, and isn't this just what the dialetheists want to argue about? Are they not still seemingly violating the laws which are supposed to be constitutive of thought as such?

I think not: Even in the extreme case of Graham Priest's acceptance of every version of the denial of the law of noncontradiction he is presented with, one can find him affirming (in "Doubt Truth to Be a Liar") that one cannot both affirm and deny a proposition simultaneously. He views this as a psychological claim, and his ultimate evidence for it is phenomenological, but he recognizes that he needs something of this sort to make his own views so much as stateable: if he has access to no such distinction as this, then he can't intelligibly say that even dialetheists believe that there are monaletheias (propositions which are only true or false, and not both): without something in his system to keep two truth-values apart in the end, there is nothing to keep his opponent from asking "Yes, yes, you accept that P is true and not false, and is a monoletheia -- but how do I know you don't also hold it to be false and not true, and a dialetheia?" (The context in which Priest invokes an absolute psychological-phenomenological distinction between affirmation and denial is one in which he has rehearsed the many "revenge" Liars that a dialetheistical treatment of the Liar leads to; Priest needs some way to handle "This sentence is false and is a monoletheia", which on his treatment is both true and false and has both one and two truth-values, without undermining his own view that dialetheism is compatible with classical monoletheic logic "mostly" holding in everyday reasoning. I would look at my copy of the book to confirm this and find references, but I am lazy and as far as I know I'm the only person who feels a need both to defend the constitutive view of the laws of logic and to make sense of Graham Priest.) So I think that Priest can be seen as still abiding by (and affirming and not denying) the distinction between truth and falsity at the level of generality at which transcendental logic needs to deal with such truth-values: he just has peculiar views about how truth-predicates should be used in formal languages, etc. When it comes to knowing that when a question has been settled in inquiry it is not also still open, or that we can err when we hold that a question has been settled or not and might need reopening, Priest (and I think any other dialetheist who considers the issue) says nothing but what the transcendental logician says we all know qua rational beings: what is, is, and what is not, is not.

03 June 2013

Rödl on Kant's First Analogy of Experience

Here's how Kant states the First Analogy in the A-edition of KRV, where it is labelled the "Principle of Permanence": "All appearances contain the permanent (substance) as the object itself, and the transitory as its mere determination, that is, as a way in which the object exists." (A182)

In the B-edition of KRV, Kant adds a paragraph to the beginning of this section. In "Logical Form as a Relation to the Object", Sebastian Rödl goes through this paragraph sentence-by-sentence (omitting some parenthetical remarks of Kant's and the final sentence). As Rödl presents it, Kant argues as follows (following the Kemp Smith translation, B224-5; each of these is a single sentence of Kant's German):

1) All appearances are in time; and in it alone, as substratum [...], can either coexistence or succession be represented.

2) Thus the time in which all change of appearances has to be thought, remains and does not change. For it is that in which, and as determinations of which, succession or coexistence can alone be represented.

3) Now time cannot by itself be perceived.

4) Consequently there must be found in the objects of perception, that is, in the appearances, the substratum which represents time in general; and all change or coexistence must, in being apprehended, be perceived in this substratum, and through relation of the appearances to it.

5) But the substratum of all that is real [...] is substance; and all that belongs to existence can be thought only as a determination of substance.

Claim 1) is established by the Transcendental Aesthetic. See II.ss4.1, the first paragraph of the Metaphysical Exposition of the Concept of Time, for the argument; the idea is that we can't build up to a representation of time by first perceiving items as simultaneous or sequential and then abstracting "time" out of those perceptions, since perceiving them in that way requires already seeing them as in time: so the representation of time is a priori, as it is only against the background of time that we can represent things as simultaneous or sequential.

2) follows from 1): if we need to use time as a background against which to represent things as happening simultaneously or sequentially, then representing anything as changing will also require this (as changes are sequential: there is something changed from and something changed to). The background against which change is represented is not itself represented as changing, but is what makes the change intelligible as a change. As this background is time, time must be represented as remaining without change. Rödl here notes that it seems that the way to represent the logical form of thoughts of items in time is as "determinations of time", as Kant says: something like "A exists at t1", with the time "t#" being part of what the thought determines. Simultaneity would then be existence at the same t#, succession would be existence at a later or earlier t#.

