Timeline for Substitutional modality
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 17, 2018 at 1:34 | vote | accept | Andrew Bacon | ||
| Jan 16, 2018 at 23:34 | comment | added | Peter LeFanu Lumsdaine | Ah, yes, my bad: I was overlooking that the interpretation of 2nd order prop. logic won’t give that “if and only if” when the quantification is restricted to sentences. | |
| Jan 16, 2018 at 19:25 | comment | added | Andrew Bacon | @PeterLeFanuLumsdaine If you understand the propositional quantifiers substitutionally, and you have propositional operators in the language, the constraints on truth are no-longer well-founded (and the circularity is potentially problematic, as demonstrated by things like Prior's paradox). It's not in general obvious that the unproblematic semantic interpretation of the propositional quantifiers corresponds to the substitutional interpretation: i.e. that $\forall p\phi$ is true in the semantic sense if and only if $\phi(A/p)$ is true for every sentence $A$. | |
| Jan 16, 2018 at 17:24 | answer | added | Emil Jeřábek | timeline score: 8 | |
| Jan 16, 2018 at 13:11 | comment | added | Peter LeFanu Lumsdaine | I’m not sure whether a proper valuation exists, but: if something is vicious here, I don’t think it’s the circularity.. Second-order propositional logic has essentially the same circularity, and quite unproblematic semantics. Rather, the issue in your setup is the (lack of) treatment of bound variables: the semantics of $\Box \varphi$ quantifies over propositional variables in $\varphi$, but syntactically, $\Box$ doesn’t bind them. E.g. any proper valuation must have $v(\lnot \Box A) = \top$ (so one might think $\lnot \Box A$ was “logically true”) but also $v(\Box \lnot \Box A) = \bot$. | |
| Jan 16, 2018 at 1:16 | history | edited | Andrew Bacon | CC BY-SA 3.0 |
added 2 characters in body
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| Jan 15, 2018 at 22:10 | history | asked | Andrew Bacon | CC BY-SA 3.0 |