Wednesday, June 25, 2008

More notes on MacFarlane

MacFarlane's project in his dissertation requires that he make sense of quantifiers in terms of his presemantics. The initial suggestion is to assign quantifiers the type ((O => V) => V), where O is the basic object type and V is the truth value type. Two problems arise for this. The quantifiers could receive interpretations that are sensitive to the domain of objects. Regardless the quantifiers receive different interpretations as the domains vary. This leads to the second problem. What do the variable domains represent? Why do we use them? MacFarlane follows Etchemendy here. Etchemendy says that there are two competing ways of understanding the variable domains, neither of which is satisfactorily captured by variable domains. The first is understanding the variable domains as representing the things that exist at each possible world, with models representing worlds. Three objections to this are given, only two of which I will mention. One is that it seems to make the strong metaphysical claim that for any set of objects at all, the world could have contained just those objects. There might be ways to respond to this; MacFarlane cites a couple of attempts, one of which appeals to "subworlds." The other objection, which seems promising, is that this is hard to square this with the use of frames in modal logic. If the various domains are parts of worlds in different frames, then we must make sense of ways the very structure of possibility could have been, in MacFarlane's phrase. This seems like a problem. I think some people have objected along these lines to David Lewis's modal realism. Making sense of the moving parts of modal logic is hard. Read more

The other way of understanding variable domains is as picking out different meanings for terms and quantifiers. This gets around some of the problems with the possible worlds understanding. It runs into a problem with cross-term restrictions, restrictions put on one class of terms by another class. It becomes unclear on this understanding why the same domain is used for both the universal and existential quantifier. It seems like one could stipulate the usual interdefinability. Etchemendy's point leads to the question of why the same domain is used for singular terms as well as the quantifiers. It isn't clear what a principled response to this would be. MacFarlane and Etchemendy both seem to find it decisive.

In response to these worries, MacFarlane suggests that the proper way of understanding the variable domains is not either of these two. Rather, he thinks that it is as "a specification of a presemantic type: the type O, from which semantic values for singular terms is to be drawn in an interpretation." (p. 199) Using variable domains is just using different basic presemantic types. Before proceeding to the point that I really wanted to focus on, I want to comment on this. This is a better explanation of the variable domains if we have a good grip on what a presemantic type is. I'm not sure that I've enough of a grip on it that it would explain variable domains. The types are sets of things (or functions, constant functions or not) and functions on those (or sets of those). Does this really explain or give us a better understanding of the use of variable quantifier domains? It seems like we would want to appeal to the same intuitions used in variable domains, i.e. the possible meanings and possible worlds intuitions above, for the basic types. I'm not sure how the types fair with respect to the objections to the possible worlds view. It seems to get around the objections to the possible meanings view since all the types are defined with respect to the basic types and so build in the cross-term restrictions.

MacFarlane continues by saying that the basic types are indexical. That is, the basic types in the presemantic ontology are functions from contexts (or points of evaluation; MacFarlane does not use this term.) to domains or types. This is clearly based on work in the philosophy of language, such as Kaplan's work. The basic type O is relative to an index, namely the sortal or set of sortals that specify object in a given interpretation. The idea, from Brandom, being that "object" and "thing" are presortals, which depend on context for complete specification since they do not carry with them the criteria of individuation that other sortals carry with them. I think this is something that he picks up from Quine, Evans, Strawson, Gupta, and others. The understanding of variable domains then depends on this view about sortals. Granted, MacFarlane points out that treating all the basic types as indexical in this way results in a simpler presemantic theory. Other logics can vary the type V of sentential values, say, assigning different sets of propositions in different settings.

Later on MacFarlane says that semantics should answer to postsemantics, which is rooted squarely in the notions of assertion and inference, again notions from the philosophy of language. MacFarlane suggests that a coherent, useful philosophy of logic should be rooted in other philosophical views, in particular from the philosophy of language. It might be useful to extend this to the philosophy of mind since logical notions on MacFarlane's view are supposed to be normative for thought as such. I'm not sure how one would not hae to engage in some philosophy of mind with a claim like that.

