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.
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.