Sunday, June 12, 2011

Rough Thoughts about Boole and (x)(1-x)(1+x)=0

Boole bases his logic on the equation x2=x (of course, an algebra in which this is true is famously an algebra of 1's and 0's). From this he derives the equation (x)(1-x)=0, which represents in his system the law of noncontradiction: the left-hand side is interpreted as a combining of a term or claim (x) and its negation (1-x) or, alternatively, in terms of true and false. Of course you don't have to treat x2=x as the 'fundamental law of thought' and derive (x)(1-x)=0 from it; you can do the reverse, and take (x)(1-x)=0 as fundamental, getting x2=x from it.

But if you can take x2=x as your starting point, why can you not equally take some other starting point, such as x3=x? Boole considers this at one point. He notes that x3=x is equivalent to (x)(1-x)(1+x)=0, and rejects this as a viable way to go because (1+x) has no interpretation.

But perhaps he was too hasty here. You can associate interpretations with the equations in the following way (A):

x : x is in the universe of discourse
1-x : x is not in the universe of discourse (or more accurately, non-x is in the universe of discourse)

Boole usually takes 1 to symbolize the universe of discourse; If we take this, is there any room for an interpretation of (1+x)? Perhaps something like this would work:

1+x : x is outside the universe of discourse (i.e., x is in addition to everything in the universe of discourse)

Given the way Boole understands the concept of a 'universe of discourse' I'm not sure he'd think this makes much sense. But it does seem useful in logic to compare universes of discourse with each other sometimes, in which case this interpretation might be useful.

Alternatively, if we think in terms of propositions rather than terms, we could think of it in this way (B)

x : x is true
1-x : x is false

In which case it would seem reasonable to think of (1+x) as:

1+x : x is neither true nor false

This would make an algebra in which x3=x a three-valued logic. I'm not really sure how close it is to any standard three-valued logic, though. In a x3=x system, x2=x still holds, as does (x)(1-x)=0, but so does (x)(1+x)=0 and (1-x)(1+x)=0. Both of these make sense under interpretation of (B) (they essentially treat as contradictory the claims 'x is true and also neither true nor false' and 'x is false and also neither true nor false'); and under (A) if 'outside the universe of discourse' and 'inside the universe of discourse' are treated as mutually exclusive and if we don't interpret (1-x) to be synonymous with 'not in the universe of discourse'. Perhaps we can generalize both interpretations to treat (1+x) generally as stating that x is irrelevant.

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