3) I am sure Kant argues for somewhere, but I'm not finding a reference at the moment. That time is not itself an empirical intuition follows from the argument for 1), but I don't see where Kant actually makes that inference explicit. But it's a trivial enough point that he hardly needs to: once a distinction is noted between perceiving objects in time and perceiving time itself, it is easy to grant that we do the former and not the latter. (There's a reason we need clocks.) Rödl notes that this simple point poses a problem for the idea that the way to represent an item in time is "A exists at t1": we are trying to articulate the logical form of an appearance of something in time. So if the logical form of an item with a temporal position (simultaneous with, earlier than, or later than some other item) is "A exists at t#", then we cannot be given any items with temporal positions in perception, as perception does not provide us with a time to put for the schematic "t#".

4) From this Kant concludes that "there must be found in the objects of perception [...] the substratum which represents time in general": Rödl puts the point thusly: "what is given in intuition—appearances—as such contain something that represents time in the sense that something is conceived as a determination of time in virtue of being apprehended as a determination of it. Apprehending A and B as determinations of this thing, we apprehend A and B as succeeding, or as simultaneous with, one another." (p.365) The problem with thinking that "A exists at t1" could represent the logical form of a thought of an appearance having a temporal position was that nothing was given which could stand in for a "t#": the solution is to see A as a determination of time (as having a temporal position determined in the thinking of it) in virtue of it being represented as a determination of something which is given in perception, and which functions as a substratum against which simultaneity and succession can be represented.

5) And here Kant draws the conclusion of the Principle of Permanence: All appearances are in time, as the Transcendental Aesthetic established; this entails that they are given to us as simultaneous with one another or earlier or later than one another. To be represented in this way (which they must be to be given to us as appearances in time), there must be found in these appearances something which is grasped as a substratum against which temporal positions can be apprehended, and the various temporal positions must be represented against the background of such a substratum. But such a substratum in appearance is just what the Principle of Permanence calls "substance": there is in all appearances in time a distinction that can be made between substance and "mere determination", which Rödl calls "state". Rödl puts it this way: "We perceive that A succeeds or is simultaneous with B, as we apprehend A and B as determinations of time [by perceiving them thus]. And we apprehend A and B as determinations of time, not by predicating A and B of a time as in “[A exists at t1 and B exists at t2] ”, but by predicating A and B of a substance as in “S was A and is B” . Temporal thought bears a predicative structure. It is not articulated into a time and what is at this time, but rather into a substance and its states. It is in virtue of being thus articulated that a thought distinguishes a time from what is at this time and thus represents its object as temporal. This completes the proof." (p.365) "S was A and is B" represents a substance, 'S', which can be in different states at different times (is now B, was previously A) while remaining the same substance. Thoughts which represent substances as substances exhibit this form: the substance thought of is known as something which can bear contrary predicates at different times, as Aristotle put it; it is a perceptible item which can be seen to change or remain the same at different times. Being able to perceive such items is what enables us to have appearances which have a temporally order internal to how they are given to us: thinking thoughts of the form "S was/is F" is how we can represent some appearances as coming before, after, or alongside others. A Hume-style skeptic about substance needs to make this intelligible by thinking thoughts of the form "A exists at t1" and "B exists at t2", and this cannot be done, as time is not perceptible. Thus the proof against this sort of skepticism is complete: in all appearances substances are given, as the substrate of changeable states.

Rödl continues on to note a few things from the later pages of the Analogy, but the impression given is that the rest is a mop-up operation: the important work is already done in that first five sentences. There is a small puzzle about this, given that these sentences were added in the B-edition, and replaced a short section that doesn't contain this argument. But I think it's plausible that the argument Kant puts more clearly in those five sentences can be found in the first few paragraphs of the A-version Analogy, spread out more widely and unclearly.

