Partial logic

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It is a desirable feature in developing formal logics that the rules governing the logic fully determine the logic in a fairly precise way. Loosely speaking, this means that everything you might want to do with any symbol or connective in the system can be done. So, for example, in a natural deduction system, you want introduction and elimination rules for every connective and quantifier.

More precisely, the aim of defining a logic is normally that the logic be complete in a truth functional sense. That is, given a valuation that assigns all components in a language an appropriate value, then every sentence/well-formed-formula (up to certain constraints like Goedel’s Theorem) has a truth value.

A partially determined logic is simply a logic that doesn’t satisfy this requirement in either of the above definitions. A very simple example would be a logic which includes an Elimination rule for AND, but no introduction rule. Thus, if we have proven A&B, we can derive A or derive B; but if we know C and we know D we cannot conclude C&D. While this is a somewhat trivial example, it illustrates the idea.

I have been playing with the idea on this blog that natural languages may in fact determine partial logics of this sort, rather than the fully determined logics we are more familiar with. One motivation for this is that we get natural agreement on a set of logical rules in natural langauges (e.g. Modus Ponens, AND rules) but there is less agreement on other rules. Instead of looking for the ‘real’ rule that governs natural languages, it is maybe the case that natural languages do not in fact fully determine the rules of a logic.

Another motivation is that human reasoning is often driven by pragmatic, as well as logical, considerations. For example, it is plausible to argue that human reasoning relies on a principle that each step should be more informative that the previous. If someone argues in a way that appears to reduce the amount of information known, or does not increase it, we do not consider it to be a useful argument.

While this is a plausible pragmatic consideration, some valid logical rules do not adhere to this principle. The most obvious example is OR-introduction in most natural deduction systems. The following valid argument does not add information, but rather reduces it:

1. It rained this morning.

2. It rained this morning or it snowed yesterday.

OR-Introduction is therefore not ‘pragmatically valid’ in this case anyway even if it is logically valid. Any version of OR introduction is ‘pragmatically invalid’ and therefore a pragmatically valid logic would need to be partial in the sense discussed here.

Partial logics can have very interesting properties and I hope to post a draft paper on some of these shortly. Philosophically though, the present an interesting way of resolving some of the issues on the interface of natural language and logic.