Issue #13 -- October 2001. Pp. 33-47
The Justification of Deduction
Copyright © by SORITES and Silvio Pinto
The Justification of Deduction
It has been a long-standing claim of Michael Dummett that deduction poses a philosophical problem of its own justification. According to him, when philosophers set out to look for a justification for our practice of inferring, they want an explanation of the validity and fruitfulness of the rules of inference that we accept as valid and fruitful. The fruitfulness of valid deductive argument simply means that new knowledge is gained in the transition from the argument's premises to its conclusion. Another way of expressing this is to say that knowledge of the conclusion of the argument is not entailed by knowledge of its premises.
Probably the first philosopher to recognize the essential fruitfulness of deduction was Frege.Foot note 3_1 For him, understanding a proof always requires a creative act of forming new concepts via the process of carving the thoughts expressed by its various steps in ever novel ways. Frege's insight was meant to accommodate the undeniable fertility of mathematical proof but it was never developed into a systematic explanation of fruitfulness. This may be due to the fact that his recognition of both the validity and the fruitfulness of logic created a tension in his philosophy. For crucial to the Fregean explanation of validity is the idea that articulated thoughts exist independently of us so that their logical relations (for instance, entailment) obtain or not no matter whether we grasp them or not. But if a pair of thoughts is to stand in the relation of logical consequence then not all of their possible partitionings into constituent senses will be allowed. Validity must therefore impose certain ways of carving each of the thoughts in a proof into its respective constituents. That this is so in Frege's view, however, seems to go against his own explanation of fruitfulness since this latter forces upon us the opposite, apparently very anti-Fregean picture of concept formation, namely, that of proofs as producing new concepts--or senses, as Frege would say.
Following Frege, Dummett also claims that deduction is essentially fruitful. As I understand it, his argument for this claim is the following. If we do not assume that each case of deduction is--even if sometimes infinitesimally--fertile then there would be no way of explaining the numerous obviously fruitful proofs in mathematics.Foot note 3_2 One of Dummett's preferred examples is that of Euler's famous proof of the impossibility of an uninterrupted route that crosses over all the 7 bridges in eighteen century Königsberg without crossing any one of them twice.Foot note 3_3 Before the proof was found, there was already a way of checking the conclusion of Euler's argument, which proceeded by enumerating all the finite many possible routes and showing that each one of them required that at least one bridge be crossed twice. What Euler discovered was a new and more elegant method for proving this. He gave us a new way of representing an arbitrary seven-step route. Similarly, Cantor invented a new method for constructing a real number which is different from all the denumerably many rational numbers between 0 and 1. But could we not equally well explain all these cases of undeniable fertility of deduction in terms of the non-preservation of the triviality of the basic deductive steps when they appear linked together in an innovative proof? Or maybe by claiming that what makes them really surprising is the introduction and later elimination of unexpected premises?
I take it that Dummett would deny that such proposals were really offering any explanation of fruitfulness. For if the notion of a trivial deductive step is to mean that the step is such that there is no epistemic gap between the step's premises and its conclusion, then, to use the analogy between epistemic and spatial distance, we are left with no clue as to how a proof can be fruitful--that is, how there can be an (in many cases, considerable) epistemic distance between its assumptions and conclusion--given that it is assumed to consist of a sequence of deductive transitions whose premises-conclusion epistemic length measures zero. On the other hand, if the explanatorily relevant notion for fruitfulness is that of unexpected premises, then it is not at all clear how the presence of these premises could account for the epistemic gap between the premises and the conclusion of many deductive arguments. What needs clarification is the epistemic distance between two locations in the deductive chain. A fruitful assumption or definition is just a location on our geometrical picture of fruitfulness. But we want to know what makes the distance between two locations in the chain possible.
