HOW LOGIC SPEAKS One of Hilary Putnam's most profound and important essays, "Rethinking Mathematical Necessity", was inspired, he tells us, by a desire to understand an idea of young Wittgenstein's (henceforth YW), an intuition that I had never shared. For the early Wittgenstein it was somehow clear that logical truths do not really say anything ... (1994: 246) Whereas it had seemed clear to Putnam that sentences of pure logic are statements with content ... ; if proved, they are moreover true statements, and their negations false statements. (Ibid) !e initial problem was to understand what YW could have meant by this. !e route Putnam found to an answer went through an idea he found in Kant: Logic is not a description of what holds true in "metaphysically possible worlds', to use Kripke's phrase. It is a doctrine of the form of coherent thought. Even if I think of what turns out to be a "metaphysically impossible world", my thought would not be a thought at all unless it conforms to logic. (1994: 247) How to understand this idea? Does it mean that what is, in fact, a law of logic simply could not have failed to be no matter how the world were? Kant, as Putnam reads him, tells us: !e negation of a logical truth is, in a sense, unthinkable; and it is unthinkable precisely because it is the negation of a logical truth. Explanation goes no further. (1994: 255) Trivially, thought, or its logic, could not have turned out to be such that 'blah' where that 'blah' is unthinkable: there is as yet no way it could have turned out. But then, exactly what sort of notion is unthinkable? Here, then, is Putnam's concluding way with the 'Kantian' idea: My suggestion is not, of course, that we retain this idea of a nature of thought (or judgment, or the ideal language) which metaphysically guarantees the unrevisability of logic. But what I am inclined to keep from this story is the idea that logical truths do not have negations that we (presently) understand. It is not that we can say that the theorems of classical logic are "unrevisable"; it is that the question "Are they revisable?" is one which we have not yet succeeded in giving a sense. (1994: 256) !is, I will argue, is just the right way with it-as one can see in comparing Frege with YW. 1. Sätze and Gedanke: In 1919 Frege wrote, What is distinctive about my conception of logic is "rst recognisable by the fact that I place the content of the word 'true' in lead position, and then by the fact that I let thoughts follow immediately as those things by which being true can come into question at all. (1919: 273) To grasp Frege's conception of logic, then, one must grasp his notion of a thought. Perhaps the most striking thing about that notion is its contrast with something like Russell's (early-20!" century) conception of a proposition, inherited YW's of a Satz. Russell, and YW, thought of a proposition (or Satz in relevant sense) as something sentence-like in ways a thought in Frege's sense is not. Sentences, or at least sentences of a language, have two conspicuous features. First, they have some perceivable form (visual, auditory, whatever); a form by which they are recognisable as the sentences they are (without resort to knowledge as to what it is they say). !ey are the sorts of things suitable for work as vehicles in the expression of our thought; content-bearers. Second, a sentence is something structured-syntactically, and thereby semantically-in some one particular way (bracketing ambiguity). For our purposes (and Russell's and YW's), a sentence can be thought of as generated from a given vocabulary by given syntactic rules-ones by which inde"nitely many di#erent sentences are generable from that given vocabulary. It is structured by its derivation in the relevant syntax. Its semantics-what it means, or as such, says-is structured accordingly. !e English sentence 'Penguins waddle', for example, speaks of penguins and describes them as waddlers. It is intrinsic to it to do this. A sentence, thus, belongs to a system of sentences. !at system as a whole is structured by what generates it. !e structure of a given sentence locates it in that system. One, if not the main task of a syntax is to identify each sentence within some given language. To do that is to assign that sentence some structure which is its alone; thus which distinguishes it from every other sentence generated, and, given the complexity of structure here, marks the syntactically (and semantically) relevant similarities between it and given ranges of other sentences which it is marked as resembling.) It is sometimes unclear whether Russell or YW made that "rst feature of sentences part of their conceptions of a proposition. Both certainly exploited the second. Every proposition is to be conceived as having some unique representational structure. It is some particular structured way of representing things as some way there is for things to be. So it has (to adumbrate) a unique logical form. Every proposition belongs to a system of propositions, in which its structure positions it, and by which it shares features with given ranges of other propositions. (Wittgenstein espoused this view as late as January 1930. Cf. Waismann, 1979: 89-90.) For both, such relations between propositions are to be the very foundation of logic. Frege's notion of a thought, however, "ts neither of these ideas. It is essential to a thought that it is something invisible (not perceivable by the senses). One reason for this is that a thought is to be precisely that by which truth can come into question at all. It is thus to be identi"ed by just those features, and no more, which identify something on which truth is liable to depend. And whether it is true that penguins waddle does not turn on whether a sentence which so says is written in lower case or capitals. !e second idea above-what is most essential to Russell's and YW's, notions of proposition-collides, for a start, with Frege's insistence that whole thoughts come "rst. We do not, he tells us, begin with concepts and put them together so as to construct a thought. Rather concepts arise through the decomposition of thoughts. It is a central part of this idea that the same thought can be decomposed in many di#erent ways; that semantically, and structurally, di#ering sentences can all be expressions of the same thought. As Frege also puts it, I do not think that for each judgeable content there is just one manner in which it can be decomposed, or that one of the possible manners may always claim an objective priority. (1882: 118) ('Judgeable content' was Frege's "rst try at the notion 'thought' ('Gedanke') came to stand for.) Grasping Frege's notion of a decomposition is made easier by conceiving a thought in terms of its de"ning task: to bring truth into question; to make it turn in some determinate way on how things are. Like any task, this one can be broken down into sub-tasks. Serving the drinks can be broken down into serving the martinis, serving the mai tais, serving the margaritas, serving the mojitos, etc. Or it can be broken down into "lling cocktail shakers, shaking them, dusting rims of glasses, pouring, etc. Similarly, making truth turn on whether Sid waddles can be broken down into making truth turn (in part) on how Sid is and making it turn (in part) on what waddles. Or it can be broken