There are two more puzzling things about Rödl's handling of the First Analogy. One is that it might seem that Rödl's way of handling perception of time-positions can't handle the relative positions of states of distinct substances. I will handle this indirectly, by first looking at something Rödl points out. It might be thought that one advantage of representing temporal thought by "A exists at t#" is that the various numbers which slot in for '#' will all line up of themselves: t1 is before t2, which is before t3, etc. But this is an illusion: "the things to which “t1” and “t2” refer, and the unity of these, cannot be perceived. Here nothing satisfies “the condition of the empirical unity of time” (A 188/B 231). By contrast, in “S was A and now is B”, there is no need to connect two things determined by A and B respectively, for there is only one thing, the substance, determined by both. Its unity represents the unity of time. In this way is the “condition of the empirical unity of time” satisfied." (p.366) That is, the attempt to represent temporal thought by "A exists at t1" and "B exists at t2" etc. fails to satisfy a demand established by the Transcendental Aesthetic: there is only one time, and all times are limitations of it. There is nothing in the representation of temporal appearances by "A exists at t#" which guarantees that everything which stands in for "t#" will be part of a single time. Rödl's way does meet this demand: If S was A and is now B, this can only happen in a single time through which S persists, as is represented by the single symbol 'S' in the notation.

Now, what does that have to do with the following worry: Rödl's way can't handle "S is A" and "P is B" being simultaneous, since those two thoughts don't share a substance? Here I think I see why Rödl talks of "states" and not "properties" or "determinations": it is tempting to think that there is something special about monadic properties or determinations, but it is much less tempting to think this about states. The 'A' in "S is A" can perfectly well be "near the P which is B"; this is a state a substance can be in, and the thought of it includes the time-determination of the other substance, and represents the empirical unity of the times in which these substances exist. Here I am speculating as to how Rödl handles this question; it seems like something he should address, but I haven't read anything where he takes it up. But I don't see any reason my way wouldn't do fine for him: relational states are perfectly good states, and an appearance of multiple substances in relational states will satisfy Kant's demand to respect the empirical unity of time. (The Transcendental Aesthetic can, I think, already be taken to have shown that all appearances will be related in a single time; it will thus not be necessary to further guarantee that all states of substances will be so related, as these are merely a species of the genus "appearance". That is to say, I do not need to establish that all substances will stand in relational states to one another which determine their time-relations; the Aesthetic does the needed work. What is needed is only to provide a way of presenting substances in thought which does not violate the condition of the empirical unity of time, as "A exists at t#" did. Provided that substances are in fact related to one another in time, polyadic state-variables represent them as in a single time.)

There is another puzzling thing about Rödl's way of handling the First Analogy. The five-sentence proof from Kant he looks at is only in the B-edition, but he only presents the A-edition's statement of the Principle of Permanence. Related to this, he does not discuss the last sentence of the paragraph added to the B-edition: "And as it is thus unchangeable in its existence, its quantity in nature can be neither increased nor diminished." This sentence goes along with the B-edition statement of the (slightly renamed) "Principle of Permanence of Substance", which says "In all change of appearances substance is permanent; its quantum in nature is neither increased nor diminished."

Rödl mentions this sentence only in a footnote: "We disregard the last sentence of the proof (“Da diese also im Dasein nicht wechseln kann, so kann ihr Quantum in der Natur auch weder vermehrt noch vermindert werden.”), which does not pertain to anything stated in the First Analogy in the A-edition. It is a further thought, with its own difficulties, which lie beyond the scope of this essay." (p.369)

Now, it is curious that Kant changes the statements of the principles of the Analogies in the B-edition. But he leaves so much of the argumentation unchanged in these sections that it seems hard to deny that he thought he merely reworded them, and left their substance unchanged. But I think Rödl is simply right about this much: Nothing in the First Analogy in the A-edition supports the claim that the quantity of substance in nature is constant. Substances are things which can change in various ways while remaining the same substances; Kant tells us nothing here about why "quantity in nature" is something unchangeable. More problematically, I don't think Kant has given a sense to "quantum" here: Does he mean that substance in nature does not change in total mass, or in total energy, or in total extension, or in some other quantity measured in some other way? There are many quantities of substance in nature which do change: the number of dinosaurs is a quantity in nature. As far as I've been able to tell, at this point in the Transcendental Logic Kant has no grounds whatsoever for speaking of a single quantity of any sort which is constant for all substance at all times: the Principle of Permanence is entirely compatible with an Aristotelian world of many finite substances with different natures and different ways of being. (The only thing I can find which can even pretend to be an argument otherwise is the Anticipations of Perception, with its talk of intensive magnitudes of reality -- but this section also does not establish that there is a single scale of reality-magnitude, but only that any reality given in sensation is given in a scaleable way.)

