Monday, December 03, 2007

The Humpty-Dumpty Game of Logic

I have previously noted that it is possible, with a few suppositions, to do propositional logic with Lewis Carroll's Game of Logic, which uses a board consisting of a literal diagram. Once propositional connectives are defined, each sentence in propositional logic can have a corresponding configuration of the board. The Game of Logic understood in this way is truth-functional; all the truth-functional relations and equivalences between propositions are expressible in the relations between configurations of the board, because every configuration of the board is logically isomorphic to a truth table. Thus this is the biliteral configuration of (P & Q):









 x
xx

This is the biliteral configuration of (P v Q):









  
 x

And so forth; we can see immediately, for instance, that (P v Q) is weaker than (P & Q), and how.

But this is the Game of Logic, and not only can we have configurations of the board, we can move from configuration to configuration. There are three types of operations that allow us to move from configuration to configuration: information-adding, information-preserving, and information-subtracting. We can, for instance, use an information-subtracting operation to move from:









 
xx

to the (P v Q) configuration noted above. When you look closely at these operations, you begin to realize that they are rather familiar; and this is because they are applications of inference rules. When we realize this we find some obvious patterns. Assumption of premises are information-adding operations; elimination rules are all information-subtracting. The standard introduction rules are all information-preserving, with the exception of addition, which we usually take to be disjunction introduction. If we used Tom's suggested candidate for disjunction introduction, this anomaly would be eliminated, and the elegance of the Game would be greatly increased.

What has happened, though, is that the rules of inference, on the one hand, and the truth-functional relations between sentences in the system, on the other hand, have come apart; while the Game is truth-functional, the rules and the sentences are represented by different aspects of it. The truth-functional relations between sentences are found in the relations between possible configurations of the board; the rules of inference are found in the movements from configuration to configuration. On seeing this, we can start playing with the rules of inference. The possible configurations of the board remain what they are; the truth-functional definitions do not change. But movement from configuration to configuration can be modified. The roads remain the same; only the traffic is diverted.

The same can be said of the propositional logic that the Game is here modeling; implication, for instance, is not inference, or vice versa, as the Tortoise taught us, so the fact that a sentence is in a certain truth-functional relation to other sentences doesn't make it a rule of inference. As a matter of fact, for most purposes we take derivable sentences in propositional logic to correspond to rules of inference. The reasons for doing this are rather obvious: it allows us to have derived rules of inference, which can immensely simplify things. But our doing this is not intrinsic to the truth-functional relations among sentences, but the result of a rule-forming rule that allows us to convert sentences into rules of inference. This is so useful that we sometimes don't make much of a distinction. Thus, for instance, we sometimes use 'derivation' to indicate a truth-functional relation among sentences, in which case sentences no one has ever or will ever derive are 'derived', and we sometimes use it to indicate actual derivation. In other words, we sometimes mean a truth-functional relation between configurations, and we sometimes mean actual operations on one configuration to make it other configurations.

Thus the truth-functional relations among configurations do not establish our rules of inference; they merely indicate that there is no truth-functional impediment to them, when we are interested in truth functions (which we virtually always are). That A implies (A v B) may be in the relevant truth-functional relation to other sentences in the system, but that does not follow that we have to reason, "A, therefore A or B." We can simply deny its usefulness as a rule of inference, derived or otherwise, if we have reason to do so; we can accept it but only under restrictions (e.g., that it not be immediately followed by certain sorts of eliminations); we can accept it wholeheartedly. There is no reason, further, why we can't (as Russell has for the Principia, if I recall correctly) have a rule of inference that has no exactly corresponding sentence in the system (although perhaps a particular instance of it, e.g., modus ponens, might); likewise, there is no reason why we have to accept as a rule of inference each one that corresponds to an implication in the sentence-system. There is strictly speaking no truth-functional reason why we have to have only inference rules that pay attention to truth functions at all. Trying to represent tonk in the Game makes for strange and rather pointless play, but play you can (since tonk is simply defined by rules of inference, it's not necessary to change the way the board can be configured in order to introduce it, but only how you move from configuration to configuration).* We can do these things systematically and consistently; we aren't confined merely to allowing or disallowing rules of inference for particular proofs unless we want to be so confined. And through all of this the truth-functional definitions of the connectives remain exactly what they were, the whole truth-functional system remains exactly what it was. All that changes is the rules of inference, and these can be whatever we have reason to make them. It's a Humpty-Dumpty world.

Of course, from this it doesn't follow that we have good reasons for making them this way rather than that. There are very good reasons, reasons that fit very many circumstances, for making the assumptions we normally make.

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* In a sense Prior's tonk argument at least suggests the other side of the point I'm making here. We might say that regard for truth-functional relations without regard for rules of inference is useless; while regard for rules of inference without regard for truth-functional relations is pointless. This does not require a conflation; relations among sentences are not inferences, and inferences are not relations among sentences. Of course, we may deliberately allow movement in one or both directions. But you need both the possible configurations of the board and ways to move among them; and they are not the same. (In actuality, Prior's tonk argument goes farther than this and holds that even these together are not perfectly adequate for telling us what we are doing in logic, and I would agree with Prior; but that is not relevant here.)

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