Let us take stock and evaluate the dialectical situation between the intuitionist revisionist and the classicist. Is there any revisionary mileage in PTS and the dif- ferent proof-theoretic constraints inspired by harmony? We have seen a number
of axiomatisations of classical logic which bypass different difficulties presented by the Dummett-Prawitz camp. Obviously, harmony-as-normalisation does not con- stitute any real challenge to the classicist: There is a number of classical systems that are normalisable. More contentious is the issue of legitimate conversion steps and the relation to the Subformula and Separation properties. It is also possi- ble that normalisation—where the notion of maximum formula is altered—can be had without these properties, so an option for the revisionist is to require this higher standard from the classical opponent. However, even with the bar raised accordingly, the classicist has the resources to persist with a PTS account of her logic.
With respect to well-behaved conversion steps and the Separation Property the above discussion indicates that the approaches that fare the best are (a) multiple- conclusion natural deduction, NC, and (b) classical bilateralism, Ncp+−. This is no coincidence, for as we shall see, there is an intimate relationship between these two frameworks. Put more succinctly, the structural resources of multiple- conclusion and bilateralism leave the two frameworks with the same expressive power.56 In fact, for the above systems, it is straightforward to define a faithful
translation between the multiple-conclusion framework, call it SET-SET, and the signed single-conclusion framework, call it SET-FRML+. For example, if A1, ...,
An `N C B1, ...,Bm, then+A1, ...,+An, −B1, ..., −Bm−1 `N c+ +Bm. Conversely, if +A1, ..., +An, −B1, ..., −Bm `N c+ ±C, then Γ`N C ∆ = A1, ..., An`N C B1, ..., Bm, C, if +C; A1, ..., An, C `N C B1, ..., Bm, if−C.
The real question is whether there are significant philosophical arguments that can distinguish between the two approaches. There are two separate issues here: First, the burden is on the intuitionist revisionist to convince us that they are in fact 56The formal details of this ‘expressive power’ will be made more precise in Chapter 6, see especially6.2.5.
both defective as vindications of classical logic. Second, the classical inferentialist could embrace both, or argue that one approach is superior. Rumfitt, himself a classicist of the bilateralist leaning, appears to go for the second alternative:
The close formal relationship between bilateral calculi and their multiple- conclusion cousins, however, should not blind us to what is, for present purposes, a crucial philosophical difference. (Rumfitt 2000, p. 810)
What is this difference? In order to make headway we need to look at the argu- ments against multiple-conclusion in the literature. Incidentally, the classically- minded Rumfitt shares his dislike for multiple-conclusion with revisionists such as Dummett and Tennant. Let us have a look at the different objections against the SET-SET framework raised in the literature. First out, Dummett:
Sequents with two or more sentences in the succedent, by contrast, have no straightforwardly intelligble meaning, explicable without re- course to any logical constant. Asserting A and asserting B is tan- tamount to asserting pA and Bq; so, although the sentences in the antecedent of a sequent are in a sense conjunctively connected, we can understand the significance of a sequent with more than one sentence in the antecedent without having to know the meaning of ‘and’. But, in a succedent comprising more than one sentence, the sentences are connected disjunctively; and it is not possible to grasp the sense of such a connection otherwise than by learning the meaning of the constant ‘or’. (Dummett1991, p. 187)
This is where Dummett’s reliance on an analysis of denial as an assertion of nega- tion takes center stage. Of course, Dummett is correct in claiming that con- junction and disjunction are asymmetric with respect to assertion. However, this does not show that conjunctive commas are conceptually kosher prior to their
object language counterparts, whereas disjunctive commas are not. Unlike the antecedent-side commas, the disjunctive (right-hand) commas ought not to be treated assertorically. Rather, we think of the rational (normative) constrain im- posed by a multiple-conclusion consequence relation as saying that one cannot
assert all premises and deny all conclusions simultaneously. With such a reading there is not necessarily a genuine disjunctive element: hΓ, ∆i is invalid iff we as- sert A1, and we assert A2, etc. for each Ai ∈ Γ, and we deny B1, and deny B2,
etc. for eachBi ∈∆. Interestingly, we can dualise and say, equivalently, that the
argument hΓ, ∆i is valid iff we deny A1, or we deny A2, etc., OR, we assert B1
or assert B2, etc. Thus, Dummett’s insight is turned up side down: We now have