This idea strikes me as quite sensible but I want to register a concern. If the justification for some views in the philosophy of logic come from some particular views in philosophy of language, then there is a worry about circularity arising when logical views are used to adjudicate disputes in the philosophy of language. Quine comes to mind as someone whose logical views figured prominently in his views on language. It is not necessary that a circularity arise; the justifying views in the philosophy of language may not be the ones that the particular philosophy of logic is being used to defend. Of course, this sort of dependence might not strike one as bad at all if one adopts a more coherentist outlook on things. It had surprised me since I had thought that logic and by extension the philosophy of logic were more foundational. As such, this sort of dependence on other areas of philosophy did not arise. If the foundational aspirations for the philosophy of logic are abandoned, then this is even less of a problem. This requires allowing more possible answers to the question: what can justify a view about the nature of logic? Possible answers to this question, both historically and in MacFarlane's work seem rather restricted though so there can't be that much widening. One could respond that while logic might play some foundational role, the philosophy of logic need not, and so justification in that domain can come from a much wider range of sources, even though it has not historically. To abuse a metaphor, while logic is near the center of one's web of belief, the philosophy of logic stands farther out. I'm doubtful this can be right since one, especially post-Tarskian philosophers, would expect the philosophy of logic to answer, or address, the demarcation question: what is a logical constant? The answer to this changes what the logic at the core of that web looks like, and by continued abuse of metaphor, how large chunks of the rest of the web look. This suggests a bit more of a foundational role for the philosophy of logic.

Tuesday, June 24, 2008

Sellars on meaning and language

I just came across something on YouTube that I feel I must share. Someone put up a seven part clip of Sellars giving a talk on meaning and language. Thank you unknown stranger. Part of the audio on the first clip is missing unfortunately. I'm kind of amazed that this exists. It is unfortunate, however, that it doesn't use a more flattering picture of Sellars.

Monday, June 16, 2008

Thoughts on Inventing Temperature

I just read Inventing Temperature by Chang. It is, as may be expected from the title, a book on the history of temperature, focusing on the development of thermometry. Every chapter is divided into two parts: historical narrative and philosophical analysis. There are elements of each in both parts of the chapters though. I am going to comment on a few themes from the book. Read more

One is the epistemic problem of setting up a scale on which to measure temperature. This requires fixed points to calibrate against. Knowing that a certain phenomenon always happens at a certain temperature would require knowing what temperature that phenomenon happens at. This requires having a calibrated thermometer already to hand since exact temperature is not an observable phenomenon and outside a fairly narrow range it isn't observable with any sort of even rough accuracy. The response to this circularity that Chang finds in the historical narrative is a process of iterative improvement. First some substances are found that roughly agree and we can calibrate according to our senses. Based on this more precise devices can be constructed, still using ordinal comparisons. If things go well, new devices can be constructed on the basis of those with a numeric scale that has a physical meaning. Chang is hesitant to follow Peirce in taking this iterative development to be linked to truth although he notes the similarity.

I want to note about this is an affinity with some of Mark Wilson's views. In particular, Wilson's suggestion to view agents as measuring devices themselves. They don't have nice numeric scales associated with the various physical properties that they are responding to, but the measuring capabilities are enough to cope with the world. Chang's suggestion goes along with this and ties it in a clear way to the development of science: our rough measuring capacities are sufficient to start the iterative development of measuring devices as needed by various scientific enterprises. (This sounds somewhat commonsensical.) It is an aspect that seems to go missing somewhat in discussions of perception.

