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It is simply not true that multi-valued logics collapse into two-valued logics. While it is true that you can assign indeterminate statements {true,false} and

It is simply not true that multi-valued logics collapse into two-valued logics. While it is true that you can assign indeterminate statements {true,false} and call it a third value, logic is more than assignments of truth-values to statements, it is also rules of inference for that formal system. The rules of inference are the driving force behind paraconsistent logics and the rules of inference are very different. One motivation is the explosivity of contradictions in a formal system. Consider a paradox of humility: I believe of each of my individual beliefs that it is true, I believe all of the logical consequences of my beliefs because I am rational, but I can't rule out the possibility that two of my beliefs (or their consequences) are contradictory. Treating beliefs as statements, under classical logic, any contradictory beliefs would imply that you believe (or would assent to) anything and everything is true (and false). Here is how: 1. P & ~P (premise) ... 2. P (1 simplification) 3. ~P (1 simplification) 4. P or Anything (2, addition [i.e. the self-evident principal that if P is true, P or anything else is also true]) 5. Anything (4, 3 and the principle that if a or b is true and a is not true, then b must be true). In other words, contradictory beliefs imply anything and everything is true (and false). This is an odd result especially since anyone with an ounce of humility will admit that there is a reasonably high chance that there is a contradiction in their beliefs or in the logical consequences of their beliefs. Now for a 'more practical' example. Imagine you have a key-value store that represents people and properties. Suppose your database says that Jane is 34 years old and later it says Jane is 35. If you ask the database for all of the logical consequences of this people/property store, under classical logic you will get every statement and its negation. Weird huh?

Strilanc 12y ago
They don't collapse into two-valued logics, but they can be encoded into two-valued logics in the same way that numbers can be encoded into sets. The simplest way to see this is to think of a computer. Its operation can be described by a function in two-valued logic, taking the bits making up the state of the CPU and memory as input then returning the next state. And you can write programs that do multi-value'd logic... so within the two-value system you can make a little multi-value system. In terms of the database analogy: you have to translate the data and the queries. So you can't query "what is their age", you can only query things like "the current time is X, what is their age".
- dpierce9 12y ago
  I agree that you can model a multi-valued logic system on a universal Turing machine but you are wrong to think that doing this is the same as mapping to classical logic. TMs do not intrinsically follow the rules of classical logic, rather, they obey classical logic (when they do) because they have instructions written into their head. Similarly, the marks on the tape do not intrinsically represent two-valued logic even if there are only two kinds of marks on the tape. This has nothing to do with succinctness or making Turing Machines better and everything to do with 'what do the marks represent?' and 'what is the best way to reason with the marks?' which are definitely interesting questions in my book.
eli_gottlieb 12y ago
>In other words, contradictory beliefs imply anything and everything is true (and false). This is an odd result especially since anyone with an ounce of humility will admit that there is a reasonably high chance that there is a contradiction in their beliefs or in the logical consequences of their beliefs. Yeah, but we deal with this very simply: by using probability (conceived as a real-valued logic) instead of boolean logic, at which point any contradiction in our axioms gets us an "exception signal" of some probability somewhere going to 0.0 instead of an equal belief in all possible propositions. Or we can use constructive logic instead of classical, in which case we'll be able to actually find a contradiction simply by looking for assumptions that lead to a proof for the proposition False.
- dpierce9 12y ago
  There are definitely other ways to handle this same problem, I was just pointing out one counterintuitive conclusion of classical inference rules as one motivation for non-classical logics. And, since both of your examples also involve non-classical logics, I think we generally agree that there is motivation here (setting aside the viability of your solutions)

Strilanc 12y ago

They don't collapse into two-valued logics, but they can be encoded into two-valued logics in the same way that numbers can be encoded into sets. The simplest way to see this is to think of a computer. Its operation can be described by a function in two-valued logic, taking the bits making up the state of the CPU and memory as input then returning the next state. And you can write programs that do multi-value'd logic... so within the two-value system you can make a little multi-value system. In terms of the database analogy: you have to translate the data and the queries. So you can't query "what is their age", you can only query things like "the current time is X, what is their age".

dpierce9 12y ago
I agree that you can model a multi-valued logic system on a universal Turing machine but you are wrong to think that doing this is the same as mapping to classical logic. TMs do not intrinsically follow the rules of classical logic, rather, they obey classical logic (when they do) because they have instructions written into their head. Similarly, the marks on the tape do not intrinsically represent two-valued logic even if there are only two kinds of marks on the tape. This has nothing to do with succinctness or making Turing Machines better and everything to do with 'what do the marks represent?' and 'what is the best way to reason with the marks?' which are definitely interesting questions in my book.

eli_gottlieb 12y ago

>In other words, contradictory beliefs imply anything and everything is true (and false). This is an odd result especially since anyone with an ounce of humility will admit that there is a reasonably high chance that there is a contradiction in their beliefs or in the logical consequences of their beliefs. Yeah, but we deal with this very simply: by using probability (conceived as a real-valued logic) instead of boolean logic, at which point any contradiction in our axioms gets us an "exception signal" of some probability somewhere going to 0.0 instead of an equal belief in all possible propositions. Or we can use constructive logic instead of classical, in which case we'll be able to actually find a contradiction simply by looking for assumptions that lead to a proof for the proposition False.

dpierce9 12y ago
There are definitely other ways to handle this same problem, I was just pointing out one counterintuitive conclusion of classical inference rules as one motivation for non-classical logics. And, since both of your examples also involve non-classical logics, I think we generally agree that there is motivation here (setting aside the viability of your solutions)