Ramble, Part 76: True/False/?
Nov. 24th, 2009 06:35 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowTo accept Intuitionism is, among other things, to accept that some meaningful mathematical statements are neither true nor false. From the standpoint of symbolic logic, this entails introducing additional truth values beside True and False; in other words, to contemplate "multi-valued logics".
The first person to devise a three-valued logic was the Polish mathematician Jan Lukasiewicz. (Lukasiewicz, by the way, also devised the logically simple "Polish notation" for representing mathematical expressions; computer buffs will be familiar with "reverse Polish notation", which is Lukasiewicz's idea, only backward.) His version introduced Unknown as a third truth-value, and specified how Unknown interacted with the various logical operators: NOT(Unknown) = Unknown; Unknown AND True = Unknown; and so on.
There are many decisions that have to be made in constructing a three-valued logic; Lukasiewicz's choices give one such, but there are a huge number of other possibilities. How can we restrict these? There are two major sources of constraints. On the purely formal side, we'd like to hang on to as many of the laws of Boolean logic as we can - familiar rules like P AND Q = Q AND P or NOT (P AND Q) = (NOT P) OR (NOT Q); the Law of Excluded Middle and things equivalent to it have to be abandoned, of course, but as many as possible of the rest would be nice. From a more practical viewpoint, the choices should "make sense"; if you have two statements P and Q, one of which is Unknown and the other True, surely the combination P AND Q should be Unknown, while if one is Unknown and the other False, then P AND Q must be False. This cuts down the number of possibilities, but not enough.
But why should we stop at three? If there are more than two truth-values, why can't there be four? Ten? Infinitely many? Uncountably many? Strange as it may seem, there is good reason to contemplate all of these possibilities, and the collection of conceivable logics is unmanageably large, at least in this way.
It was Arend Heyting, a disciple of the Intuitionist L. E. J. Brouwer, who proposed a set of logical axioms inclusive enough to allow as many truth-values as one might desire, but restrictive enough that reasonably powerful statements could be made about all of the resulting logics. (Yes, "logics", strange though that word might sound.) The logics that satisfy these axioms are called "Heyting logics", and we'll run into them again later. (Heyting's formulation of these axioms led to a breach with Brouwer, whose version of Intuitionism was highly skeptical of formalizing logic to this extent. Most later Intuitionists, though, are - as far as I can determine - reasonably happy with Heyting's work.)
But enough, for now, of Intuitionism. We'll return to the more traditional philosophies of mathematics in the next Ramble - which, hopefully, will appear in less than two months!
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The first person to devise a three-valued logic was the Polish mathematician Jan Lukasiewicz. (Lukasiewicz, by the way, also devised the logically simple "Polish notation" for representing mathematical expressions; computer buffs will be familiar with "reverse Polish notation", which is Lukasiewicz's idea, only backward.) His version introduced Unknown as a third truth-value, and specified how Unknown interacted with the various logical operators: NOT(Unknown) = Unknown; Unknown AND True = Unknown; and so on.
There are many decisions that have to be made in constructing a three-valued logic; Lukasiewicz's choices give one such, but there are a huge number of other possibilities. How can we restrict these? There are two major sources of constraints. On the purely formal side, we'd like to hang on to as many of the laws of Boolean logic as we can - familiar rules like P AND Q = Q AND P or NOT (P AND Q) = (NOT P) OR (NOT Q); the Law of Excluded Middle and things equivalent to it have to be abandoned, of course, but as many as possible of the rest would be nice. From a more practical viewpoint, the choices should "make sense"; if you have two statements P and Q, one of which is Unknown and the other True, surely the combination P AND Q should be Unknown, while if one is Unknown and the other False, then P AND Q must be False. This cuts down the number of possibilities, but not enough.
But why should we stop at three? If there are more than two truth-values, why can't there be four? Ten? Infinitely many? Uncountably many? Strange as it may seem, there is good reason to contemplate all of these possibilities, and the collection of conceivable logics is unmanageably large, at least in this way.
It was Arend Heyting, a disciple of the Intuitionist L. E. J. Brouwer, who proposed a set of logical axioms inclusive enough to allow as many truth-values as one might desire, but restrictive enough that reasonably powerful statements could be made about all of the resulting logics. (Yes, "logics", strange though that word might sound.) The logics that satisfy these axioms are called "Heyting logics", and we'll run into them again later. (Heyting's formulation of these axioms led to a breach with Brouwer, whose version of Intuitionism was highly skeptical of formalizing logic to this extent. Most later Intuitionists, though, are - as far as I can determine - reasonably happy with Heyting's work.)
But enough, for now, of Intuitionism. We'll return to the more traditional philosophies of mathematics in the next Ramble - which, hopefully, will appear in less than two months!
Previous
Ramble Contents