In Wandering Significance, Wilson says that he thinks Gupta and Belnap's revision theory of truth and theory of circular definitions can be used to explain various episodes in the history of science. He doesn't provide examples, which is a pity since my historical ignorance left me wondering what he had in mind and how that story would go. Chang's picture of iterative development provides a clear example. The concept of temperature used, the operative core of it, is clearly circular. Starting with some initial hypotheses about values based on our perceptual capacities, an extension is roughly determined which forms the new basis for further revisions. Repeat. There are some rough edges to this though. This revision process is clearly not taken to the transfinite, or even that far into the finite. The range of starting hypotheses and values is fairly constrained, so the extension, if any, that is constant under all initial hypotheses will not be determined. Despite these incongruities, the theory of circular definitions looks to be applicable. If this is a paradigm case, then it would vindicate Wilson's claim since similar incidents of setting up a system of measuring devices, measurement operations and theoretical concepts arise often enough in the recent history of science, I would expect. Even if the particular sort of system involving measuring devices does not, Chang indicates that systems of circular concepts arise and are developed through something like the iterative process that he sketches. If this is right, then it seems like one should pay more attention to circular concepts and conceptual development than has been done lately.

A semantic trichotomy

MacFarlane presents a trichotomy within the discipline of semantics: presemantics, semantics proper, and postsemantics. MacFarlane understands presemantics as Belnap presented it in his article "Under Carnap's Lamp." Presemantics is a theory of the available semantic values and their relations. Linguistic expressions do not enter into presemantics, unless they are themselves some of the objects under consideration as possible values. In a phrase, the point of presemantics is to make it clear upon what truth depends. Semantics proper, in MacFarlane's words, "brings together grammatical and presemantic concepts to give an account of how the semantic values of expressions depend on the semantic values of their parts." (p. 187) Semantics proper is the more or less familiar enterprise of computing semantic values of large expressions or sentences from the values of their atomic parts. Read more

There seems to be a question as to whether semantics proper is supposed to be restricted to just compositional semantics.
If the parts of a sentence are all that one can use to compute the value of a sentence, then it would be restricted to compositional semantics. The phrasing given leaves open the possibility that the semantic value could depend on things in addition to the values of the constituent parts. On this broader reading would it still have to be recursive? I think so although I can't supply a good argument at present. It seems that the incompatibility semantics from Brandom's fifth Locke lecture, which is noncompositional yet recursive, fits into this picture. That would be an example.

Semantics proper imposes a constraint on presemantics by requiring that there be appropriate presemantic types to assign to the linguistic expressions. As MacFarlane puts it, the more expressive the language, the more fine-grained semantic values will need to be to preserve compositionality. (The phrasing here and a later discussion of postsemantics hints that semantics proper is supposed to be compositional.) This point interests me but it is unclear what the relation is exactly between expressive power and the grain of the semantic values. An example: to preserve compositionality in a language with the modal operators 'possibility' and 'necessity' we move from using just truth values using propositions as sets of worlds. However, if we switch to a tense logic, adding the binary operator 'until' increases expressive power but does not require more finely grained semantic values. As another example, the addition of an infinitary conjunction to a first-order language increases expressive power but doesn't require new semantic values. There seems to be some dependence there but it isn't straightforward. (As an aside, the notion of expressive power interests me a lot but I haven't found any extended discussion of it. There is some in Anil Gupta's Revision Theory of Truth and I've come across smatterings in different logical sources, e.g. Blackburn's Modal Logic. I hear there are some discussions in the paradox literature. I don't think I know of anything beyond those though.)

Postsemantics is the mediator between "the semantic values required for the purposes of compositional semantics and the fundamental semantic notions in terms of which the use of language (e.g., proprieties of assertion and inference) is to be explained." (p. 227) (This is the other place at which it seems like semantics proper is intended to be compositional.) There are two examples given that illustrate this nicely. In classical logic, postsemantics is what specifies implication. The compositional truth-values of sentences are not enough. A further stipulation, which is a part of postsemantics, that implication is truth-preservation from premises to conclusion in all interpretations is needed. The other example, due to Dummett I think, is that of multi-valued logic. Suppose one has a set of multivalues with a proper subset of designated values. The semantics provides for each sentence a multivalue that depends on the multivalues of its component parts. The validity of inferences from those sentences, however, depends not on the multivalues determined by semantics but on the designatedness of those sentences. This depends on the multivalues, but there are no operations directly on designatedness. Consequence is defined on designatedness, which is something that semantics seems not to be interested in. Postsemantics then gets to impose further restrictions on presemantics, e.g. by requiring presemantics to provide distinctions of designation. The other important role that postsemantics fills on this picture is that it is what is supposed to interface with pragmatics. This tantalizing suggestion is undeveloped. It would be probably be instructive to go through some of the philosophy of language literature to see to what extent this distinction is in play or could be drawn. I wonder how much of MacFarlane's postsemantics is incorporated into pragmatics elsewhere in the philosophical literature. Although, I'm not entirely sure how much pragmatics on this picture coincides with pragmatics as used, say, in Grice.