down into making truth turn on whether the concept being a waddler is satis"ed by everything which satis"es the concept, being that very item, Sid. And, I stress, so on. For a set of subtasks to be a decomposition is just for their joint performance to be (no more nor less than) performing the whole thought's task. A thought-element, on this way of thinking, is, "rst of all, an element of some decomposition of a thought, and then an element of it on that decomposition. For it to be an element on that decomposition is for it to be part of some set of subtasks which, jointly, just are that whole thought. No more is required of a decomposition for it to be a decomposition than just said. Such is what opens the door to multiple decompositions. Making truth turn in a particular way on how things are is not like serving drinks in all respects. In the last case, one can parcel out the tasks: Pia mixes the martinis, Sid serves them. For a thought there is no parallel parcelling out of (proper) subtasks. Making truth turn on what waddles can only be done at all as part of making truth tout court turn on how things are. !ere is no such thing as making truth turn in part on how things are, punkt. If Pia "lls the shakers and Sid omits to shake, at least we have full shakers. To make truth turn on what waddles and omit to do the rest would be to do nothing at all. If Sid gets a Lego set for Christmas, he might build a garage. Pia may then decompose it into parts. Or her nephew, !or, might just oblige. !or's parts may be very di#erent from what Pia's would have been. None of his would have "t together with any of hers. You can break up a Lego model in many di#erent ways into sub-parts. But-assuming there are no plastic shards about-there is one decomposition of the garage that may claim objective priority. It is the decomposition of the garage into those parts in which the Lego set came. !is is the sort of decomposition one expects of a friend when he returns the Lego set he borrowed. Some philosophers have understood Frege's idea of multiple decomposability on this Lego model. Such is not Frege's idea. Between about 1902 and 1904 Frege and Russell debated Frege's notion of Sinn. A thought is one sort-the central sort-of Sinn. So they were debating, inter alia, Frege's notion of a thought. Russell rejected that idea. He (and then YW) preferred their notions of proposition. One notorious point of disagreement concerned whether objects-Sid, or Mont Blanc-could themselves be thought-elements. As Russell seemed to understand this issue, the crucial question was whether the relation between thought-elements which made a thought about an object and the object the thought would thus be of was, or could, be manyone. Russell also seemed to think that Frege's case for Sinn was banking on this being many one. Perhaps Frege invited such misunderstanding. Perhaps he was occasionally unclear in his own mind on the point. (See, e.g., 1906.) But this is a misunderstanding. !e trouble with Mt. Blanc, shared by Sid and the foam on Pia's cappuccino, is that neither these things nor anything to be found in investigating them, can make truth turn on anything. Nor is Sid a way for truth to turn on anything. !e truth of a thought may well turn, inter alia, on how Sid is. But then the relevant element would be one which made the thought do this. Such an element would not be Sid. He is the wrong sort for such feats. Frege did think that the relation in question is, in general, many-one. But the existence of Sinne depends on no such fact. Rather, many-oneness serves a di#erent end here. Facts as to where there is one thought, where two, need to mesh with facts as to where there is need for proof (and then what proof). If the thought that A is the thought that B, then, trivially, anything which is proof of A is proof of B. So there had better be di#erent thoughts wherever proof of B from A would not be immediate, or wherever some proof of A might fail to be proof of B. We might look at arithmetic, for example, to see where proof of one thing from another, or of one thing, but not yet of another, would need to be recognised. Such will settle, for arithmetic, just where there are two arithmetical thoughts, where one. For Frege the bare notion of a thought should leave it open for arithmetic to settle such questions. Similarly for other areas of thought. In general, as he puts the point, !e principles of concepts, and of judgements, serve only as preparation for the theory of consequence. (Kernsatz 14 (Nachgelassene Schri"en, p. 190)) Such, I think, is one of Frege's greatest insights. Compare thoughts and propositions on this point. When are two di#erent decompositions of a thought (each a decomposition of some thought) a decomposition of the same thought? For Frege di#erent decompositions, each of some thought, are to be recognised as decompositions of the same thought where needed so as to represent correctly the facts as to what would be proof of what. !at a thought is decomposed in some one way cannot on its own determine what another decomposition of that very thought might be. Nor does the notion of a thought provide us per se with any e#ective condition on thoughtidentity. Contrast this with the answer to the question when two decompositions of a proposition would be decompositions of the same thing. Excluding the Lego model of decomposition, the answer is: only where they decompose it in the same way. A proposition is identi"ed as the proposition that it is by a particular decomposition: that which decomposes it into the structure assigned it by its generation in the system to which it belongs. !e structure thus assigned it identi"es it as the one it is in the same way that the structure the syntax of a language assigns a sentence in generating it identi"es that sentence as the one it is. !us, when YW tells us, !at the truth of one proposition follows from that of others, we see in the structure of the propositions. (5.13) He is telling us that, for any stock of questions of truth-ways there are of making truth turn on how things are-there can be no identifying which questions, or ways of turning, these are without ipso facto answering the question what, within the stock, would be proof of what. For YW, whatever allowed truth to come into question at all-any actual determinate way for truth to come into question-could, per se, allow for only one role for what was thus in question in the phenomenon of proving and being proved. It is just this idea of a unique way of generating any given proposition which, I hope to show, Putnam cannot accept. In his rejecting it we "nd his deepest insight about the inevitability (such as it is) of logic. 