3.2 Classical Logic Revamped
3.2.2 Multiple-Conclusion Natural Deduction
We return to the system NC, and another proposal for how to circumvent Dum- mett and Prawitz’s attack on classical logic. We saw in Section 2.3.2 that Read argues that classical logic could regain total harmony (conservativeness) by going multiple-conclusion in natural deduction. In a sense, this is adapting the sequent calculus profile of classical logic to a natural deduction environment.27 Since the positive fragment is sufficient to prove Peirce’s Law in sequent calculus, the hope was that it would be the case in multiple-conclusion natural deduction as well. And, indeed, Peirce’s Law can be proved in the positive fragment of NC.
We might continue by asking about intrinsic harmony in the context of NC. In particular, we might return to the question of the legitimacy of classical negation in a setting where multiple-conclusion is allowed. It turns out that we can get much more out of this analogy than total harmony. Recall that we were able to prove Peirce’s Law in the positive fragment of NC because of the following multiple-conclusion rules (together with structural rules):
Example 3.1. Multiple conclusion implication:
[A] .. .. Γ, B Γ, A→B I → Γ, A ∆, A→B Γ,∆, B E →
Applying a similar modification of the negation-rules I¬ and E¬ we get the fol- lowing result:28
Example 3.2. Multiple conclusion negation
[A]u .. .. ∆,⊥ ∆,¬A (I¬N C)(u) Γ, A ∆,¬A Γ,∆ (E¬N C)
27Compare natural deductionin sequent calculus style. See Dummett (1977). 28Similar modifications apply to the other rules. See AppendixA.9.
What about normalisation? Even though Boričić (1985) reports that normalisa- tion holds for the system NC, we are not given a direct proof of the theorem. Neither he nor Read provides information about how conversion steps are sup- posed to work with multiple-conclusion rules; rather, they rely on an indirect demonstration through a translation between NC and sequent calculus, and an accompanying preservation result between normalisation and cut elimination due to Zucker (1974).
The main challenge with multiple-conclusion rules is that there are, strictly speak- ing, two sorts of commas around: Conjunctive commas—corresponding to sequent- style left-side commas—anddisjunctive commas—corresponding to right-side com- mas. In a sequent, such commas have a location (right or left of the sequent ar- row) to distinguish between them, whereas in natural deduction, this is no longer the case. Of course, in Nip, the problem never arises since the the calculus is single-conclusion (and commas are always unambiguously conjunctive).29 But, by
acknowledging that PTS for classical logic is best dealt with in NC, the equivo- cation is introduced.
So far we have mostly surpressed conjunctive commas (and open assumptions in the rules), but the issue will resurface in Section 4.5.1.30 Until then, we ignore
the complication and give rules where commas are always disjunctive. We then get a conversion step for negation which is reminiscent of the one for intuitionistic negation, albeit it involves the added complexity of multiple-conclusions:
Π2 Γ, A [A]u Π1 ∆,⊥ ∆,¬A (u) Γ,∆,⊥ Π2 Γ, A Π1 Γ,∆,⊥
Note that in the converted derivation Π2 ends in a disjunctive pΓ, Aq, so Π1 in
turn will preserve Γ as a disjunctive context. This is a way of saying that either 29That is, unless the system is in sequent style (see footnote27).
30See also Section4.2.2where we discuss an example of connectives that are distinguished by the presence of conjunctive contexts in the inference rules (specifically, in quantum logic).
some member of Γ obtains, or A obtains. If the latter, either ⊥ or some member of ∆ obtains (according to Π1). The result is the same end-line as in the original
non-normal derivation.31
What is more, we can use the new rules to prove LEM without the use of any of the old characteristically classical rules:
[A]1 A∨ ¬A A∨ ¬A,⊥ A∨ ¬A,¬A (1) A∨ ¬A, A∨ ¬A A∨ ¬A
Alternatively, we can have ⊥as primitive and prove LEM without rules for nega- tion: [A]1 A,⊥ (K) A → ⊥, A (I→)(1) A∨(A→ ⊥), A ( I∨) A∨(A→ ⊥), A∨(A→ ⊥) ( I∨) A∨(A → ⊥) W
Read makes the pertinent remark that under these circumstances, i.e., with nega- tion defined, it is the →-rules that are doing the work (together with weakening). As a consequence, both intuitionistic and classical negation can be defined with
⊥, →, but the → rules will distinguish between the systems. In fact, the above derivation does not even apply the ⊥-rule! Nevertheless, the rule is required for the derivation of DNE.32
It is worth mentioning that NC upstages other normalisable natural deduction systems for classical logic because it also gives neater corollaries. In particular, we get a Subformula Property unnegotiated by E⊥C, and thus also the Separation
31Multiple conclusion in natural deduction will receive more detailed attention later in Section
4.5.1.
Property. This is not unexpected given the close relation between NC and the sequent calculus G1c, where both properties also hold as corollaries of the cut elimination theorem.33 Furthermore, unlike Prawitz’s result, the normalisation theorem for NC is not restricted to the ∨, ∃-free fragment of the language. Again, these modifications of classical logic render it unclear at best whether Dummett and Prawitz’s objections still have any bite. In NC, classical logic has both total and intrinsic harmony. Even more, with full Subformula and Separation Properties, the situation is indistinguishable from that of intuitionistic logic. If the revisionist still wants to maintain that there is a reason to reject classical logic, there appear to be only two strategies available. Either new proof-theoretic constraints must be forthcoming that will reinstate the relevant difference between the logics, or—and this is the option preferred by Dummett—some argument must be given why multiple-conclusion systems are not in good standing. We will have ample opportunity to explore the debate about the second approach below (Section 3.3).