On MacFarlane's picture, presemantics and syntax feed into semantics proper which outputs some values. Postsemantics takes these values, possibly doing something to them, before passing them off to pragmatics. The interface between postsemantics and pragmatics isn't developed, so the rest of the paragraph is pretty speculative and rough. Is the directedness presented here required? Suppose one is a wild-eyed Wittgensteinian or someone with inferentialist sympathies. Then one would try to start on the pragmatics end of things. The natural interface would be postsemantics, which, while imposing some restrictions on presemantics, doesn't seem to say much about semantics proper. The discussion of Dummett following the introduction of postsemantics makes this clear. Starting with pragmatics or the use of a language would provide one with some material for developing a postsemantics. To some extent this might feed into a presemantics. It seems like it would only provide some points to check the semantics against: however the semantics is developed it can't contradict this stuff. That isn't much constraint. Although, if one wants to start at the pragmatics end of the picture (the literal picture is on page 188) one might be using a different sense of pragmatics. While there do seem to be some points of interaction, semantics and postsemantics seem like they can be developed independently although ideally there would be a nice story linking them. The values taken by postsemantics are already present in the presemantics, None of presemantics, semantics proper, or postsemantics determines either of the others. Although this paragraph started by describing the relationships between them with a definite order of dependence, this doesn't seem to be mandated. MacFarlane's ideas are then compatible with those of our Wittgensteinian friend. [Edit: This paragraph is admittedly a bit lame. I think my desire to use the phrase "wild-eyed Wittgensteinian" overwhelmed my sense for when a paragraph is sufficiently developed to show others.]

At the outset I said that MacFarlane presents a semantic trichotomy. I don't think MacFarlane anywhere claimed this was an exhaustive distinction, although things proceed as if it were. Prima facie it seems to be exhaustive. There are probably further distinctions to draw within postsemantics. The interface between it and pragmatics seems a little hazy. Even if it isn't quite exhaustive, the modularity of these distinctions is quite appealing and is put to some good work by MacFarlane.

Saturday, June 14, 2008

Cartesian surprise

I seem to be having some difficulty putting together a post lately. Possibly related to this, I have been reading Descartes' Discourse on Method. I came across a passage that I never expected to see in Descartes, or really any philosophers from the early modern period forward. This is mainly because I thought of Descartes as a mathematician and physicist in addition to a philosopher. This was a mistake since the work is prefaced with a paragraph saying what the general content of each part is. The fifth part, whence the passage, includes an explanation of the movement of blood. Here is the passage:
"And so that there may be less difficulty in understanding what I shall say on this matter, I should like that those not versed in anatomy should take the trouble, before reading this, of having cut up before their eyes the heart of some large animal which lungs (for it is in all respects sufficiently similar to the heart of a man), and cause that there be demonstrated to them the two chambers or cavities which are within it."
Challenge: to write a philosophical book which asks the reader to dissect a large animal before continuing; bonus points if it is a logic book.