The problem of the justification of deduction, as Dummett sees it, is therefore that of providing a philosophical explanation for the validity and the fruitfulness of those forms of argument which are valid and fruitful. It is incumbent on the theory of meaning for the language, in his view, to supply such an explanatory argument. The difficulty is that the requirements of either one of them seem to conflict with the requirements of the other: validity seems to demand that the conclusion brings in nothing new with respect to its premises whereas fruitfulness seems to require exactly the opposite.Foot note 3_4 This is how Dummett expresses the puzzle:
For it [deduction] to be legitimate, the process of recognising the premises as true must already have accomplished whatever is needed for the recognition of the truth of the conclusion; for it to be useful, a recognition of its truth need not actually have been accorded to the conclusion when it was accorded to the premises. Of course, no definite contradiction stands in the way of satisfying these two requirements: recognising the premises as true involves a possibility of recognising the conclusion as true, a possibility which will not in all cases be actualised. Yet it is a delicate matter so to describe the connection between premises and conclusion as to display clearly the way in which both requirements are fulfilled. (Dummett 1973b, p. 297)
Dummett maintains that only a molecularist theory of meaning can deliver such a harmonious explanation of validity and fruitfulness. Generally speaking, a meaning molecularist believes that the meaning of a word or a sentence is fixed by a linguistic unit which is well short of all the language. His opponent in this debate is the meaning holist, someone whom Dummett represents as denying that for each word or sentence of a language there is a group of sentences the knowledge of whose meanings is sufficient to determine the meaning of the mentioned word or sentence. In other words: the meaning holist does not believe that each expression of the language splits the set of sentences of which it is a constituent into a constitutive and a collateral group. Clarification of the constitutive/collateral terminology as applied to words and sentences as well as of Dummett's reason to maintain that the holist cannot reconcile the validity with the fruitfulness of deduction now requires that we get into the detail of the controversy between molecularism and holism.
§2. Molecularism versus Holism
The terms `molecularism' and `holism' will be used here to designate two mutually exclusive positions in the philosophy of language concerning what should be taken as a sufficient basis for fixing the meaning of each expression of a language. But how can the notion of the basis for fixing meaning or its equivalents--for instance, unit of meaning--be cashed out? Quine,Foot note 3_5 in his well-known criticism of the logical empiricists' criterion of empirical significance, made a very strong case for the thesis that linguistic content cannot be attributed to sentences in isolation but only to a larger linguistic unit, which he sometimes identifies with our language and some other times with our present total scientific theory. This is so because in Quine's own terminology sentences do not face the tribunal of experience--which in agreement with the logical positivists he takes to be the source of linguistic content--piecemeal, i.e. one by one or even in small groups. Hence, the unit of meaning is, according to Quine, that minimal linguistic whole to which empirical content can be attributed independently of any content attribution to any larger linguistic unit.
Another way of spelling out the notion of meaning unit and one in which the constitutive/collateral dichotomy comes out quite naturally was suggested by Dummett himself. In the opening pages of his first book on Frege,Foot note 3_6 Dummett mentions a circularity (certainly known to Wittgenstein in the Tractatus and to Russell) which appears to threaten any view that takes sentence meaning to be explanatorily prior to word meaning. If the latter is explained in terms of sentence meaning by saying that it consists of the contribution the word makes to the meaning of the sentences in which it occurs,Foot note 3_7 then sentence meaning would be circularly characterized were it also proposed that the meaning of a sentence is to be explained in terms of the meanings of its constituent words. A possible way out of the circle is the one which, according to Dummett, Frege uses, namely: to elucidate sentence meaning in terms of truth-conditions. In this early book, Dummett did not think there was any danger of circularity at the level of understanding; knowing the meaning of all the constituent words in a sentence is obviously necessary, and therefore in this sense prior, to knowing the meaning of the sentence.
In his later work,Foot note 3_8 however, Dummett suggests that the circularity problem discussed above can also arise at the level of understanding for the anti-atomist conceptions of linguistic content recognition (i.e. for those who regard as absurd the idea that grasping the meaning of words is prior to understanding whole sentences in which these words appear). The circle seems very present, for example, in the famous Tractarian passages where it is said that understanding sentences requires understanding their respective names and that knowing the meaning of each name presupposes understanding the sentences containing it.Foot note 3_9 The problem is then how to spell out a competent speaker's knowledge of the meaning of a word in such a way as, on the one hand, to express it in terms of his primary understanding of sentences (as should be expected from an anti-atomist about content recognition) and, on the other hand, not to fall into the sentence/word meaning circularity. Dummett proposes to tackle this problem by distinguishing two groups of sentences for each expression of the English language. Let us call them, respectively, the expression's constitutive and collateral group.Foot note 3_10 The first consists of all the sentences containing the expression which together provide the basis of our understanding it. The second group comprises all the other sentences containing the expression; it is called collateral group because understanding any sentence belonging to this second group requires a prior understanding of the expression in question. Here is what Dummett says about the constitutive/collateral distinction:
the priority of sentence-meaning over word-meaning requires the understanding of a word to consist in the ability to understand certain sentences, or more exactly, at least some of the sentences of a certain range, in which it occurs. (...) The compositional principle demands that, for any given expression, we should distinguish between two kinds of sentence containing it. An understanding of the expression will consist in the ability to understand representative sentences of the first kind and does not, therefore, precede the understanding of sentences of that kind. By contrast, an antecedent understanding of the expression will combine with an understanding of the other constituent expressions to yield an understanding of a sentence of the second kind, which demands an understanding of the expression but is not demanded by it. (Dummett 1991, p. 224)
Later in the same text, Dummett adds that the constitutive group of a word must contain the simplest sentences containing it, whereas there is no limit to the complexity of the sentences that can figure in the word's collateral group. He illustrates this with the word `fragile': while the sentence «this plate is fragile» should be part of its constitutive group, «I'm afraid I forgot that it was fragile» should belong to `fragile''s collateral group. Dummett equates understanding a word with the ability to understand the simplest sentences containing the word. According to him, this is the picture one gets if he takes the molecularist standpoint on the issue of recognition of linguistic content. That is, if he insists that knowledge of the meaning of a word does not require more than understanding a sufficiently small fragment of the language. All the sentences containing the word which are situated beyond the boundaries of this fragment do not contribute to fix its meaning; on the contrary: a previous knowledge of the word's meaning is necessary in order to understand them.
