Unfortunately, although the new 1⁄2 truth value can resolve some paradoxes, it introduces new paradoxes. “This sentence is either false or 1/2” cannot be consistently assigned any of the three truth values.
Under some plausible assumptions, Lukaziewicz shows that we can resolve all such paradoxes by taking our truth values from the interval [0,1]...
Well, a straightforward continuation of paradox would be “This sentence has truth value in [0;1)”; is it excluded by “plausible assumptions” or overlooked?
Excluded. Truth-functions are required to be continuous, so a predicate that’s true of things in the interval [0,1) must also be true at 1. (Lukaziewicz does not assume continuity, but rather, proves it from other assumptions. In fact, Lukaziewicz is much more restrictive; however, we can safely add any continuous functions we like.)
One justification of this is that it’s simply the price you have to pay for consistency; you (provably) can’t have all the nice properties you might expect. Requiring continuity allows consistent fixed-points to exist.
Of course, this might not be very satisfying, particularly as an argument in favor of Lukaziewicz over other alternatives. How can we justify the exclusion of [0,1) when we seem to be able to refer to it?
As I mentioned earlier, we can think of truth as a vague term, with the fuzzy values representing an ordering of truthiness. Therefore, there should be no way to refer to “absolute truth”.
We have to think of assigning precise numbers to the vague values as merely a way to model this phenomenon. (It’s up to you to decide whether this is just a bit of linguistic slight-of-hand or whether it constitutes a viable position...)
When we try to refer to “absolute truth” we can create a function which outputs 1 on input 1, but which declines sharply as we move away from 1.[1] This is how the model reflects the fact that we can’t refer to absolute truth. We can map 1 to 1 (make a truth-function which is absolutely true only of absolute truth), however, such a function must also be almost-absolutely-true in some small neighborhood around 1. This reflects the idea that we can’t completely distinguish absolute truth from its close neighborhood.
Similarly, when we negate this function, it “represents” [0,1) in the sense that it is only 0 (only ‘absolutely false’) for the value 1, and maps [0,1) to positive truth-values which can be mostly 1, but which must decline in the neighborhood of 1.
And yes, this setup can get us into some trouble when we try to use quantifiers. If “forall” is understood as taking the min, we can construct discontinuous functions as the limit of continuous functions. Hartry Field proposes a fix, but it is rather complex.
Note that some relevant authors in the literature use 0 for true and 1 for false, but I am using 1 for true and 0 for false, as this seems vastly more intuitive.
I’m confused about how continuity poses a problem for “This sentence has truth value in [0,1)” without also posing an equal problem for “this sentence is false”, which was used as the original motivating example.
I’d intuitively expect “this sentence is false” == “this sentence has truth value 0″ == “this sentence does not have a truth value in (0,1]”
“X is false” has to be modeled as something that is value 1 if and only if X is value 0, but continuously decreases in value as X continuously increases in value. The simplest formula is value(X is false) = 1-value(X). However, we can made “sharper” formulas which diminish in value more rapidly as X increases in value. Hartry Field constructs a hierarchy of such predicates which he calls “definitely false”, “definitely definitely false”, etc.
Proof systems for the logic should have the property that sentences are derivable only when they have value 1; so “X is false” or “X is definitely false” etc all share the property that they’re only derivable when X has value zero.
Understood. Does that formulation include most useful sentences?
For instance, “there exists a sentence which is more true than this one” must be excluded as equivalent to “this statement’s truth value is strictly less than 1″, but the extent of such exclusion is not clear to me at first skim.
Well, a straightforward continuation of paradox would be “This sentence has truth value in [0;1)”; is it excluded by “plausible assumptions” or overlooked?
Excluded. Truth-functions are required to be continuous, so a predicate that’s true of things in the interval [0,1) must also be true at 1. (Lukaziewicz does not assume continuity, but rather, proves it from other assumptions. In fact, Lukaziewicz is much more restrictive; however, we can safely add any continuous functions we like.)
One justification of this is that it’s simply the price you have to pay for consistency; you (provably) can’t have all the nice properties you might expect. Requiring continuity allows consistent fixed-points to exist.
Of course, this might not be very satisfying, particularly as an argument in favor of Lukaziewicz over other alternatives. How can we justify the exclusion of [0,1) when we seem to be able to refer to it?
As I mentioned earlier, we can think of truth as a vague term, with the fuzzy values representing an ordering of truthiness. Therefore, there should be no way to refer to “absolute truth”.
We have to think of assigning precise numbers to the vague values as merely a way to model this phenomenon. (It’s up to you to decide whether this is just a bit of linguistic slight-of-hand or whether it constitutes a viable position...)
When we try to refer to “absolute truth” we can create a function which outputs 1 on input 1, but which declines sharply as we move away from 1.[1] This is how the model reflects the fact that we can’t refer to absolute truth. We can map 1 to 1 (make a truth-function which is absolutely true only of absolute truth), however, such a function must also be almost-absolutely-true in some small neighborhood around 1. This reflects the idea that we can’t completely distinguish absolute truth from its close neighborhood.
Similarly, when we negate this function, it “represents” [0,1) in the sense that it is only 0 (only ‘absolutely false’) for the value 1, and maps [0,1) to positive truth-values which can be mostly 1, but which must decline in the neighborhood of 1.
And yes, this setup can get us into some trouble when we try to use quantifiers. If “forall” is understood as taking the min, we can construct discontinuous functions as the limit of continuous functions. Hartry Field proposes a fix, but it is rather complex.
Note that some relevant authors in the literature use 0 for true and 1 for false, but I am using 1 for true and 0 for false, as this seems vastly more intuitive.
I’m confused about how continuity poses a problem for “This sentence has truth value in [0,1)” without also posing an equal problem for “this sentence is false”, which was used as the original motivating example.
I’d intuitively expect “this sentence is false” == “this sentence has truth value 0″ == “this sentence does not have a truth value in (0,1]”
“X is false” has to be modeled as something that is value 1 if and only if X is value 0, but continuously decreases in value as X continuously increases in value. The simplest formula is value(X is false) = 1-value(X). However, we can made “sharper” formulas which diminish in value more rapidly as X increases in value. Hartry Field constructs a hierarchy of such predicates which he calls “definitely false”, “definitely definitely false”, etc.
Proof systems for the logic should have the property that sentences are derivable only when they have value 1; so “X is false” or “X is definitely false” etc all share the property that they’re only derivable when X has value zero.
Understood. Does that formulation include most useful sentences?
For instance, “there exists a sentence which is more true than this one” must be excluded as equivalent to “this statement’s truth value is strictly less than 1″, but the extent of such exclusion is not clear to me at first skim.