Saturday, June 07, 2008

Tractarian formality

In MacFarlane's thesis, he distinguishes three related but distinct notions of formality that have been important in the evolution of the conception of logic, 1-formality, 2-formality, and 3-formality. The first is defined as being normative for thought as such. The second is defined as being insensitive to distinctions amongst objects, usually cashed out in terms of permutation invariance. The third is defined as abstracting from all conceptual or material content. In Kant these three notions are equivalent due to his other commitments, notably theses connected to his transcendental idealism. In Frege, the first and third come apart and Frege thinks the second does not characterize logic. Tarski and those writing after him focus mainly on the latter two and it seems that the second has been given pride of place since it admits of such a crisp mathematical formulation. Read more

It is unfortunate that MacFarlane's thesis does not cover the Tractatus. The omission is entirely understandable since one can only cover so much in a dissertation and this one already covered a great deal. MacFarlane's dissertation covers in detail Kant's, Frege's, and Tarski's (although 'Tarskian' may be a better way of putting it) views about logic. The appendices go on to discuss developments in ancient logic. Fitting the Tractatus into MacFarlane's story would be interesting and surely fill it out. The philosophy of logic found in the Tractatus is a bridge from Frege to the Vienna Circle and their descendants. Granted, by the time we get to the Vienna Circle, especially post-Logical Syntax, Tarski's work has been absorbed. However, the Tractatus was skipped over, so we jump from Frege straight to Tarski, Carnap, and Quine.

Which notions of formality does the philosophy of logic in the Tractatus exemplify? It certainly is 3-formal. The rejection of Frege's claims that logic has a certain subject matter, namely the logical constants, is one of its notable features. To supply a quote, 6.124 says: "The logical propositions describe the scaffolding of the world, or rather they present it. They 'treat' of nothing." It takes 3-formality as one of the essential features of logic, not a consequence of some other essential feature. Thus it breaks with Kant as well. I think it adopts 2-formality, although I am not sure if this is a definitional feature or not. In the 4.12's Wittgenstein says that variables are the signs of formal concepts, which present a form that all its values possess. Although, I'm not sure how correct it is to say that logic in the Tractatus is all that much involved with that sort of form. The logical sentences, the tautologies, are certainly going to be insensitive to variation in the names of objects appearing in them. In that sense one could say that the Tractatus is 2-formal. Nowhere, as far as I remember, does Wittgenstein talk about permuting objects in those terms, although he does talk about the range of possibilities of the existence and non-existence of facts. One is always dealing with the same objects although they may be arranged differently. (I haven't yet figured out where 6.1231 fits in. It says: "The mark of logical propositions is not their general validity. To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologous just as well as a generalized one." I suspect it is dismissing a spurious notion of formality. In context it is criticizing Russell.)

The big question, after reading MacFarlane, is whether the Tractarian view claims that logic is 1-formal. Kant does. Frege does. MacFarlane does. Wittgenstein does not seem to. There is not any talk of the normativity of logic, as far as I could discern from my quick check. When he talks about logic showing that something follows from something else, it seems to be entirely in descriptive language without any normative aspect. Despite this, logic seems to be constitutive of thought and concept use, as in 5.4731 "What makes logic a priori is the impossibility of illogical thought." In a few other places he talks about the impossibility of illogical thought. There doesn't seem to be any normative dimension to this talk at all. I'm not sure where to look for anything more normative. Although, it would not be surprising for there not to be any normative element to the philosophy of logic expressed in the Tractatus since Wittgenstein left the ethical out of the book for the most part, excepting a few mentions in the late 6's. I will hesitantly conclude that the view in the Tractatus does not claim 1-formality.

The view of the Tractatus then is an example of 2- and 3-formality. I don't remember if there is an earlier example of this combination pointed out by MacFarlane. I expect that the reasons motivating this combination in Wittgenstein differ from those in any earlier examples if such exists.