2. Frege's Logical Insights: Frege tells us two or three crucial things as to what logic is. First, he tells us, !e meaning of the word 'true' is unfolded in the laws of being true. (1918: 59) Second, he tells us, How must I think to reach the goal truth? We expect logic to give the answer to this question, but we do not require of it that it delve into the particularities of each area of knowledge and its objects; rather we only assign it the task of setting out the most general things which hold for all areas of thought. ... We can thus also say: Logic is the science of the most general laws of being true. (1897: 139) !ird, there is a suggestion in Frege of what sort of generality might be involved in such references to 'the most general.' First, then, the laws of being true (or laws of truth) are arrived at in unfolding the concept true (unde"nable, in a sense, but not thereby altogether without content). !ey are laws which hold simply in virtue of what being true is per se. Which tells us something as to how to understand Frege's second idea. He frames it in terms of what one might see as logic's universality, or topic-neutrality. Logic, the idea is, tells us how to think insofar as such advice applies to, and holds for, all thinking whatever, no matter what it is about. So its maxims contain no restrictions on the sort of thought to which they apply. Hence logic tells us only that much as to how to reach truth which is contained in what it would be, as such, for something to be true. Since the interest here is in aiming at truth, logic would be centrally concerned with relations between the truth of some thoughts and the truth of others; here centrally with truth-preservation. !e sort in question would be that ensured simply by truth's very nature-by what being true is essentially. Such is a way of understanding the idea of topic-neutrality. It is also a clue to what shape, for Frege, logic would take. What content is there in the bare idea of being true? One strand concerns a certain objectivity. ("Logic begins with the conviction that there is a distinction between truth and falsehood." (Kernsatz 12) !ere is a thought-a question of truth-just where it can be "grasped as the same" by di#erent thinkers, and, thus, agreed to or disputed. (Cf. 1919: 146) Where there is a thought, it is true, or false, independent of who thinks it or whether it is thought. (Cf., e.g., 1918: 69). Frege's propositional logic is simply a development of this idea. A thought is, or at least aims to be, either true or false. !us, the set of functions from truthvalues, or pairs thereof, to a truth-value will identify all the ways in a thought may be compounded out of others whose truth-values determine its. For each such function there is a logical form formable from any thought, or pair, according as the function takes singletons or pairs as arguments, where what has that form has the value true just where that function maps the values of its elements, so formed, into the value true. Propositional logic just maps truth-preservation across such forms. Frege's news was as to how logic could look inside whole thoughts. !ere is such a thing as a thought of an object that it is thus and so-a thought which so decomposes-only where there is an object it is thus of. For the thought to be true is then for that object to be as (so decomposed) the thought represents some particular object to be. Suppose we present the form a thought assumes when so decomposed by, say, the symbols F(a) (as Frege notes, as we would typically present the logical form of a (mathematical) function. !en, extracting the form-element F( ) from that whole, we can introduce a new logical constant to combine with that element to form a new logical form. We might write it, e.g., 'ExF(x)'. !e basic truthpreserving properties of a thought of that form (insofar as part of what being true is per se) would be: such a thought follows from any thought of the corresponding form, F(a) (I here omit details of what 'corresponding' is to mean); if, the role of 'F( )' remaining "xed, G follows from any thought which shares the form 'F(a)' stands in for, then G follows from ExF(x). Now, if you like, introduce a further way of completing 'F( )'-write it, say, 'AxF(x)'- whose basic truth-preserving properties are that it both follows from and is entailed by notEx-notF(x)-letting 'not' here stand for a standard negation operator. With which we have recognised (at "rst-order) those logic forms of concern to logic which one discovers by looking inside whole thoughts. Where logic's laws are expressed in that special way a calculus does this, the relevant properties of those just-indicated constants in logical forms will be made recognisable syntactically. !e relevant rules for constructing logical forms for proof, though, are meant to correspond to certain facts of truth-preservation. In the propositional case, e.g., to the facts of when truth would be preserved moving from two thoughts to a compounding of them which took on the value true just when both of them did, and from such a compound to some further thought. In the quanti"cational case, in the way just indicated. !ere is a logic-a construction of a special sort. And there is logic-what a logic aims to have represented rightly. Logic-the topic to be represented rightly-has its laws, just as, e.g., mechanics does. For Frege these are the laws of the phenomenon of being true (notably laws of truth-preservation). If there is a topic here, there might also be a theory of it. A theory would mention the key items that the laws govern-such things as quanti"ers, or quanti"cation. It would treat the same phenomenon as a logic does. But it would be answerable to this in a very di#erent way. !e above is a somewhat tedious expansion on-if you like, informal theory of-that to which a logic, or a theory of logic is answerable, each in its own way. !e point of the expansion is to begin to point to the di#erent strands which make up the notions logic treats of; notions of that of which its laws are to hold. Multiple core ideas interact here. !ere are ideas of objectivity. !ere is the idea of truth's bipolarity. !ere is an idea of a certain universality and authority to logic-of thought as something per se governed by given laws, no matter what, or when, or by whom, the thought is. Such a tangle of threads at least begins to make truth a notion of just that sort about which Putnam has had so much to say. !ere remains Frege's third idea. Laws of logic, since universal, have a speci"c sort of generality. !ey have consequences for the ways any thought relates to others. Frege o#ers a way of understanding the generality of a thought. On this understanding, generality is to be attained through quanti"cation. Where a thought has some speci"c content-e.g., where it si decomposable into making truth turn on how Sid is, and on who smokes-one can move from it to a thought without that content by replacing that element (in relevant decompositions) with a quanti"er. If the thought is that Sid waddles, one moves in the right direction by thus moving to a thought that something waddles (or that everything does). One continues in the right direction by moving from there to the thought that something (or everything) does something (or everything). Eventually, the idea is, one reaches a point where there are no more such moves to be made. One would then have attained to a most general thought, on this understanding of generality. Frege's idea of universality is o$en read as the idea that laws of logic (or of truth) belong to the realm of most general thoughts in this sense. 