Here I suspect Kant changed the B-edition of KRV to make it line up more smoothly with his physics, which he had in the meantime laid a groundwork for in "Metaphysical Foundations of Natural Science". But this sort of move is illicit, by Kant's own standards: principles of a special science such as physics are not established before the System of Principles of the Pure Understanding, as these pure principles are used in determining the principles of the special sciences (which are partly empirical: in MFoNS Kant relies on experience for the claim that bodies have weight, if memory serves). If Kant hasn't established that the sort of substance which must be found in the appearances to make experience possible is the kind his preferred empirical science talks about, he shouldn't pretend otherwise: and viewed from our later vantage-point, we should feel welcome to jettison the supposed necessity of Kant's Newtonianish physics, and feel no compulsion at all to read it back into the Analogies -- even if Kant himself did this while revising the B-edition. It is only Kantian for us to attempt to understand the philosopher better than he understood himself.

02 June 2013

"Kantian Humility"

I read about half of Rae Langton's "Kantian Humility"; I skimmed the chapters between the one on phenomenal substance and the one on primary/secondary qualities. Here are some thoughts I had.

She latches on to some passages that I find fairly opaque, and is able to give a sense to them (the stuff about matter being constituted by "mere relations"), but I felt like her overall interpretation was severely hindered by her unwillingness to discuss core arguments of the Transcendental Analytic. For instance, she doesn't commit herself to any view as to how the argument for the First Analogy is supposed to work. But, she's committed to reading "phenomenal substance" as akin to "wax duck": phenomenal substances just aren't substances (and in her defense, she shows that this is how Wolff used the phrase); the schematized category of substance is not a species of the pure category for her, and on her reading Kant denies that we are ever given anything in experience which "can only be thought as subject, not as predicate". She puts a lot of weight on Kant's remark that we can make anything a "logical subject" in a judgement without that saying anything about whether or not it's a substance ("Love is abiding" and "Yellow is pale" don't make love or yellow into metaphysical substances), and holds that this shows that treating "matter" as a substance is only done by Kant in a "comparative sense": that it is a logical subject relative to empirical predicates of matter, not that it can't be thought of as a predicate of the thing-in-itself. And in fact she holds that this is how it is: the only substances for her Kant are things-in-themselves, which can't be thought as predicates of anything. (She assumes this throughout, without any argument that I saw. I don't know why someone like Spinoza wouldn't deny it, and claim that these monadic "substances" are in fact mere predicates of God; I've never been clear on how Leibniz prevents his monads from collapsing into God in this way, though it's clear he wants them not to.) All of this means that, in fact, no knowledge of substance can play any part at all in the First Analogy: the subsistent in time is only a permanent predicate, not something which can only be thought as subject. This strikes me as ruling out any plausible interpretation of the First Analogy, as it makes the relationship between its principle and the category associated with it essentially null.

She constantly turns to Kant's physics when discussing what Kant means by "matter", and reads his dynamical theory of matter as providing argumentative support for large swathes of the critical philosophy. (How this doesn't render the entire project circular is a problem I don't think she ever addresses: From what I recall of the Metaphysical Foundations of Nature, Kant uses the Analogies to argue for his force-theory. So he can't presume that this is how matter works when arguing for the Analogies themselves.) But if the permanent in experience is the matter explicated by Kant's physics, then it's not something we are consciously aware of as such: attractive and repulsive forces are not something we can sense directly. She takes a very radical move here, and severs the connection between the senses and intuition: she reads the Third Analogy's principle as committing Kant to the view that all matter affects us at all times, and that it is only because most of these effects are too small ("lacking in reality") to be brought to consciousness that prevents us from being aware of all objects at all times. This puts Kant's view of experience very close to Leibniz's: every subject represents the entire world at all times. In her defense, she quotes Kant saying things that seem close to this radical a view in his reply to Eberhard in "On a New Discovery etc.", which I haven't read. (I remembered reading Allison's introduction to it years ago, and then skimming the text to confirm that it was how Allison had said it was. But all of the details are now lost to me.)

