That Logic Blog

January 04, 2006

Basic Logic

One of the common features of many philosophies of mathematics such as platonism, formalism, fictionalism and so on, is that they view mathematics as a constant, static entity. From this perspective, it makes sense claim something along the lines of "X is the correct foundation for mathematics", where X may be ZF set theory, category theory, constructive type theory or what have you.

However, a subject such as mathematics is constantly changing. So, why not allow this natural evolutionary process into your foundational doctrine? This is the view argued in a very entertaining fashion in:

Steps toward a dynamic constructivism; Giovanni Sambin.

Now, if one accepts that the body of mathematics is constantly changing and, a fortiori, that a foundational theory ought to be malleable, then the concept of mathematical truth is no longer static. This pluralist attitude towards mathematical truth may be refined to a pluralist attitude towards logic. Indeed, Sambin argues:

It is difficult to accept a plurality of logics and still believes that logic deals with a static, a priori, and hence univocal notion of truth.

Given that one accepts a pluralist position, one may seek a way in which to systematically organise various logical theories. To this end, Sambin seeks a common core for a multitude of logics, arriving at a system he calls Basic Logic, which is "the minimal core, with no structural rules". Thus, Sambin adopts a view common amongst substructural logicians of various pursuasions, though he argues via a different route. In order to understand a bit more about what Basic Logic is all about, one must turn to:

Basic logic: reflection, symmetry, visibility; Giovanni Sambin, Giulia Battilotti, Claudia Faggian.

Skipping out the ideological motivations, which are explained in the paper and implemented via a notion of "reflection", one may describe Basic Logic as being almost the logic that results from dropping weakening and contraction from intuitionistic logic. There is one additional ingredient and one technical modification. The extra ingredient is a coimplication, which is the residual or adjoint of multiplicative disjunction in exactly the same way as standard implication is the residual or adjoint of multiplicative conjunction. The technical modification is the one suppresses contexts.

While several logical systems can be seen as extensions of Basic Logic, one ought not see Basic Logic as being the ultimate core of logic and, indeed, Sambin does not claim it to be. For instance, it is an overarching assumption throughout the development that both conjunction and disjunction are commutative (so the statement that it is free of structural rules is not quite true). This is a rather strong requirement and immediately discards many interesting and useful noncommutative logics arising in linguistics and computer science.

Librorum Helluo said...

Although I am neither a logic nor a philosophy expert, I have noticed how many philosophers treat math as if it were a fixed subject, relying on it to support their theories and to compare other theories against. Although this might not be news to a mathematician, I am sure that it is news to most laymen/students. Thank you for pointing this out and for providing valuable insight.

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