3. YW On Logic: !e instigation to our present exploration was YW's idea that logical 'truths' (if such they really are) say nothing. YW, in fact, means several di#erent things by this. !e most plausible of these turns au fond on the notion of representing-as. YW, though, puts it in slightly di#erent terms: In tautology the conditions on agreement with the world-the representing relations-cancel each other out, so that it stands in no representing relation to reality. (1922: 4.462) It is the distinctive mark of a logical proposition that one can recognise in the symbol alone that it is true; and this fact contains the whole philosophy of logic. (1922: 6.113) Consider the relation of representing-as. One might see this as three-place: in the "rst place there is a representer-even if, sometimes, only a stand-in for one in the form of something like a thought or a proposition; in the second place, something which is represented as something or other-in the main cases so far, either things or a thing; in the third place, that which what is in the second place is represented as (being)-in those main cases, some way there is for things, or for a thing, to be. In a normal case where there is a question of representing truly or falsely, truth value is a cooperative enterprise: the third term in the relation "xes how truth is to turn on the second term-what is demanded of this second term if there is to be truth; the second term, how things are, delivers the outcome of such turning. Normally, the various elements in the relevant thought, or proposition, each contribute substantially to forming some substantial demand on what occupies the second place. But suppose that instead of this, in YW's terms, those would-be partial demands contained in these elements 'cancel each other out', so that really no demand is placed by the third term on the second. So that, in YW's more metaphorical terms, one can recognise in the third term itself that the would-be representing (if either true or false) must be true-or, again, must be false. !en there is no real role for the second term here. It makes no di#erence at all how, or what, it is. It might as well be anything. !e result is already determined. Such, if it happened, would be a plausible case for mere schein-representing-as; a case where nothing was really represented as anything. Perhaps there was a schein-occupier of the second place in the relation. But at best we have only a degenerate case of the obtaining of this relation. Putnam suggests that this e#ect of cancelling out shows up only in unembedded items of relevant forms, not in embedded ones. One thing this suggests is that cancelling out has to do more with force than simply with content. More generally, though, it suggests that cancelling out is somehow all relevant to context, on some notion of context (yet to be explored); that cancelling out is not something which that which brings truth into question (a thought or a propositions) does as such. !is more generally suggestion points in the direction in which we are now headed. But YW also o#ers another account of of saying nothing. Such turns more patently on what is peculiar to a proposition as opposed to a thought in Frege's sense, though YW seems to see it as merely continuous with the "rst idea, above. 6.1222 expresses it as follows: Not only must a proposition of logic be incapable of refutation by any possible experience, but it must also be incapable of being con"rmed by any such. If propositions were what brought truth into question, these two ideas might fuse. For if what brought truth into question was what was identi"ed, per se, by a proposition's structure, then wherever truth was borne on, eo ipso what was borne on could not be the same proposition as any whose structure cancelled out demands on truth as per above. !e ideas separate, though, if what brings truth into question is a thought. Putnam showed us why they must: there is no such legislating of a question of truth what will, what not, matter to its answer. A law of logic would be absolutely impervious to worldly bearing on its truth if it merely re%ected structure which made the thoughts which had it the thoughts they are. !ere could not fail to be thoughts so structured. !e law could not fail to apply to them. YW this idea in 6.341-6.342 by comparison of logic with what I will call a special system. Such a system generates a stock of propositions from given vocabulary by given syntax. !e structure thus assigned to each such proposition distinguishes it from any other proposition in the system- or from any tout court: to be that proposition is to have that structure; to be it is to be generated by that system. (Compare English sentences.) So that structure, with the contrasts it makes with other propositions, identi"es the content of that proposition as what it is (identi"es the question of truth thus posed). !e system need not generate all propositions. !e structure it assigns its may be largely proprietary. It yields a particular scheme for describing things; one, perhaps, among many possible. Its concern may be some particular subject matter. Now the crucial idea is this. On the whole, the system generates propositions whose parts do not cancel each other out. !ese are genuine descriptions of the world. But it may also generate, or "x, ones whose parts do cancel out in the above-scouted sense. !ese merely tell us what system we are dealing with; what content the "rst-mentioned propositions have in relating to each other as the syntax of the system makes them do. !e illuminating comparison is to be between such proprietary dicta of a special system and laws of logic. YW o#ers two examples. !e "rst is a hypothetical system for describing black and white patterns on a white wall. In the system such are described in terms of a (notional) net. !e net consists of labelled cells of a particular size and shape (say, hexagonal). (A label might be a pair of coordinates for row and column.) A description in the system supposes this net placed over the wall in a particular orientation. It is then a conjunction, each conjunct pairing a cell-label with one of the descriptions, 'black', or 'white'. A rule of the system is, say, that a cell is to be paired with 'black' just in case it is at least 50% black; otherwise white. Such "xes when a given description within the system would be true of a given wall. For any given wall, the system might also generate, for each cell, &, in the net, the (would-be) proposition, '& is not both black and white'. Or it might generate a generalisation of this, such as 'no cell of this wall is both black and white', or, still more generally, 'there is never a cell of any wall which is both black and white'. But such would-be propositions would say nothing as to how any white wall was in re being black-patterned. !ese pseudo-propositions merely "x how this particular scheme for describing walls works. !ey help identify the content of a genuine proposition such as '... & <<17,39>, black> ...'; what it says, insofar as to say this is just what it is to be the proposition in question. !us, the idea is, the pseudo proposition is in no way liable to proving false. YW's second example of a special system is Newtonian mechanics. Here the Newtonian laws and de"nitions are the pseudo-propositions. For example, the (would-be) proposition, 'Momentum is mass times velocity', merely tells us how the terms of the system describe, just as with the would-be proposition, 'No cell of any wall is both black and white.' On the other hand, '!at six-pack is traveling towards that windshield with momentum 200 m/hr/kg'. as generated by the 'Newtonian special