a reading of multiple-conclusion sequents that does not presuppose conjunction but disjunction. Summed up, to the extent that it is a worry at all that there are meta-level conceptions of connectives, Dummett seems to have put too much emphasis on the role played by the interaction between assertion and conjunction. This counter to Dummett can be made more precise using a framwork offered in a series of papers by Restall (2005, 2008a,2008b, 2009b). Restall says:
We can think of the rules for the connectives as giving instructions on how to treat assertions and denials—at least with regard to whether or not these assertions and denials are out of bounds or not. (Restall 2009b, p. 5)
On Restall’s approach we consider structures deceivingly similar to arguments, pairs of collections of statements, written [X : Y]. Call such a structure a state. We say further that for a state [X : Y], X is the set of asserted statements and Y is the set of denied statements. A state [X : Y] such that X ∩ Y 6= ∅ is an
incoherent state (i.e., states where the very same statement is both asserted and denied). For an incoherent state [X : Y], we write X` Y, indicating precisely the
One can then contemplate adding natural constraints on coherence. Restall sug- gests, for example, that any state [A : A] is incoherent (reflexivity); if [X : Y] is coherent, and X0 ⊆ X and Y0 ⊆ Y, then [X” : Y”] is also coherent (weakening); and, if [X : Y] is coherent, then so is either [X, A : Y] or [X : Y, A] (transitivity). Of course, one might have conceptions of assertion and denial (and associated conceptions of coherence) which drops any or all of these constraints. The differ- ence, as Restall remarks, amounts roughly to the difference between substructural and non-substructural sequent calculus.57 That observation is further bolstered
by the option of introducing pragmatic constraints on logical connectives. One might for instance say that if [X : Y, A ∧B] is coherent, then either [X : Y, A] is coherent or [X : Y, B] is coherent, and so on. As with the structural constraints above, we recognise sequent rules by taking the contraposition of the pragmatic constraints.58
Importantly, such a reading of sequent calculus in terms of assertion and denial is different in some respects from the Smiley-Rumfitt style signed calculus discussed in Section3.2.3. Although signs and the state-interpretation can be translated into each other, the former is intended as a superstructure on traditional semantics, while the latter is an analysis of multiple-conclusion consequence. The deep point here is one of foundational semantics: Restall’s strategy offers a way in which inferentialism can be combined with truth-conditional semantics by way of prag- matic roles. Smiley and Rumfitt, on the other hand, have no such objective. For them, the traditional truth-conditional semantics is still the protagonist. Later, in Chapter 6, we will return to the connection between proof-theory and truth- conditions in a systematic fashion. In that connection, Section6.5 will discuss the potential of Restall’s strategy in more detail. Before that, however, we need to attend to some more general worries about multiple-conclusion, typically raised by those whole-heartedly opposed to the classical line.
57Analogously, Restall points out that the framework builds in contraction by working with sets rather than subsets. Again, this, as in sequent calculus, affects the logic of the structural commas of the states.
For a sample of what the single-conclusion aficionado has to say against multiple- conclusion more generally, we turn to an argument of Tennant’s:
[T]he classical logician has to treat of sequents of the formX :Y where the succedent Y may in general contain more than one sentence. In general, this smuggles in non-constructivity through the back door. For provable sequents are supposed to represent acceptable arguments. In normal practice, arguments take one from premisses to a single conclusion. There is no acceptable interpretation of the ‘validity’ of a sequent X : Q1, ..., Qn in terms of preservation of warrant to assert
when X contains only sentences involving no disjunctions. If one is told thatX : Q1, ...,Qn is ‘valid’ in the extended sense for a multiple-
conclusion arguments just in caseX :Q1 ∨... ∨Qnis valid in the usual
sense for single-conclusion arguments, the intuitionist can demand to know precisely which disjunctQi, then, proves to be derivable fromX.
(Tennant 1997, p. 320)
The latter point—that multiple-conclusion is unacceptable for the intuitionist— has been dealt with in a convincing manner in Steinberger (2008). First, multiple conclusion is in no way intrinsically classical: There are well-known intuitionistic multiple-succedent calculi. Two systems, m-G1ip and m-G3ip, are obtained by the standard classical sequent calculi,G1cpandG3cprespectively, by restricting the rules for →. E.g., in the former case (with explicit structural rules) we give the following rule:
Γ, A⇒B
Γ⇒A→B,∆ (R→)
Contrast the original rule which preserves the right-side context (see Appendix A.10).59 All other rules remain multiple-succedent in the ordinary sense.