But if this sort of neo-Leibnizian view is Kant's, then it seems simply incoherent: if external bodies are given to us only by means of attractive/repulsive forces, then the fact that forces sum means that external bodies are not given to us individually: two forces of velocity X and one force of velocity 2X are not distinguishable, and so all of those remote objects which Langton's Kant has making "subconscious" effects on us are not distinguishable (in principle) from a single external object making a single impression on us whose force is the sum of those effects. It might seem that her Kant also faces the problem of how to distinguish between proximal and distal causes of the effects on us, but I think that's actually not a problem for her Kant if the issue of forces summing isn't: since Newtonian forces act instantaneously at a distance, a proximal and a distal stimuli simply produce distinct forces on us, and so if these forces can be distinguished then so can the proximal and distal stimuli.

I don't think Leibniz's view has these problem, because Leibniz thinks that forces, which are relational properties of bodies, are "well-founded phenomena" which reduce down to simple properties of monads: so the representation in a single monad of some particular lump sum of force is analyzable (by God, not by us finite provers) into non-relational properties of monads, and it is only by means of these non-relational properties that Leibniz has each monad representing the entire world. But Kant is adamant about relations not being reducible to non-relational properties, as Langton shows at length, so I don't see how her Kant can go from the forces to anything which represents the world -- even setting aside that Kant has independent arguments against Leibniz in these quarters (such as Leibniz presuming the identity of indiscernibles, which is needed to make his monads "represent" individual objects by means of non-relational descriptions of them). I don't know how her Kant is supposed to be able to represent individual objects merely by having forces impinging upon it at all, but she is explicit that this sort of physical interactionism is what drives Kant's thoughts about thought's receptivity.

I found the book disappointing overall, but if Langton's not right about what "matter being constituted by mere relations" means, I don't know what those passages in Kant mean. (Langton can here apply Modus Tollens; I apply Modus Ponens.) So the book is worth looking at just to see how she handles the passages her view handles well; it is a desideratum for any alternative view of Kant's matter-doctrine to be able to handle them as smoothly, but without sacrificing so much of the rest of Kantianism.

25 May 2013

McDowell on Cognitive Science and Epistemology

University College Dublin has put online two lectures (and an interview) McDowell gave there recently; kudos to them for doing this. It is all the to better that more talks are going online these days, so that it's easier for those who want to hear them to do so.

I just finished listening to the first one, "Can cognitive science determine epistemology?" Its topic is familiar for anyone who's read McDowell very much, and its content replicates some material available elsewhere recently: McDowell is here replying to Burge's recent attacks on him, just as he was in a talk he gave when visiting IU three years ago.

It's actually interesting to listen to McDowell's views on cognitive science while at IU: he is sanguine throughout on the current state of representationalist cognitive science, and explicitly doesn't want to make any waves or raise complaints against it. This is not exactly the attitude one finds most often at IU. (B.F. Skinner was at IU for nine years, and reportedly designed the psychology building; his influence is still felt here.) It's especially surprising to hear him speak of "representations" so blithely, though I suppose this is in accord with the "representations without representationalism" slogan he urged on Rorty back in "Rehabilitating Objectivity".

But I think that, even if he does harbor secret doubts about the current state of the science (which I have no reason to believe he does, but which I think is the horse he should bet on), it makes sense to be ameliorist in a lecture like this: there's no reason to fan the flames of people who, like Burge, see self-standing epistemology as just an armchair form of psychology by running together such a self-standing epistemology with a rival vision of psychology. This is especially true because of the claim he concludes his talk with, which I think might represent a genuine shift in his thinking: he claims that cognitive science needs epistemology (as a self-standing enterprise) to be able to identify contentful states in the way that it does.