system', is a genuine proposition. Refer to the Newtonian laws to see what it says. !e comparison is thus between laws of logic and the dicta of special systems. What is the comparison to be? One might (not entirely plausibly) think of the laws of logic as generated by, or by-products of, some vocabulary and syntax which generates all special systems. Or, less implausibly, one might think of them as things generated in generating any consistent system of propositions, or perhaps under some suitable closure of it (say, under suitable compoundings of propositions, and operations on sub-propositional parts). In any case, the idea would be that the laws of logic are pseudo-propositions in the same way that pseudo-propositions of special systems are. !ey are simply part of what "xes how any system of propositions is to work; or that structure, or content, of any proposition which is "xed independent of to what special system it belongs. Hence (the thought is) they are as impervious to being proven false (or true, for that matter) by vicissitudes of history as are the dicta of special systems. !e comparison, though, founders at at least two points. First (borrowing again from Frege), it misconstrues the nature of the authority laws of logic can claim over our thought. Second, it misconstrues the way in which it is open to a special system to identify those questions of truth towards which we stand. If the "rst point is not Putnam's in particular, the second certainly is. First point. To conceive of dicta of special systems as immune to worldly bearing, as YW suggests, is to conceive of them as something like stipulations: the descriptions of this system are to work thus. In the system a cell is to be called black just in case there is at least as much black in it as white. In another system, perhaps, not. And one could describe patterns on walls in a system that did not so work at all. Similarly YW suggests, one can capture mechanical phenomena in Newtonian mechanics, in terms of its physical quantities, or, if you prefer, within a di#erent system in terms of others. Or, omitting to speak of mechanics, one can, as one cannot for logic, duvk being subject to any mechanical dicta at all. By contrast, If the wall has a black spot and the six-pack is hurtling towards the windshield, then the six-pack is hurtling, no matter what the special system. Nor can one stipulate whether a conjunction is to be taken to entail its conjuncts; nor whether any particular thoughts we think are conjoinable. Nor would such room for stipulation "t with Frege's, and YW's idea that there is no such thing as illogical thought (an empty idea if one can stipulate how thoughts are to behave). Second point. It is simply not true that momentum is mass times velocity. In thinking that momentum is mass times velocity, one cannot be thinking something which could be made true by placing it in some special system. No proposition of any system that would decribe the mechanics of the world could connect momentum, mass and velocity in that way. Which means that to think that momentum is mass times velocity, whatever this might be, could not be simply to think some given proposition, on YW's conception of what a proposition is. Which brings us to what is most central in Putnam's thought. 4. Open Questions: !ought and proposition are two rival conceptions of a question of truth. Pro tem, abstracting from this disagreement, I will speak simply of questions of truth. A question whether the six pack is %ying towards the windshield with such-and-such momentum might be one such. !e disputed question is: what identi"es a question of truth as the question it is. Suppose there were some special system, with speci"ed vocabulary and syntax, which generated a proposition, ', that the six-pack is %ying towards the windshield with momentum 200(. To be that proposition would then be, per se, to be structured as that system structures '. If '* is structured di#erently, then, ipso facto, it is not '. If ' identi"ed a question of truth-a given way of making truth turn on how things are-then things could be made to turn in that way on how things are only in representing structured as ' is. ''s structure would be essential to representing in that way. In applying logic's laws to given discourse we look, in "rst instance, for something essentially structured in the way a proposition is to which to apply those laws. But this, by itself, does not answer the question that presses here: whether the same question of truth might be posed by representing otherwise structured; and if so, what sort of structuring might do this. !is question cannot be answered by ''s parent special system itself. !ere would be no such question if questions of truth were to be counted as (YW's) propositions are. But both Frege and Putnam give reasons why this way of counting such questions cannot be right. I will focus on Putnam's. Newtonian mechanics de#nes 'momentum' as mass times velocity. Why can it not just be understood as using that term to speak of something of which this is true? Answer: because Newtonian mechanics is, or was, to be understood as thus speaking of a notion to which many strands belong. !ese individual strands may prove not to hold together. !e Newtonian de"nition is just one such strand. It might prove (and has proven) the one that has to go. Among other strands in the Newtonian understanding are: that momentum is a physical quantity; that a rigid body has some; that momentum has a certain role in explaining mechanical phenomena. !e de"nition appears not to "t with these others. More speci"cally, if relativisitic mechanics is right, there can be no physical quantity "tting the Newtonian de"nition. On the other hand there is a physical quantity of which all of the above would have been being (quite reasonably) supposed before 1905. Now two possibilities: 'momentum' in Newtonian mechanics referred to nothing; or it referred to this last-mentioned physical quantity. On the "rst there is no such thing as momentum; so no special system ever spoke of such. On the second, there is such a thing. Many of the above strands hold good of it. What Putnam has shown is how the second can be what is correct as to what Newtonian mechanics spoke of. A simple parallel (from another familiar context). I point and say, '!e man behind the Foster-Grants is on his "$h martini.' But those 'Foster Grants' are really Maui Jims. Nor is that 'man' actually a man. If I said something, it was of someone. But if I said something of someone, that person would not "t the description, 'behind the Foster Grants'. In fact, if I did say something of someone, it is clear who it would be. (Vide my pointing.) So either I said nothing of anyone, inclusive the person I clearly meant to speak of, or I said something (false if she is wearing Ray Bans) of that person. !e reasonable choice: the last. What, now, of YW's Newtonian special system? First, if that the six-pack is %ying with momentum M is about momentum, then there is no such proposition in that system. For if it is intrinsic to that system to be governed by the Newtonian de"nition, then it is, so far, a system for speaking of what could not be a physical quantity at all, whereas momentum is one. Second, though, that same question of truth, whether the six-pack has momentum M, might