But, as Steinberger goes on to point out, even if there were no multiple-conclusion alternative open for the intuitionist, the argument is faulty for more fundamental reasons. The intuitionist revisionist and the classicist were at a stalemate: Both alleged that theirs was the One True Logic, but neither had the resources to convince the interlocutor. The genius of Dummett and Prawitz’s novel take on the debate was to relocate the debate: PTS is the correct meaning-theoretic approach, and since intuitionistic logic, but not classical logic, abides by its constraints, the latter contains unwarranted principles. If the argument had been correct—which it is not—the intuitionist could have helped herself to the Disjunction Property with a clean conscience. But, Tennant’s argument is premature: In a discussion of whether or not classical logic does in fact abide by the PTS constraints, it is a blatantly illicit move to argue from one of the properties which the revisionary argument was precisely supposed to establish.
There is another charge against multiple-conclusion in the quotation by Tennant, one which is potentially harder to adjudicate.“In normal practice”, he says, “ar- guments take one from premisses to a single conclusion”. A first remark, albeit perhaps a bit ad hominem: It is strange to have an advocate of intuitionist re- visionism recourse to an argument from ‘normal practice’. After all, their case is founded on the tenet that such a practice might be dysfunctional (see Sec- tion 2.3.1), and, thus, susceptible for revision. Why then should the classicist be hostage to the arbitrary nature of ‘normal practice’ ?
Nevertheless, the argument from our ordinary inferential practice is dubious even notwithstanding the above point.60 Making good sense the claim is not easy, but
presumably the objection is based on an observation to the effect that arguments in normal practice may contain lists of premises (without conjunction), but not lists of conclusions (without disjunctions). Normally, when we reach a conclusion which is disjunctive we express this by using disjunction explicitly. The thought, I take it, is 60Restall (2005, p. 11-12) makes the case that there are indeed multiple-conclusion arguments in ordinary reasoning.
that for premises and conjunction the same is not the case. Without any empirical data, however, such an argument appears spurious. In fact, it seems perfectly legitimate to list conclusions disjunctively without explicit use of disjunction. As with premises, the context might determine how we should interpret the list. The fact that we do not normally proceed argumentatively in such a manner is of no consequence.
Surely, upon investigating normal inferential practice we will find, for instance, that most arguments are also single-premise, either because the premises are as- serted as one statement, or because some premises are tacit (in enthymematic form). Whatever the usual form of arguments might be, it does not standardly constrain the mathematics of the logician. In fact, along these line, a neat sugges- tion by Kosta Doˇsen (1989, p. 365) is helpful: Just as true premises sometimes are enthymematic, perhapsfalse conclusions are also enthymematic.61 Only upon failing to provide a single, conclusive conclusion is there any need for multiple- conclusion.
Finally, against multiple-conclusion, Rumfitt (2000, pp. 795-96) has raised an alternative objection. Citing Kneale as one of the culprits, Rumfitt says that “not only is it doubtful whether people actually give such arguments, it is also doubtful whether we can attain any intelligible conception of them”. For Rumfitt a SET-SET framework gives a collection of arguments whose constituent state- ments are not used but mentioned. However, since our inferences are structures where statements—both premises and conclusions—are used, the formalism should mirror this fact.
But why think that multiple-conclusion arguments merely mention the involved statements? Rumfitt takes multiple-conclusion to be the confused result of amet- alogical remark about logical consequence, namely, that “if certain propositions 61Doˇsen is another of the PTS proponents of sequent calculus. Two other interesting sources for multiple-conclusion systems are Kneale (1956)—see Section2.1.2—and, of course, Shoesmith & Smiley (1978).
are true, then certain other propositions cannot all be false” (ibid.). He insists, however, that proper inferences must be expressible in the form A1, ... An `
B. However, it is unclear why single-conclusion structures are not also just ‘met- alogical remarks’, and, regardless, the assertion/denial interpretation discussed above shows that we are by no means forced to read SET-SET arguments truth- conditionally.62