Incorporating the Tractatus into the story told by MacFarlane would be interesting for other reasons. For one, it includes some criticisms of Frege, in particular on Frege's doctrines that logic's subject matter is the logical constants and that the laws of logic govern all thought. There is also a reading of the Tractatus, supported by Sullivan I think, that takes the book to be a criticism of transcendental idealism. If that is on track, then it would provide an example of a principled acceptance of 3-formality independent of transcendental idealism, whose doctrines Kant appeals to in order to support 3-formality. This, by itself, would be neat. Finally, MacFarlane does not claim that his three notions of formality exhaust the possibilities. This is a wild conjecture (that might be generous; perhaps wild hunch is better), but there might be another sort of formality available in the (sort of) algebraic approach found in the Tractatus and also found in Schroeder and Peirce apparently. I haven't worked out any of the details, but it seems plausible that there might be something there.

Wednesday, June 04, 2008

The insufficiency of permutation invariance

MacFarlane has two related criticisms of the permutation invariance criterion of logicality, which says that an operation is logical iff it is invariant under all permutations of the domain.

The first criticism is that not all permutations are considered. It is only permutations of the domain of objects and not of truth values that are considered. There are not many classical truth functions that are invariant under permutation of the domain of truth values. However, if no permutation of the domain of truth values is allowed, then all functions on just that domain are logical. This might not be so bad when considering classical logic since all classical truth functions are equivalent to some combination of conjunction and negation. It is not enough to say that we only consider permutations on the domain of objects. If we are working with a modal frame, we consider permutations of the domain of worlds just as readily as those of the domain(s) of objects. In fact, there are also not many operations that are invariant under all permutations of the domain of worlds. The objection takes an even more concrete form when considering larger domains of truth values than the classical one, {t,f}. MacFarlane's suggestion is to consider all permutations that respect what he calls intrinsic structure. This is a relation of some sort on a domain, considered as just a set. The intrinsic structure of classical logic is: null structure on the objects, f≤t and not t≤f.

The notion of intrinsic structure sets up the second criticism of permutation invariance. This criticism is that the sorts of intrinsic structure that the permutations must respect is in need of justification. Why is it, MacFarlane presses, that there should be no structure among the objects? This rules out the epsilon of set theory as a logical notion. Similarly, for modal frames, if there is no structure among the worlds then we get an S5 necessity as logical rather than the operations of any of the other modal logics. Null structure on the objects is structure, albeit of a degenerate sort, and stands in need of justification. MacFarlane seems to be doubtful that any non-question begging reason can be obtained without recourse to an antecedent view of logic. The objection underlying both of these can be summed up as: at best permutation invariance systematizes some antecedent intuitions about logicality; it does not explain the notion of logicality.

MacFarlane's own suggestion is that something more is needed, so that logicality can not be specified just in terms of permutation invariance. He suggests that the extra is an understanding of intrinsic structure that connects it to a Kantian notion of logic, i.e. as normative for thought as such. This gap is supposed to be bridged by noting that "intrinsic structure belongs to a type in virtue of the most general purpose of logical theory: the study of the semantic relationships that hold between sentences solely in virtue of their capacity for being asserted and used in inferences." He continues: "On this ground, one might say that notions that are sensitive only to intrinsic structure are applicable to thought as such..." Intrinsic structure ought to be the minimal structure demanded by the above Kantian view of logic. There is fleshed out using some distinctions from Belnap that I'm not going to go into in this post (because I need to read the relevant article, "Under Carnap's Lamp"), the distinctions between presemantics, semantics and postsemantics. Interestingly, this suggestion ends up ruling that the non-S5 modal operators are not logical because the accessibility relation on worlds is not minimal in his sense, whereas the structure in various logical lattices, like the standard one for {t,b,n,f} is minimal. Even more interestingly, it suggests that the structure needed for Prior-Thomason(-Belnap)-style indeterministic tense logics has some claim to minimality and so to putting tense operators in the logical box.