have been expressed by a proposition in such a system had Newtonian mechanics proved correct. Hence, that question, while identi"ed with a particular thought in Frege's sense, cannot be identi"ed as such with any proposition in YW's. Moreover, if we understand it as intrinsic to YW's system that it is a system of descriptions for mechanical phenomena, then as it turns out there is no such system. (!e world-dependence of a thought's existence.) So far, something on which Frege and Putnam agree. A particular expression, or presentation, of a question does not on its own "x what another presentation of the same question would be. But while for both Frege and Putnam what thoughts there are is, somehow, a world-involving matter, Frege is unlikely to have anticipated Putnam's take on the idea that so is what would count as a presentation of a given thought. Frege had his standards on proper de"nition. What he may not have anticipated is that whether, e.g., has momentum M is well-de"ned is hostage to how things happen to be. Both agree, though, on this crucial point: identifying di#erent presentations of the same question of truth involves extra-logical work-the sort of work involved in settling whether two people are disagreeing (or agreeing) about the same thing. For Frege, I suggest, it is the fate of such work which requires SinnBedeutung to be a many-one relation. But for Frege's birth there would have been no thoughts about him. Had Venus had a di#erent history, there might have been no thought about it in which it was presented as the Morning Star. !e world does that much in deciding what thoughts there are. A fortiori there are no thoughts about physical quantities without quantities for them to be about. Putnam's point: here, too, it is for the world to decide just what thoughts there therefore are; and just how any such thought makes truth turn on how things are. !e way things are is what, in a thought, we represent as some way or other. What truth thus comes to turn on is also something on which turns just how, in that representing, we made truth turn on it. Such world-involvingness of the identity of thoughts lies at the core of Putnam's response to the idea that laws of logic say nothing. But before seeing how, there is one more step to take. 5. Logic's Topic: A proposition, for YW, structures elements each of a type to which truthpreservation is sensitive, insofar as such preservation is part of being true as such. So what logic says about, e.g., the relation of a conjunction to its conjuncts applies directly to propositions themselves: what is a conjunct, what its conjuncts, can be read directly o# of the structure by which these are to be identi"ed. YW's propositions, however, are not, as we have seen, what identify questions of truth as the ones they are. Frege's thoughts do that. Finding conjunctions and conjuncts within Fregean thoughts, though, involves extra-logical work; inter alia, working of seeing in just what ways the same thought may be decomposed. Fregean thoughts are not each built in a given way from some given stock of building blocks. A thought, as opposed to a proposition, is not something to which logic applies in just one way. Each law of logic, or tautology, as YW sees things, embodies a particular way for elements in a representing to cancel out. !ereby it re%ects the structure of some given system or class of systems. A domain of Fregean thoughts, though, does not as such form any such system. Decompose a thought in some given way and one may arrive at something to which laws of logic apply directly. !ere then remains a substantial question: how else that same thought might be identi"able. Identify a proposition by the structure conferred on it by its derivation in its system, and no such substantial question remains. Each element in a decomposition of a thought brings it under a certain generality; presents it as the same as some range of thoughts in some given respect. Some such samenesses bring us to that to which logic as such is sensitive. What Frege saw as essential to a question of truth, though, is that for any given such question, there is no one right way of doing this. A decomposition of a thought cannot on its own provide us with a determinate notion of same thought. To which, thanks to Putnam, we may add: for any given candidate way of presenting a thought-even by stipulation-whether this is a way of presenting that thought, or any thought, is liable to be a substantial, sometimes world-involving, matter. How, then, could logic tell us how one must think to reach the goal truth? How could there be such a thing as what follows by logic from what? Well, how do we get from a whole thought to a structure to which logic speaks directly? Let the whole thought be that Sid slurps. !is decomposes into (roughly) an element which makes the thought one whose truth turns on whether Sid is the ways it speci"es, and an element which makes truth turn on whether the object it speci"es slurps-as it were into a naming element, making the thought hostage to how some given object is, and a predicative element, making it hostage to which objects are some speci"ed way-here, such as to slurp. Each of these elements is of a given type, instanced in an inde"nite range of other thoughts. !ere is, familiarly, the thought that Sid waddles, etc; and there is the thought that Pia slurps, etc. Where there is such a structuring of elements, there is also a truth-preserving inference to a related thought with a di#erent structure: colloquially, from Sid slurps to someone does. Such is the sort of thing logic tells us. On what information does the result depend? Logic takes an interest in the occurrence of predicative elements, and of naming elements, and in their distribution within a given decomposition, or corpus of them. It is not interested in whether a thought is about Sid, nor in whether it is about slurping. Whatever is proprietary to such notions is not part of what follows from the notion being true as such. It is interested in an element's recurrence. It matters that it is the same way someone is represented both in Sid slurps and in Someone does. Whether this happens to be slurping is beside the point. Let us call what I have just abstracted from that decomposition of the thought that Sid slurps a logical form. Logical forms can serve as logic's primary interest. We can think of logic as generating some "xed stock of them. Each form would be a construction of indexed thought-element types, each index marking a distribution of some given element of that type. Such a stock of forms would re%ect the most fundamental features of being true as such. !e rules for such constructions would generate such structures as, e.g., ones in which some concept (more properly what it was a concept of) would be predicated of some object. Laws of logic would then be dicta identifying which transitions from some logical forms to others were truth-preserving. Logic treats the phenomenon logical form. A standard logical calculus presents the details (within its scope) in that special way peculiar to such calculi. We began from a conception of logic on which the distinctive feature of laws of logic, aside from their truth, was their maximal generality, in a sense in which such generality is achieved through maximal quanti"cation. On