That there is for each word such a fragment well short of the totality of language is what the meaning holist wants to deny. Although he may accept that sentence meaning is--at least partially--determined by the meaning of its constituent words plus the sentence's syntactical structure, the holist will take issue with the thesis that word meaning is always determined by an appropriate fragment of the language consisting of logically non-complex sentences. His rejection of the thesis may take various forms. A more radical holist might, for example, hold that the meaning of any sub-sentential expression is fixed only by the whole of the linguistic network. Echoing a familiar sort of holism about belief, such a holist would say that the content of any word or sentence can only be individuated in the context of the totality of its linguistic web; any smaller unit would leave its meaning substantially undetermined. Quine's holism, for instance, has been interpreted in this way.Foot note 3_11 A moderate holist might, however, reject the molecularist thesis merely on the ground of his belief that the meaning of a new sentence is not completely determined by the constitutive groups of its respective constituent words and adequate hypotheses about the sentence's syntax. This kind of holist will be willing to allow for the possibility that the new use of the sentence be tracked down to a new use of one or more of its constituent words. Davidson's linguistic holism seems to me to be of this latter moderate variety.Foot note 3_12
Dummett's purely constitutive way of characterizing the constitutive/collateral distinction and therefore the molecularist/holist dichotomy does not seem to me to be entirely satisfactory. By `purely constitutive way' I mean to point out that his rendering of the distinction is in terms of what constitutes a speaker's grasp of the meaning of a word. His answer to this constitutive question appeals to an ability to understand certain simple sentences containing the word. My uneasiness with Dummett's route to establish the constitutive/collateral distinction relates to what I take to be a gap in his explanation of the priority of sentence meaning over word meaning for the members of a word's constitutive group. If for the members of this group sentence meaning is prior to word meaning, how can a speaker ever get from the former to the latter? In other words, how can his grasp of the meaning of a finite number of sentences ever be sufficient for the speaker to derive the meaning of a word they all have in common? After all, one might say, the kind of understanding of sentences that is at stake for the members of the constitutive group cannot be further analyzed in terms of the understanding of these sentences' constituent parts. But, he might insist, if compositionality and structure do not play a role in the grasp of the content of any member of the constitutive group, then the non-analyzed content of any finite group of sentences, however numerous, cannot on their own ever yield the content of any of these sentences' constituent parts.Foot note 3_13 Within the framework of Dummett's explication, the constitutive/collateral distinction threatens to collapse.
I reckon the gap in the account of the constitutive/collateral dichotomy offered by Dummett can be bridged if we switch from his constitutive perspective to a radically interpretive one. The suggestion is that we consider the possibility of drawing such a distinction from the perspective of someone who is trying to learn the language for the first time. A very illuminating description of the interpretive framework has been provided by Neil Tennant in the following passage:
When we discern meanings as we learn the language, it is primarily whole sentences whose meanings we work out in context. (...) Sufficiently many sentences tentatively grasped allow meaning to coagulate upon the words they have in common. This coagulation of meaning is constrained by conjecture as to the structure of each sentence grasped thus far. The structure of a sentence is a matter of how the words and phrases and clauses have been composed within it. Once global sentence meanings plus structural hypotheses have conferred meanings upon individual words, the process is then reversed. New sentences are understood on the basis of one's assignment of meanings to words, and one's analysis of how the words are put together to make the sentence. The possibility in principle of the eventual success of the compositional method is a methodological principle guiding the language learner. (Tennant 1987, p. 31)
In this account, the language learner (or radical interpreter) starts with hypotheses about the content of the speech acts of his subjects which involve whole sentences (his assertions, for example) and about the syntactical structure and composition of these initial sentences. From these, he derives hypotheses about the meanings of the sentences' component words. Once the latter hypotheses have been firmly corroborated by the data, then the initial hypotheses about sentence meaning are no longer needed. For, at this stage (when the interpreter has learned the language), the meaning of a new sentence can already be obtained from word meaning plus the hypotheses about the sentence's structure. In Tennant's picture, this is the stage at which the initial procedure of inferring claims about word meaning from hypotheses about sentence meaning and about syntactical structure is reversed.