Logicality and permutation invariance

Chapter 6 of MacFarlane's dissertation is a study of the permutation invariance criterion of logicality, which says that an operation is logical iff it is invariant under all permutations of the domain. As an historical note, MacFarlane mentions that the idea of permutation invariance comes from Mauntner and Tarski and originates in Klein's Erlangen program. The Erlangen program was an attempt to classify geometries by the classes of transformations under which their basic notions were invariant. For example (examples from MacFarlane's thesis actually), the notions of Euclidean geometry are invariant under similarity transformations and those of topology under bicontinuous transformations. Since the logical is presumably broader than the geometrical, its notions should be preserved under a larger class of transformation. While Tarski was a logician and philosopher, Mautner was more a mathematician and Klein was certainly a mathematician. Resting our notion of logicality on the notion of permutation, as in the permutation invariance criterion, seems to put the mathematical before the logical, i.e. requires permutations to get a grip on (or to explain, to be a bit less loaded) the logical notions. Since this idea is a grandchild of a mathematical program, it is not surprising that it would give rise to various approaches to logic that rely so much on mathematical notions. (This is by no means a bad thing. I'm all for it.)

There is one thing I want to note about the permutation invariance criterion. If the criterion is necessary and sufficient for logicality, then the logical is tied to the representational approach, that is, there must be a domain of things represented that is invariant under permutations (at least for the first-order case; although it looks like a similar point could be made about, say, truth functions in the propositional case.). This seems to put it in tension with inferentialism. This may not be the case if the inferentialist is not giving a criterion for logicality. I'm not sure that anyone would be happy with this. An inferentialist might claim that the inferential rules give the meaning of the logical constant whereas the permutation invariance criterion tell us which operations are logical. This doesn't seem promising either for two reasons. First, this raises the further problem of explaining when a certain constant designates or otherwise represents a logical operation. Second, (and putting the first to one side) if we already have a the logical operations, what further use is there for a distinct meaning given by the inference rules? It would appear none. This leaves the inferentialist looking a little left out in the cold. Of course, it would likely be illuminating if some link between an inferentialist criterion and the permutation invariance criterion could be found.

Must there be Tractarian objects?

I was thinking about the Tractatus recently and came up with a question about it that I was unsure about the answer to. (This is not that hard to do really.) The question is whether Wittgenstein thinks that it is a logical impossibility that there could be no objects.

Why would one think that this is not an option that there could be no objects? In the 2's, Wittgenstein talks about how there must be a substance to the world, and this substance is comprised of Tractarian objects. This makes it seem like it is not a logical possibility for there to be no objects. Granted, this would only be a logical possibility if the world were connected tightly to language or logic or if objects were similarly tightly connected to language. I'm not terribly comfortable with the 2's, either alone or together with what comes later. Luckily, I don't think that we need to appeal to them specifically in order to come up with an answer, which point I'll get to below.

Why would one think that it is an option? In 5.453 Wittgenstein says: "All numbers in logic must be capable of justification. Or rather, it must become plain that there are no numbers in logic. There are no pre-eminent numbers."
If it is a matter of logic that it is impossible for there to be no objects, this would seem to make zero a distinguished or pre-eminent number, which seems to be ruled out by the above. It might look like I'm running together objects and names, and all that logic will deal with is the names. In the Tractatus, however, every name designates an object.

Alternatively, one might ask why there must be at least one thing. Is this asking for logic to give a justification? Thinking about the way that the TLP is set up, it seems not. Rather, this issue is left implicit in the propositions. Propositions consist of concatenations of names. Names designate Tractarian objects. Thus, for there to be Tractarian propositions at all, there must be names and so objects. In order to be talking about logic at all, we must presuppose that there are at least some objects. It looks like the question of why there is something rather than nothing is barred from the outset, which is probably something that Wittgenstein would've approved of.

I do want to note that things are complicated with Tractarian claims about possibility. In TLP, the objects are the same in all possible worlds. Indeed, the possible worlds, if we want to use that language, are constituted by those things in various arrangements of facts. It seems like claims such as "there could have been more or fewer things than there are," if formulable in a Tractarian proposition, must come out false. The possibilities are completely determined by the Tractarian objects that there actually are. Of course, the way that the above claim is formulated, in terms of generic things, is probably the source of this seeming weirdness. "Thing" and "object" are formal concepts in the TLP. A claim like "there could have been more espresso cups than there actually are" needn't turn out necessarily false because "espresso cup" is a proper concept, which can be expressed with a propositional function.