this conception logic speaks to thoughts through a reverse process, instantiation-what, on our present conception, would lead to a decomposition of a thought; one such among, perhaps, many. So conceiving laws of logic in this way is one way to try to capture logic's topic-neutrality; its having no special subject matter. But we now have another way of conceiving the matter. Such leaves logic both universal and topic-neutral. But it does not achieve this by quantifying away from all subject-matter. Rather, it achieves this by virtue of the special subject-matter it makes logic's topic. !at special subject-matter is the domain of logical forms. Logic unfolds the notion being true in, and by, unfolding the notion logical form. !is changed conception of logic is occasion once again to consider the idea that being true (the notion logic unfolds), like the notion momentum, is made up of many strands. Logic's concern is meant to be something intrinsic to being true itself, hence to any thought. Part of that something is now that such-and-such are the logical forms. So, "rst, any thought is decomposable into some of these. Second any decomposition of any thought is of some one of these. !ird, perhaps, for any form whose constituents are place-holders for thoughts or predicative elements thereof, any thoughts or predicative elements (as appropriate) may replace those place-holders to form a compound thought. !us, any move from thoughts of a given forms to something of a form which follows from these by logic's laws is a move to a thought, moreover one true if these "rst ones are. Only some quantifying, e.g., may preserve truth; but all of it preserve being truth-valued. Logic may never lead from thoughts to nonthoughts. Logic does not tell us what thoughts have which forms. It is thus far insulated from con"rmation or refutation by the way things happen to be. But it unfolds a notion with enough independent threads to justify Putnam's refusal to endorse "metaphysical guarantees of the unrevisability of logic." Logic may "x 'the form of coherent thought' as such; in which case there is no such thing as thinking something so, not subject to its dictates. Such is what it is for logic to be universal. Universality does not shield the question what laws have such scope from proving, like laws of momentum, hostage to how things are. 6. Truth's !reads Revisited: Having pointed out that laws of logic were not laws of holding true (when thoughts would be held true), hence not psychological laws, Frege continued, It is because of this that they have authority for our thought if it would attain to truth. (1893: xvi) But, as MW later wrote, neither, equally, is when to count something as being red a matter of when it would be held red. (Cf. Zettel: §§429-432) !e facts of what would count as red (or green, etc.) hold a certain authority over all thought about colour (or those colours). But it is not that special authority logic holds over all thought. Such, then, must have another source. Objectivity is per se authority external to us. But one can omit attitudes towards colour, or at least towards being red or green. !ere is no such opting out of logic's laws, or none we currently understand. Also, while we do understand how the world may reveal what momentum really is, we as yet have no idea what it would be for it to reveal to us what (the) logical forms really are. Logic holds a special authority over us. Such cannot derive just from the fact that its laws are not psychological. Frege's idea was: logic is not just universal, but also ineluctable. !ere is only one thing it could have been, no matter what. Logic (for Frege) unfolds what belongs to being true as such. To be a thought, for Frege, is just to "x (or be) some given question of truth. ("I place the content of the word 'true in lead position ...and I let thoughts follow immediately as that by which truth can come into question at all." (1919: 273) Logic thus could not but be universal. Further, if to think is (inter alia) to think thoughts, then logic thus governs all thinkers. !ese points do not depend on what logic's dicta are, or on just what they dictate. Still, being true is a notion made up of separable threads; as are concomitant notions such as logical form. We have seen how such a feature mattered to what momentum is. How might it matter here? In search of an answer one might "rst look more closely at the notion ineluctable. When Frege tries to imagine not being bound by our familiar laws of logic (see 1893, xvi), what he thinks of is thinkers who $out these laws. Another idea would be: the thoughts they think (or some) are-despite what seems to be just intrinsic to being true as such-simply not articulable into those logical forms in terms of which our familiar laws are de"ned. So our familiar laws would be, for part of their thinking, at least, not %outed, but inapplicable. Here is a new understanding of being ineluctable. !ose laws we know may well be inescapable for thought insofar as it articulates into those forms in which the logic we know trades, and yet not of force where (if anywhere) thoughts did not assume those logical forms for which the laws were designed. Such an idea "ts with the way in which we have seen YW's two di#erent ideas of logic's laws as saying nothing come apart. One idea is that these laws say nothing in that their elements cancel each other out. We might thus think of logic's dicta as di#ering from other thoughts in that, since their role is to govern any though whatever, no provision has been made for them to be hostage in any way to how things turn out to be. In other cases, to put it in YW's terms, lack of provision is within some special system. !at system of Newtonian descriptions of mechanical phenomena provides no way for the Newtonian de"nition of momentum to prove either false or senseless. It stands within the system as a "xed point. But the life of the thought that momentum is mass times velocity is not con"ned to its place in any given special system. Logic is a di#erent story. !e universality attaching to its role means that in the case of its dicta cancelling out is not just within any special system. YW's other idea of saying nothing (6.1222) is: being neither supported, nor called into question, by anything experience may show as to how things are. But saying nothing on the "rst idea of doing so does not seem to entail saying nothing on this second. Laws which, if laws, hold of all thoughts are plausibly ones whose holding is not hostage in any determinate way to what, given them, is liable to be true or not. !ey are precisely what would hold independent of the truth-value of such things. YW's idea of cancelling out is a not implausible way of thinking of this sort of insulation from worldly vicissitudes. Such laws, as Putnam's Kant has it, merely limn the form of thought as such (for Frege, of being true as such). But what would be so of laws which held cannot, it would seem, by itself select which laws do hold. Or not unless the ways in which the world is liable to bear on truth and falsity are identi"ed, uniquely, by some structure intrinsic to the domain of thoughts as such-an idea which Putnam has certainly given us reason to reject. Such abstract rumination begins to gain content with our shi$ed understanding of what logic is