Two weeks ago, I wrote a post proposing a dialogical perspective on structural rules. In fact, at that point I offered an analysis of only one structural rule, namely left-weakening, and promised that I would come back for more. In this post, I will discuss contraction and exchange (for both, I again restrict myself to the left cases). (I will assume that readers are familiar with the basic principles of my dialogical approach to deductive proofs, as recapped in my previous post on structural rules.)
Contraction, in particular, is very significant, given the recent popularity of restriction on contraction as a way to block the derivation of paradoxes such as the Liar and Curry. What does contraction mean, generally speaking? Contraction is the rule according to which two or more copies of a given formula in a sequent can be collapsed into each other (contracted); in other words, the idea is that the number of copies should not matter for the derivation of the conclusion:
A, A => C
A => C
(If two copies of A give you C, then one copy of A will give you C just the same.) Logics that reject contraction are based on the idea that the number of copies does matter; in the case of linear logic, the most notable non-contractive logic, formulas are seen as resources, which are no longer available once they are used (not that particular copy, in any case), and are sensitive to the number of copies available. (The standard example for resource sensitivity in linear logic is money required to buy a pack of cigarettes, which I think reveals something about the time and place in which the system was developed… Two 5-franc notes do not buy you the same amount of cigarettes as one 5-franc note!) In other words, linear logic has a plausible story to tell on why, given the purpose of the logic (to keep track of resources), the number of copies does matter.
Prima facie, in a dialogical setting, once a proposition is stated and granted, it becomes part of the public domain, as it were, and may be used as many times as necessary – at least by those who explicitly committed to it. So contraction may appear to be unproblematic in this setting. What a proposition stated or granted does is to produce a discursive commitment for the speaker in question, but it also licenses her to refer back to this commitment whenever necessary – in other words, it also creates an entitlement that can be ‘used’ as many time as one wishes. (I’m deliberately using Brandomian terminology here.)
However, one may well conceive of particular kinds of dialogical interaction where, every time a premise is required so as to license a conclusion, a fresh copy of it is required. We would need a story on why, once having granted a particular premise to proponent, opponent might then refuse to grant the same premise when it is asked again by proponent; if opponent will always have to grant premises he has granted before, in practice there is no need to go through the procedure of actually generating the new copies (new commitments).
One reason why discursive commitments may have to be modified during the interaction is if the reasons one had to commit to a statement at a given point no longer hold (say, due to changes in the world, or incoming new information); in that case, the possibility of retracting a commitment may seem plausible after all, in particular if discursive commitment is time-relative. But notice that this is a very different phenomenon from the idea of formulas being used as resources in linear logic; a given statement may no longer be ‘available’ even if it hasn’t been used yet, but simply because there are good reasons to revise one’s prior discursive commitments.
In a similar vein, my friend and former colleague Dora Achourioti has been developing a (thus far unpublished, I think) account of the truth operator where its function is precisely to turn a given statement into something that can be used as many times as one wishes; it becomes a limitless resource (she explicitly adopts a multi-agent perspective, and uses notions from linear logic to formalize this insight). So the presupposition is that this does not hold for other, ‘regular’ occurrences of statements not affected by a truth operator, and thus that contraction does not hold unrestrictedly.
So I conclude that, while contraction is prima facie a very plausible principle in a dialogical setting, there may be purely dialogical reasons to restrict contraction, but which are different from linear logic’s rationale for contraction restriction.
What about exchange? It is not a structural rule that is much discussed in the paradoxes literature, but it is interesting in its own right for different reasons. While contraction entails that the number of copies does not matter, exchange entails that the order in which formulas are presented does not matter.
A, B => C
B, A => C
In a purely model-theoretic conception of (logical) consequence, it is indeed the case that order does not matter, as the sequence (A, B) has the same models as the sequence (B, A). In a dialogical setting, however, it is not at all obvious that order should not matter. This is because every new discursive commitment – every new premise granted by opponent – creates an update in his commitments; naturally, dialogues are intrinsically dynamic processes (and here you see that I am a real Amsterdam child!). Indeed, depending on the specific rules for different kinds of dialogical interaction, a premise A may be granted if it is proposed at a given stage of the debate, but rejected if it is proposed at a different stage (not in the very same interaction, but in an alternative interaction involving the same statements).
For example, the regimented kind of disputations known as obligationes (very popular among Latin medieval logicians) is inherently dynamic. If the starting point of the disputation is the proposition ‘Every human is running’ (which should be accepted if it is possible, even if it is not true, given the rules of the game), and then ‘You are a human’ is proposed, the player (in this case called ‘respondent’) must grant it as irrelevant for the starting point (it is not entailed by it or incompatible with it) and true. Then, if ‘You are running’ is proposed, the player must now accept it, as it follows from her two previous commitments, even though it is false (presumably, she is not running while disputing!). If however, given the same starting point ‘Every human is running’, ‘You are running’ is proposed first, respondent should deny it as irrelevant and false. Then, if ‘You are a human’ is proposed, it should be denied, even though it is true, because this follows from accepting ‘Every human is running’ and denying ‘You are running’. So different responses are required to the same statements depending on their order of presentation.
(However, in other dialogical situations, the order of presentation of premises may not matter; in Aristotle's Topics, for example, the recommendation is that questioner gets answerer to commit to all the premises he will need before he starts drawing conclusions (Book VIII, chapter 1).)
So I conclude that exchange is not a plausible principle from the point of view of the dialogical conception of proofs if we take into account the inherently dynamic nature of dialogues. In dialogues, the order of presentation of premises may well matter.
(I had intended to talk about cut too, but this post has again reached the reasonable length for the genre. Cut is complicated because it is related to the fundamental property of transitivity, so it cannot be discussed in haste. Maybe in another post?)