I know McDowell has long claimed that alternative accounts of perceptual knowledge make the very idea of perceptual knowledge unintelligible (this is a central claim of "Knowledge and the Internal", and is tied to the central arguments against Davidson in "Mind and World"), but I can't think of another place where he so straightforwardly says that an empirical science (as opposed to a (discardable) alternative account in philosophy) relies on something that his sort of account of perceptual knowledge provides. I think that's a stronger claim than anything he made up through the 90s, at least; I would need to reread the exchange with Dreyfus to see if there's a similar claim made there, given how Dreyfus views his own relation to psychology. If this sort of thing is true, then philosophy can't be a purely therapeutic enterprise: the sciences need it to do something else in addition to showing flies the ways out of flybottles, in order that they may be sciences. A certain sort of constructive work (in some sense of the term "constructive") is needed from philosophy to allow a properly empirical science to proceed. I don't think that McDowell would have been willing (or happy, at least) to say this at all points in his career, but he seems to have said it explicitly here. I have to wonder whether he thinks claims of this sort generalize beyond the field of representationalist cognitive psychology -- might physics rest on principles like the analogies of experience, or biology on a logical account of the lifeform? If there are some dependencies between empirical sciences and accounts of rational self-consciousness, how deep (far?) do these dependencies go? (This way lies Naturphilosophie, which is worrying and at least a little heady.)

Side-note: Michael Friedman argues for something like the claim McDowell made in his "The Dynamics of Reason", that philosophy needs to provide (and has, in the past, provided) the sciences with something they need in transitions between paradigms during periods of revolutionary science. I was unconvinced that Friedman had shown anything stronger than the claim that, in fact, work done in philosophy was instrumental in making turn-of-the-century revolutions in physics possible; I'm not sure how strong of a case can be made for the usefulness of philosophy in the other scientific revolutions he briefly discusses, and am skeptical of a general claim he argues for, that philosophy in general provides the connections between mathematics and mathematical physics that physics needs. He clearly wants to put forward a vision of scientists and philosophers working in tandem in a certain way, but I came away from the book with the impression that he's too weak to a nostalgia for logical positivism in its heyday. I couldn't see a future for that sort of thing, because I couldn't see how his account generalized beyond the weird combination of Neokantianism and crises in physics that lead to the syntheses he spent so much time looking at. On the authority of Einstein, work in philosophy really was important in certain reformulations of physics -- but it's in the nature of revolutionary science that we can't tell in advance what sorts of reformulating will be called for, and so I don't see how philosophers could intentionally try to provide it. It was just a happy accident that it did in that case, it seems to me. But it was a fun little book regardless.

18 May 2013

Dummett's Frege and Rödl

I have not been good at blogging recently; I have neglected comments for about a year, and have written nothing. Apologies to those who I did not respond to (which by my records include Daniel Nagase, N.N., Charles Wolverton, Evan Kuehn, and Duck; I could make excuses for this bad behavior, but they would be of merely psychological importance, and that's a poor way to start a Frege post).

But if I cannot manage to blog well, then I should at least blog badly more often; here then are things I typed to try to get clear to myself on what Rödl is getting from the Dummett essay he leans on in the first chapter of "Categories of Temporality".

In "The Context Principle: Centre of Frege's Philosophy" Dummett claims that Frege tried to use the context principle to justify his "realism", his treatment of numbers as objects, while simultaneously using it to answer the question of how numbers can be given to us ("epistemologically", in Dummett's term). Dummett's idea is that Frege tried to answer this question (and establish his realism) by fixing the truth-values of every sentence in which a number-word appeared; in the Grundgesetze, this is fixing the truth-value of every sentence in which a term for a value-range appears.

The line of thought seems to be this:
1. Frege can fix the truth-values of sentences by (in part) stipulation, going through each possible combination of a value-range term with a primitive concept-term and assuring that it has a truth-value.
2. For Frege truth-values are the Bedeutungen of sentences.
3. The context principle: the Bedeutung of a subsentential term is determined only by the way it contributes to the Bedeutungen of the sentences it appears in.
Which gives Dummett's Frege the conclusion that by stipulating truth-values for each (atomic) sentence in which a value-range term appears, he has also settled what the Bedeutungen of value-range terms are.