To see how Dummett's distinction is presupposed by Tennant's account, consider again the sentences with which his compositional method starts. These are the sentences without which, according to Tennant, the hypotheses about the meaning of the words such sentences have in common would find no support. The content of these sentences constrained by suitable further hypotheses about their structure will eventually converge into the content of the words in question. Hence the set of initial sentences make up their shared words' respective constitutive group. Once the sentences of a word's constitutive group have fully determined its meaning, which is guaranteed by the success of the compositional method, then any new sentence containing the word will have its meaning determined by the meaning of the word and the meanings of the sentence's other constituent expressions, as these latter meanings are already given by the expressions' respective constitutive groups; all such new sentences will belong to the word's collateral group. Tennant's description of the compositional method therefore corresponds to a molecularist view of language. For only a molecularist would maintain that finitely many hypotheses about sentence meaning could ever come to suffice for the determination of word meaning.
Tennant's approach also suggests a way of escaping the difficulty mentioned five paragraphs back about the transition from sentence meaning to word meaning. The problem, let us recall, was that if understanding the sentences of a word's constitutive group does not involve discerning in these sentences' contents any structure then it appears that no finite amount of constitutive sentences understood by an interpreter will suffice for him to derive the content of the word common to them. According to the compositional method, what enables him to infer word meaning from hypotheses about the word's constitutive sentences' meanings are the cited additional hypotheses concerning the syntax of these sentences. Once word meaning is firmly determined, then this kind of understanding of sentences (i.e. unstructured grasp of their content) is no more necessary as, with the reversal of the process, sentences can now be understood on the basis of their constituent words' meanings and of their composition (i.e. grasp of the sentence's content as structured in a certain way). The first kind of sentence understanding (unstructured grasp of the sentence's meaning) is only prior to word understanding until the interpreter finds his way into the alien language, at which point it ceases to have any application and the direction of priority runs from word understanding to the second kind of sentence understanding (structured grasp of the sentence's meaning). This is how a molecularist account can avoid the sentence/word meaning circularity.
By contrast, this discussion of Tennant's compositional method also helps us see what a holistic picture of meaning would be like. For all the holist needs to reject is the molecularist assumption that the compositional method will eventually succeed. According to such a holist, the interpreter will never reach a stage where sufficient contact with sentences containing a word uttered by the alien speakers will enable him to settle on a meaning for that word which will determine the meaning of any new sentence containing it exclusively from the content of its constituent words and the way they are put together. The reason for this, claims this kind of holist, is that no matter how many sentences containing the word an interpreter considers, there will always be new sentences whose understanding cannot be reduced to the old meaning of their parts and the way they are combined together. In order to understand these new sentences, the interpreter will have to revise his previous hypotheses about word meaning and structure in the light of this new datum: a new use of a word. For this sort of holist a new use of a sentence is not always reducible to the previous uses of the words in it. This means that no fixed group of sentences is fully constitutive of the meaning of any sub-sentential expression, so that the rest is collateral. This holist can make no sense of the constitutive/collateral distinction. As seen above, one need not be more than a moderate holist in order to adopt the view of Tennant's opponent.
§3. Dummett on the Justification of the Deduction
The holist cannot, according to Dummett, offer a harmonious explanation of the validity and fruitfulness of deduction since he fails to account for validity. And this latter cannot be explained within a holistic approach to language, Dummett insists, because in the context of this approach there is no room for the constitutive/collateral distinction.Foot note 3_14 The connection between the availability of this distinction and the possibility of a philosophical elucidation of validity is nonetheless not straightforward. Below, I will attempt to make it explicit by considering Dummett's account of validity.
§3.1. Dummett on Validity
As our interest here is in deduction, let us start with the question of how, for the molecular verificationist, the constitutive/collateral distinction applies to the logical constants. His claim is that the constitutive group of a logical constant comprises just its introduction rules.Foot note 3_15 In a system of natural deduction, each logical connective is characterized by a set of introduction and elimination rulesFoot note 3_16--a characterization which can easily be shown to be equivalent to one in terms of the connective's truth-table.