about. Suppose that logic's laws have an identi"able subject matter: logical forms-not what forms given thoughts assume, but what forms there are for thoughts to assume, and how these relate to one another. Here there seems, at "rst blush, something for a would-be law to get right or wrong. Would not any claim as to what forms there are (in Frege's phrase) expose itself to risk of error-a risk, of course, which some such claims would escape? But when we look more closely, perhaps such dealing in abstraction is mere word play, a %ight of fancy. For what do laws of logic say? What features of being true do they unfold? Well, for example, what is truth-valued is either true or false; where which of these it is is independent of what we think. !ere are thus an easily surveyable variety of ways for the truth-value of a thought to be "xed by the truth-values of others. Logic provides us with logical forms corresponding to these ways; and then rules determining how truth is preserved in moves from some forms to others. So, for example, logic speaks of a way for a thought to be formed from, A&B. It tells us that a move from such a thought to the corresponding thought, A, is truth-preserving. Call a thought which assumes this form relative to some thought, A, and some thought, B, a conjunction of A and B. Which thoughts are conjunctions of which others? Such is not for logic to say. What it does say is when a thought would be a conjunction of two others: for a start, when that "rst thought is true just where those two others are. Now look at that fact about truth-preservation. How minimal can a fact get and still be one? Such, I think, exempli"es the sort of thing laws of logic (viewed one way) say. But there is more. Logic tells us what logical forms there are: any thought must assume some of these. !ese then identify the ways for thoughts to relate logically to one another. Moreover, moves which logic tells us preserve truth it also tells us preserve thought-hood. So, too, with all ways of compounding thoughts (or making thoughts of predicative elements) always preserve thought-hood. So, for example, though identifying thoughts is extra-logical work, for any two thoughts, A and B, there is a third, A&B, and a fourth, and etc. Any two thoughts can be compounded or disjoined, disjunction distributing over conjunction and vice-versa. On the other hand, for a given sort of thought-say, of given water, at a time, that it is boiling-there is, on the one hand, the way a thought of that type would represent things as being, and there is what it so represents: some particular case, that water's being as it then is. To be able to think water to be that way is to be able to recognise, of particular cases (of suitable sort) when they would, when not, be cases of things being the way in question. Such exempli"es abilities we have to recognise the obtaining of a relation to which logic does not speak: a relation between something governed by its laws, and something-bits of history- not. Such an ability can at least be exercised as a control on what logic tells us here. Is it really so that wherever there is a conjunction of a thought with a disjunction there is a disjunction of the conjunction of that thought with each disjunct? Perhaps. But might there not be something here for logic to be hostage to? We can now understand how Putnam stands in relation to both YW and MW. Putnam can accept the universality of logic. Universal is just what it is designed to be. Logic, as Frege put it, unfolds the content of the notion being true. A thought is precisely 'that by which truth can come into question at all'. So logic applies to all thoughts. Concern with any thought is just its business. !ere is no pressure for things to be otherwise. Nor need Putnam dissent from YW's "rst notion of saying nothing. Logic's concerns (truth-preserving) touch one side of the representing-as relation: ways to represent things; not (directly) things so to represent. Truth belongs to representation. Normally it is a joint product of both sides of that relation: one side makes truth turn in a certain way on the other; the other then yields a product of so turning. What represents a law of logic to hold, though, cancels out work of that other side; makes truth turn in the null way on it. Or better: it provides no way for truth thus to turn. Putnam need not balk at this either. Putnam must, though, with MW, resist YW's second account of saying nothing. A logical truth, like any thought, represents things as a certain way there is for things to be. !ere are things to be understood as to just what way that is. But it is a world-involving matter what ways for things to be there are (and what each is). So that what is to be understood as to what any given one is is relative to a perspective on it and on the world. What is (now) to be understood as a law of logic would be might, from a more revealing perspective on the world, be to be understood di#erently, or, perhaps, prove simply ununderstandable, as the world has proven to be. Which rules out YW's idea of absolute immunity to worldly bearing. It may belong to a law of logic that for it to hold is for the world to be denied a role in deciding that it does. Whether there is such law, or such a thing to hold or not, is another matter. I have here championed one idea of Frege's at the expense of another. Frege was right, and Russell and YW not, about what questions of truth there are, hence about what the truths are across which truth is to be preserved. !oughts are distinguished from propositions in the room a decomposition leaves for di#erent applications of the notion same thought. Just this leaves the space Putnam has charted for the world's role in "xing what ways there are for things to be and how any given one in fact makes truth turn on how things are. Frege's proving right on this point, though, brings with it a conception of logic decidedly not his. A law, so logic's, holds. It is neither true nor false. So it is not a thought. But it could relate to what it governs as a quanti"cation to its instantiations. For the project Begri%sschri" addressed it was important for logic's laws to do so. On the present view, they do not. !ey govern relations between thoughts. But in "rst instance they speak of forms abstracted from thoughts. How they apply to given thoughts is thus not "xed until extralogical work is done. (Compare the way mechanic's laws apply to given rigid bodies.) !is idea of application-at-a-distance is sometimes said just to be our current conception of logic; a conception re%ected in that peculiar form of expression of what logic says, a calculus. If this is right, though, the current conception is one current philsoophers all too o$en forget. Charles Travis 25/01/14 Bibliography: Frege, G., (date uncertain): "17 Kernsätze zur Logik", in Nachgelassene Schri"en, pp. 189-190. _____, 1882: Letter to Anton Marty, Gottlob Freges Briefwechsel mit D. Hilbert, E. Husserl, B. Russell, sowie ausgewählte Einzelbreife Freges, Hamburg: Felix Meiner, 1980, p. 118. _____, 1893: Grundgesetze der Arithmetik vol. 1, Jena: Herman Pohle, 1893. 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Wittgenstein, L., 1922: Tractatus Logico-Philosophicus, London: Routledge and Kegan Paul, 1922. _____, 1967: Zettel, Oxford: Basil Blackwell, 1967.