Dummett contrasts this to a way of providing terms with Bedeutungen which would go against the context principle: first establish the domain over which the variables of the language can range, and then determine for each term of the language which needs a Bedeutung which of the items from that domain is to be its Bedeutung. As this initial domain-determination requires a grasp of the possible values of variables anterior to the securing of Bedeutungen to the sentences of the language in which those variables appear, it violates Dummett's version of the context principle.

Dummett spends some time on the objection that one might ascend to a metalanguage to avoid this violation of the context principle: if the domain-determination for the object language takes place in sentences of another language, then the context principle is not sinned against. But Dummett argues that Frege did not mean to be giving Bedeutungen to the terms of Begriffschrift which were already understood by anyone who could read his German: this would make some of Frege's prose a necessary element of his logic, which Frege clearly wants to avoid. The Begriffschrift is supposed to stand on its own, with the German prose serving only as a propaedeutic to its understanding; in particular, if the question of whether numbers are objects or how they may be given to us is to be solved by the Grundgesetze project, then it cannot rely on the German-reader's already knowing these objects and including them in the domains over which Begriffschrift variables are allowed to range.

Dummett's objection to Frege here is put rather tersely, and I am writing this post because I need to do some work to unpack it for myself: "The fallacy appeared at the very first step. The stipulations governing the primitive functors [what I called concept-words, above], including the criterion of identity for value-ranges embodied in Axiom V, could be determinate only if the domain, consisting wholly or largely of value-ranges, was determinate; but the domain was in the process of being determined by fixing the Bedeutungen of the value-range terms, and so the procedure went round in a circle." (p.18)

The circle seems to be this: Frege tries to establish that among the objects are value-ranges (the Bedeutungen of value-range terms) by stipulating truth-values for each atomic sentence which results from combining a value-range term and a primitive function of Begriffschrift. But the sentences which are here stipulated to have truth-values can only have truth-values if the domains over which their variables range is determinate; an indeterminacy in the domain means an indeterminacy in what can count as the sentence being true or false, and Frege's stipulations do not settle this question about domains. In fact, Frege wants to settle questions about domains by securing Bedeutungen for his primitive terms, so that he can settle the question of whether numbers are objects, and how numbers can be given to us, by making clear how sentences featuring number-words function in inference. So he both needs to settle the domain issue prior to his procedure and only by means of it, which is contradictory. This is why he is able to think he has given Bedeutungen to all of his Begriffschrift expressions such that all of his Basic Laws are true, when in fact they are jointly inconsistent (because of Basic Law V, whose Bedeutung was supposed to be settled by the procedure of securing Bedeutungen for value-range terms).

Dummett's verdict here is despair. His closing paragraph:
"The realist interpretation could be jettisoned without abandoning the context principle itself, but only if that principle, as here understood, can be shown to be coherent; and this remains in grave doubt. And yet it is hard to see how it can be abandoned, so strong is the motivation for it. The alternative is an apprehension of objects, including abstract objects, underlying, but anterior to, an understanding of reference to them, or, indeed, a grasp of thought about them; and this is a form of [what Putnam calls] external realism too coarse to be entertained. I am therefore forced to conclude without either endorsing the central feature of Frege's philosophy or rejecting it; I can do no more than to say lamely that the issue if one whose resolution is of prime importance to philosophy." (p.19)

This is where Rödl intervenes: he claims that the context principle can be saved (and should be saved), but only by rejecting the idea that a logic such as the Begriffschrift can be said to present us with the form of thought as such. As Dummett argued, a coherent version of Frege requires something other than Begriffschrift to settle the question of what objects Begriffschrift is about; Begriffschrift cannot take care of itself, but needs the "pinch of salt" Frege infamously asks his readers for. The distinguishing characteristic of Begriffschrift Rödl picks out for blame here is that Begriffschrift expressions are characterized only by their inferential structure: a Begriffschrift expression is to have its meaning fixed solely by determining how it figures in lines of a Begriffschrift proof. This is what Frege really fixes by his procedures: how Begriffschrift expressions are to be used in constructing Begriffschrift proofs. But Rödl claims that this fails to settle the question of how Begriffschrift expressions relate to their objects, for the reasons Dummett gave: and this is why Frege fails to notice that his logic cannot take care of itself, as part of what determines the thoughts expressible in Begriffschrift is the relation of these thoughts to their objects, and not merely the relations of these thoughts to each other, and this is why Frege fails to determine thoughts expressible in Begriffschrift in the way he believed he had. So the form of thought as such cannot simply be an inferential order, but must already determine the relation of thoughts to their objects in a way that Frege's logic did not.