But why does the verificationist privilege I-rules? The reason is this. It is a basic tenet of molecular verificationism that the meaning of a sentence consists in the canonical method of verifying it. This method--the sentence's direct means of verification--will surely involve deductive inference if the sentence is not atomic. But in this case we need only apply the I-rules for the sentence's respective logical constants. For instance, a sentence like «6 is even and is perfect» can be verified directly via the separate computations of the two conjuncts and posterior application of the I-rule for conjunction. There may be other methods of verifying this sentence which use logically more complex sentences--for example, the sentence «some perfect numbers are not odd»--or other auxiliary sentences. Think, for instance, of the verification of Euler's proof's conclusion (the sentence «any uninterrupted minimal route through Königsberg's 7 bridges contains more than 7 bridge-crossings»). The proof gives us a new method of establishing the truth of this sentence which proceeds from the auxiliary sentence «consider any route through Königsberg's bridges which is uninterrupted, minimal and crosses all the 7 bridges». By contrast, the canonical method of verifying the conclusion of Euler's proof exploits only its internal logical concatenation of smaller sentence units; it makes exclusive use of the introduction rule of `any', which in this case is equivalent to a finite conjunction. Any non-canonical method for establishing the truth-value of sentences--i.e. those which appeal to elimination rules as well--Dummett labels an indirect means of verification.
The constitutive/collateral distinction with respect to sentences corresponds in Dummett's view to the dichotomy between direct and indirect means.Foot note 3_17 According to him, to know the meaning of a sentence is to know how to verify it directly. All the other means of sentence verification do not contribute to constitute its meaning and therefore belong to the sentence's collateral group. Thus, Euler's proof offers an indirect means for verifying its conclusion; the canonical method of establishing its truth consists of enumerating and demonstrating all its conjuncts and then using the I-rule of conjunction.
Now, what does the characterization of the constitutive/collateral distinction for sentences have to do with the homonym distinction for the logical constants? The following is the verificationist story. First of all, as Frege had already realized, in order to give the meaning of a logical constant it is enough to present the truth table of an arbitrary sentence for which it is the main connective. This is equivalent to taking all the sentences for which the constant is the main connective as members of the latter's constitutive group. Secondly, according to the verificationist, the meaning of all these sentences is identified with a direct means of verifying them and their main constant contributes to this means in so far as its respective I-rules warrant the transition from the direct methods of verification for both of the sentence's immediate constituent sentences to the direct method of verification for the sentence itself. Thus, if we know the meaning of «6 is perfect» and «6 is even», then this together with the I-rule for conjunction will tell us how to verify «6 is perfect and even» directly. Hence, within the molecular verificationist framework, the meaning of the logical constants is fixed by its respective I-rules.
Having established this, our next task is to clarify the connection between the verificationist characterization of the constitutive group for the logical constants and his explanation of validity. Dummett claims that this latter explanation must appeal to the notion of a harmony between the two main aspects of our practice of uttering sentences to make statements. Deduction is valid, according to the verificationist, if the practice of offering grounds for an asserted sentence--that aspect which is verificationally constitutive of its meaning--is in harmony with that of drawing consequences from it--the verificationally collateral aspect of the meaning of a sentence. Thus, Dummett says:
For utterances considered quite generally, the bifurcation between the two aspects of their use lies in the distinction between the conventions governing the occasions on which the utterance is appropriately made and those governing both the responses of the hearer and what the speaker commits himself to by making the utterance: schematically, between the conditions for the utterance and the consequences of it. (...) Plainly, the requirement of harmony between these in respect of some type of statement is the requirement that the addition of statements of that type to the language produces a conservative extension of the language. (Dummett 1973c, p. 221)
The quotation is explicit of how Dummett thinks a verificationist should cash out the vaguer notion of harmony, namely: in terms of the more precise notion of conservative extension. The latter has a precise sense, however, only when applied to formalized languages. Thus, of two formalized mathematical systems A and B expressed respectively in languages LA and LB we can say that A is a conservative extension of B if and only if LA extends LB and no sentence belonging to LB could be deduced from A that was not already deducible from B. Nothing like the concept of deducibility exists for natural language though; our reasons for asserting the truth of non-mathematical sentences are normally defeasible to a lesser or greater extent. But if we cannot speak for natural language of conservatism with respect to provability with the resources of the non-extended language, how else could conservatism in this domain be spelled out?