Monday, June 02, 2008

Views of logic

In his dissertation, MacFarlane presents some reasons for thinking that the view that Kant was the first to hold that what is distinctive of logic is that it is normative for thought as such and that logic's abstraction from content is a consequence of this. To do this, he goes through some of the views of logic held by Kant's predecessors. He wants to show that Kant agreed with his predecessors to some degree about logic so that his own views about logic would not be seen as merely changing the subject. Part of what made this exercise interesting for me was seeing what other views of logic were in play.

We start with the Kantian view that logic is normative for thoughts as such. Logic consists of a bunch of norms or rules for concept application. The judgment that all A are B says that in applying the concept A one ought to apply the concept B. Frege likewise thinks that logic is normative for thought. This drops out of the picture by the writing of the Tractatus. I'm fairly certain that there isn't any normative status attributed to logic in the Tractatus. I'm not sure if, say, the Principia mentions norms at all.

Next, there is the Wolffian view of logic which holds, like Kant, that logic is normative for thought and also that logic can tell us substantive things about the world. It should, quoting MacFarlane, "be grounded in ontology and psychology." One derives the rules of logic by examining psychology. The rules are also supposed to "be derived from the cognition of being in general, which is taken from ontology." (Quoting Wolff.) Based on the examples of Kant's criticisms of this view in MacFarlane's dissertation, it seems like the adherents of it do not distinguish between rules applying to concepts and those applying to objects.

The Wolffians held that the form of thought was the same as the form of being, to use MacFarlane's characterization. They thought that the relations among concepts were an accurate guide to the relations among beings. Possibly because of this they seemed to think that logic could be a guide to ontology. I'm not sure to what extent this meshes with contemporary views about logic, if at all. Alas, I'm not likely to read a bunch of Wolff in order to find out. This also, I expect, runs roughshod over distinctions that could be, and need to be, made about the phrase "guide to ontology".

Third, there is the view held by Locke and Descartes: logic is a set of rules "which teaches us to direct our reason with a view to discovering the truths of which we are ignorant."Interestingly, Descartes criticizes what he characterizes as Scholastic logic for being about mere forms of reasoning, unconcerned with the truth of the premises of the arguments. This view of logic holds that it is normative, not necessarily for thought as such, and must be grounded in empirical psychology. There could be thought that does not follow the norms of this logic; it just wouldn't be particularly useful for generating knowledge. Logic is supposed to extend our knowledge in substantive ways. It is also supposed to help us avoid sophisms since we are not concerned just with mere forms.

It isn't clear what the continuity is between these views of logic and the current one(s). A link might be obtainable with the Kantian one via Frege. Frege thought that the laws of logic were the laws of thought, but his laws of logic took a form more familiar to us than Kant's. This seems like a family resemblance sort of transition since we lose the normativity in the process. Setting that aside, if something of the older Kantian view of logic is lingering in the modern conception of logic, then it seems like there is plenty of room for there to be logical notions, completely distinct from mathematical ones. Although, this notion of being completely distinct from needs clarification, possibly along the lines sketched by Colin for the notion of dependence in the comments a few posts ago.

Sunday, June 01, 2008

Foundations of semantics

It occurred to me that I don't have a very good idea about either the origins or the foundations of semantics, primarily logical. In particular, why something counts as a semantics and whether this exhausts the possibilities for the concept are both a bit opaque to me. This is a bit open ended and vague, but that is roughly where I'm at. I'm not sure where to look for insight into this topic. A quick Google search reveals that, as expected, Tarski has an article on the matter, entitled "The semantic conception of truth and the foundations of semantics". There is something on this topic in one of the early chapters of Marconi's book Lexical Meaning. I think somewhere Brandom says something short about this. Apart from that, I have very little idea about where to look. It seems like Montague might have some thoughts on this. Possibly Kreisel or Carnap as well. Although, I am not sure where to look for any of those philosophers. Any suggestions?