The option Dummett's despair overlooks is that of determining the truth-values of sentences not in a way anterior to their relatedness to their objects, but only by already having in view these sentences' relatedness to their objects. Dummett's Frege erred in trying to secure relatedness to objects only indirectly, by means of securing the inferential functions of thoughts, and Dummett as sees the only alternative to secure the relatedness to objects of thoughts to be that of grasping thoughts and objects independently and then bringing them together (in some medium other than thought, which yet stands in need of relatedness to objects). Rödl's excluded third alternative is to promote a logic which determines thoughts only as already related to objects which are given by these thoughts: such a logic is what Kant called "transcendental logic". Kant distinguished this from the sort of logic he infamously claimed to have been settled since Aristotle, which he characterized as a "general logic" that abstracts from all objects of thought and deals with modes of inference independently of the relatedness of thoughts to any objects (which is why for Kant general logic can lead to transcendent metaphysical thoughts by means of fallacious inferences, but transcendental logic does not give any "Sinn oder Bedeutung" to the "thoughts" of transcendent metaphysics).

To present a transcendental logic is to present a logic which includes within it an account of the objects which can be given to thought; non-transcendental logic can omit this only because it treats of thoughts without inquiring into their relatedness to objects. If a logic is not to leave the question of the relatedness of thougts to their objects outside of itself (as a topic for something other than logic), i.e. if a logic is to take care of itself, to be able to present the form of thought as such, then the objects which are given in thoughts must be treated of by logic itself: thus transcendental logic must be metaphysics. This is just what we find in Kant: the Transcendental Logic is just where Kant establishes the principles of his "metaphysics which is to come forth as a science", that all appearances are substances undergoing lawful changes in mutual interaction etc. As general logic determines the forms of thoughts which are capable of figuring into inference, so transcendental logic determines the forms of thoughts of objects: and so it determines the ways in which objects may be given to us, the way in which objects which can be given to us may (or must) be. And unless there is a distinction made between objects which can be given to thought and objects which can be given to our thought, transcendental logic will not have as a consequence transcendental idealism: if the metaphysics of transcendental logic determines how objects must be to be given to thought, then to speak of objects which are not (or might not be) thus is to speak of something contradictory, or to put forward a thought which has no relation to any object: thus it cannot have as its object any "thing in itself" which is unknowable by thought. Kant's transcendental idealism arises because of his accounting our form of sensibility as not the only logically possible one: hence his transcendental logic does not present the form of objects which can be given to thought, but only the form of objects which can be given to spatiotemporally-formed thinkers; the "thing in itself" thus remains as something which (as far as logic allows) might be given to thought, but cannot be given to our thought, and so is for us unknowable and undetermined by Kant's metaphysics. If Kant's forms of sensibility can be shown to be the only logically possible ones, to be demanded by transcendental logic and not merely an addition from a "transcendental aesthetic", or if Kant can be shown to have erred in his claiming that the (logically contingent) forms of space and time are the forms of our sensibility, then Kant's transcendental idealism can be excised from his system. This is part of Rödl's project, in line with earlier German Idealists such as Ficthe and Hegel: to carry out transcendental logic without an independent transcendental aesthetic, so as to avoid transcendental idealism and the "thing in itself". They do not reject the division between transcendental logic and transcendental aesthetic because of ignorance of the importance of a transcendental aesthetic, of an account of the form of objects which can be given to thought, but because they seek to have transcendental logic alone provide for it, as it should be able to if transcendental logic is a logic which can take care of itself.