Dummett claims that the verificationist has a plausible natural language surrogate for deducibility, namely, the notion of verifiability via whatever means is available for truth-value attribution.Foot note 3_18 He suggests that an extension (NLf + e) of a certain fragment of natural language (NLf) is a conservative extension of NLf if the incorporation of new methods of verifying sentences propitiated by NLf + e does not change the truth-value of any sentence already verifiable with the resources of NLf. In other words, the new language which results from the enrichment of a fragment of it by new vocabulary is conservative relative to the latter fragment if any sentence which is verifiable in the scanter language, and hence has already a determinate truth-value assigned to it, does not have that truth-value altered as a result of the application of any new means of verification made available by the richer language. Still another way of putting the verificationist account of conservative extension is this: NLf + e extends NLf conservatively if the new vocabulary introduced by NLf + e does not conflict with that which determines the meaning of the expressions and sentences of NLf. Now, we know that for the verificationist sentence meaning is fixed by its direct means. Hence, the conflict we alluded to must be between an eventual indirect means of verification of a sentence of NLf introduced by NLf + e and its direct means supposedly available in NLf: these two means will be in conflict if they attribute different truth-values to the sentence.
Let us illustrate the Dummetian concept of conservative extension with a counter-example. Think of NLf as any fragment of natural language and of NLf + e as this fragment together with Arthur Prior's famous logical constant `tonk' which he defined in the following way:
where `p' and `q' are sentences of NLf or NLf + e.Foot note 3_19 Tonk does obviously not extend NLf conservatively since if we are justified (by a direct verification, say) in attributing the truth-values true and false to two sentences `r' and `s' of NLf respectively, then with the introduction of `tonk' the truth of `s' can be (indirectly) established as well. The presence of `tonk' would provoke a disharmony between the practice of grounding asserted sentences--for instance, `¬s' through its direct means--and that of drawing consequences from asserted sentences--for example, `s' as consequence of `r'. The lesson to be learnt from `tonk' is therefore the following: the I- and E-rules of logical vocabulary which does not extend its respective language conservatively in the verificationist sense are to be deemed invalid. As the logical constants are also an essential part of the indirect means of verifying sentences (via their respective I- and E-rules) and as for the verificationist sentence meaning consists in the direct means of establishing the sentence's truth-value, non-conservative logical constants--that is, those whose I- and E-rules are invalid--might easily upset the meaning of a sentence. Such a modification in sentence meaning would consist in the alteration of the truth-value conferred to the sentence before the logical constant's incorporation into the language. Alteration in sentence meaning would in turn be translated into an alteration in the meanings of their constituent words. So, if we do not want meaning to change in the transition from suitably rich fragments of a language to a larger one--i.e. if we side with the meaning molecularist--this must be our reason for banning non-conservative logical constants from our linguistic practice.
This is the molecular verificationist account of validity. According to Dummett, if one denies the constitutive/collateral distinction with respect to sentences on the grounds that their various indirect means also contribute to fix the sentences' respective meanings--i.e. if one adopts a verificational holistic position--then he would be left with no constraints to impose on a certain linguistic practice. He would be in possession of no criterion to exclude disharmonious practices of inference as unacceptable and could not explain why some other deductive practices are in order. I have argued elsewhere against Dummett's claim that a holist cannot provide a satisfactory alternative explanation of validity.Foot note 3_20 Here I will review some of the difficulties the molecular verificationist account of validity faces.
The first has to do with the applicability of the modified notion of conservative extension to natural language. One might complain that, as the truth-value of natural language sentences can normally not be established definitively, the verificationist version of conservative extension inevitably loses the classificatory power of the initial one. Thus, it is highly doubtful whether we could classify `tonk' as verificationally non-conservative if an application of the direct means of verifying `q' could only guarantee that we are fallibly justified in denying `q', while the use of a certain indirect means--via the defeasible direct verification of `p'--established that we are justified in asserting `q'; the specific use of the two methods might be blamed instead for the apparent quasi-conflict. The point is that a defeasible ground for assertion does not make room for a sufficiently precise concept of lack of conflict between, on the one hand, reasons to assert and, on the other, reasons to deny. One application of a method of verification may give us a reason to assert which a later application of the same method (in a situation where our set of background information has changed) might withdraw.
A second difficulty with the verificationist notion of conservative extension concerns the issue of whether the meaning of a word can be completely determined by an appropriately specified fragment of its respective language. As discussed above, a moderate holist denies this and one could manifest his agreement with him by citing the constant modification of the meaning of natural language terms by new uses of sentences containing it or by the discovery and confirmation of new scientific theories. An example of the latter would be, say, the word `dog' before and after the atomic theory became common currency. Or think of the word `funny' just before and after it started being used to signify an odd person. Examples like these can be multiplied almost indefinitely. In my opinion, they highlight the point that the thesis according to which the meaning of a word is totally fixed by a certain group of logically non-complex sentences containing it is, at the very least, highly controversial. I do not want to proffer a final word on this matter here. But if the holist is right about the irreducibility of the new use of a sentence to the old meanings of its constituent words plus the sentence's compositional structure, then there will be no room for a constitutive/collateral distinction neither for sentences nor for words. And without such a distinction no sense could be made of the verificationist concept of conservative extension. We must conclude hence that a big question mark still hangs above Dummett's explanation of validity.
§3.2. Dummett on Fruitfulness
As far as the explanation of the fruitfulness of deduction goes, it must incorporate some sort of epistemic gap between premises and conclusion. Dummett claims that this epistemic gap can only be accounted for by theories of meaning which allow for a distinction between truth and the recognition of truth. According to him, radical verificationism offers the perfect example of an approach which is unable to explain fruitfulness because of its identification of the truth of a statement with its actual verification by whatever means (direct or indirect). The problem is that the explanation of validity will require the preservation of some property of statements which this verificationist claims to be that of actual verification. But if verification of the conclusion reduces to the verification of the premises of an arbitrary valid deduction--as the radical verificationist would have it--then there can be no epistemic gap between them.
Dummett acknowledges that verificationism runs the risk of being unable to explain fruitfulness. He suggests that, in order to avoid the difficulty discussed in the last paragraph, the reasonable verificationist should move some way towards realism, that is, he must replace in the characterization of truth the implausible notion of actual verification by that of verifiability.Foot note 3_21 Consider, for example, one of Dummett's partial characterizations of truth:
True statements must comprise, though they are not necessarily confined to, all those which would have been established as true had the relevant observations been made; `observation' is, as before, not to be taken as mere passive exposure to sense experience but to include physical and mental operations and the discernment of structure (of patterns). In particular, we are able to say that a statement is true, in this sense, whenever we can show how observations that were made could have been transformed into ones that would have established it. (Dummett 1991, p. 181)
Truth, in Dummett's view, is to be elucidated counterfactually: a sentence S is true if were we in a position to verify other sentences (by observation or by some mental operation like computation or proof), we could thereby transform these verifications into a verification of S. In other passages, Dummett identifies this possibility of transformation with a method of verifying S.Foot note 3_22 Hence, truth for Dummett must be predicated of those sentences concerning which at least a method is known to us for transforming the direct verification of other statements into a direct verification of S.Foot note 3_23 The necessary gap between truth and knowledge of truth is preserved, he insists, because a competent speaker who understands S--i.e. knows a canonical method of verifying it--may still never have, or could never have, verified S directly; besides, if he knows of any indirect means of establishing the truth of S, he may have equally never applied it.
Once truth has been described in the above manner, Dummett can then explain fruitfulness in the following way. He will say that knowledge of the truth of a sentence (S) always goes beyond knowledge of the truth of the premises S is inferred from even when S's truth is established by a direct means. Of course, there are cases of the use of the direct verification procedure where the gap between knowledge of the sentence's truth and knowledge of the truth of the procedure's premise's may be infinitesimal as, say, for the sentence «6 is even and a multiple of 3». Other cases of application of the direct verification procedure are not so trivial and it would be plausible to say that there the mentioned epistemic gap is a bit wider. For instance, in the case of the sentence «101000 + 1 is prime». But the really interesting cases, the ones where the gap is much wider according to Dummett, are those where deduction gives us a totally unexpected indirect way of verifying S. Examples of these cases abound in mathematics. One could mention Cantor's proof about the cardinality of real numbers or Andrew Wiles' recent proof of Fermat's last theorem. For Dummett, there is a continuum of possible cases covering the whole scale measuring epistemic distance between premises and conclusion; in the most trivial cases such a distance will be extremely small, whereas the most ingenious proofs will be located at the other extreme of the scale.Foot note 3_24 The important point is that all cases of the application of deduction involve some epistemic distance, however small, between premises and conclusion. This distance is to be explained in terms of how novel the deductive transition under consideration is; that is, how much it diverges from the trivial cases where the canonical method of verifying the conclusion is what connects deductively premises and conclusion and from the even more trivial cases where such a canonical deduction is quite easily applicable. The distinction between truth and knowledge of truth guarantees that, although truth is automatically transferred from premises to conclusion in a correct deductive transition, there may be a quite pronounced epistemic gap between them.
Euler's proof nicely illustrates Dummett's explanation of fruitfulness. Recall that before the proof was found, there was already a canonical means of determining the truth-value of its conclusion--the sentence «any uninterrupted minimal route through the 7 bridges in Königsberg contains more than 7 bridge-crossings». Euler discovered a new, indirect means for transforming the canonical method of verifying the premise--the sentence «consider any route over the Königsberg's bridges which is uninterrupted, minimal and crosses all the 7 bridges»--into a method of verifying the conclusion. The proof--Euler's non-canonical method--is our warranty that the transition from premise to conclusion is truth-preserving. Moreover, knowledge of the proof together with the verification method for the premise provide us with a new route to the recognition of the truth of the conclusion. Imagine, for example, that all the bridges were numbered and that a device at each bridge detected only that this bridge was crossed but did not keep track of how many times it was crossed. Suppose also a central device which controlled all the bridge cross-detectors and signaled when all the bridges were crossed, and also whether the route was uninterrupted as well as minimal. This seems a good method for verifying the premise (MVP). Now, by Euler's proof, each step in MVP becomes a step in a new method of verifying the conclusion (NMVC). Application of MVP and of Euler's elegant procedure to MVP allows one to establish the truth of the conclusion, eliminating thereby the need to verify the latter by means of its cumbersome direct method. It is precisely this newness that makes Euler's proof fruitful.
So, the fertility of a deductive argument is to be explained, according to Dummett, in terms of how much it, when applied to the direct methods of verifying the premises, departs from the direct method of verifying the conclusion. In the Logical Basis of Metaphysics,Foot note 3_25 he claims that this explanation agrees with Frege's insight that the novelty of a fruitful deduction lies in the fact that a new pattern--a new procedure for connecting methods of verification, Dummett would add--has just been discerned; a pattern that was already there to be discerned. Now, we have seen that this supposed insight creates a major tension in Frege's philosophy: a tension between that which according to him validity requires--that is, that the conceptual framework we employ be prior to our deductive practice--and that which he rightly saw as essential to the fertility of deduction--that our deductive practice be partly responsible for the creation of new concepts. The question is: would Dummett's agreement with the Fregean picture of the fertility of deduction not create a similar tension in his philosophy as well?
§4. Concluding Remarks
In one of his most recent papers on the nature of deduction,Foot note 3_26 Dummett explicitly defends an externalist account of deductive consequence. He stresses his agreement with the Fregean picture of the practice of inferring as that of discerning changes in the patterns we impose on reality, where the patterns consist in our judgements about this very reality. The mentioned externalism translates into the idea that the possibility of transformation of one judgement with a certain discerned pattern into another--i.e. the possibility of a given deductive transition--is intrinsic to the judgements in question and not created by our inferring activities (like, for instance, a new proof). The patterns of our judgements are not imposed by us; their multiplicity is already contained in the judgements themselves. Dummett believes such a dose of externalism is necessary if validity is to be accounted for. Otherwise, as he claims, deductive transitions would not be meaning preserving; the judgements on which the patterns must be discerned would themselves change in the course of the transition from premises to conclusion.
Dummett's externalism about deductive inference matches very well with his explanation of validity in terms of conservative extension. Remember that, according to him, the requirement that our logical vocabulary be conservative with respective to verifiability is a necessary condition for valid transitions to be meaning-preserving. But this requirement, as we already know, rules out the possibility that the meaning of sentences and words might be changed by correct deductions. That is, if the explanation of validity is to appeal to conservative extension then the discovery of a new deductive argument cannot alter our conceptual framework. Here, I think, is where Dummett's view diverges from Frege's. For, in order to avoid inconsistency, Dummett must deny that the process of devising and understanding a new proof requires the deployment of new concepts. The conceptual apparatus does not change, he would probably insist; what varies is our command of such a conceptual network.
The problem with this way of accommodating validity and fruitfulness is that the explanation of the former is achieved almost at the expense of rendering the account of the latter very unsatisfactory. For to say that the fertility of deductive inference resides in its often contributing with new methods for verifying sentences whose possibility is nonetheless provided for by our language amounts to admitting only a very mitigated notion of newness: one that rests solely on our cognitive limitations with regard to seeing pre-existing conceptual connections. On Dummett's view, a being not subject to human epistemic constraints who could nonetheless make sense of our deductive practice would not find it useful at all. Put like this, the molecular verificationist account of deduction's usefulness does not look any more promising than the one in terms of the non-triviality of concatenations of trivial inferences. Perhaps Dummett's constraint on a plausible explanation of validity--i.e. that deductions should be meaning-preserving--was too strict. However, if that constraint was relaxed, the molecularist framework would have to be abandoned in favor of a holistic justification of deduction. Dummett, as noted above, is pessimistic about the prospects of such a justification. Many people, including myself, do not share his pessimism. Expounding my reasons would nevertheless deserve another paper.