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Semantics: Difference between revisions
Explain how properties are represented
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Both of the last two options will work, and we should ensure that our semantic notation can accommodate either of them as resolutions to the paradox. This is where the second interpretation comes in: '''propositions as individuals'''. The idea is to let some individuals stand for propositions, and use the functions <math>\text{juna}</math> and <math>\text{sahu}</math> (both of type <math>\left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle</math>) to access their semantic content. There could also be a function <math>\text{prop}</math> (type <math>\left\langle \left\langle \text{s}, \text{t} \right\rangle, \text{e} \right\rangle</math>) which lets you convert propositions in the other direction, from functions to individuals. With this approach, quantifying over propositions, as in {{Derani| |Dua jí sía raı}}, looks like this: <math>\neg\exists a.\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{dua}_{\text{w}}(\text{ji}, \text{juna}(a))(e)</math>. Note the use of <math>\text{juna}</math> to convert the variable <math>a</math> into an <math>\left\langle \text{s}, \text{t} \right\rangle</math>, which enables us to reuse the same version of <math>\text{dua}</math> that takes <math>\left\langle \text{s}, \text{t} \right\rangle</math> propositions. | Both of the last two options will work, and we should ensure that our semantic notation can accommodate either of them as resolutions to the paradox. This is where the second interpretation comes in: '''propositions as individuals'''. The idea is to let some individuals stand for propositions, and use the functions <math>\text{juna}</math> and <math>\text{sahu}</math> (both of type <math>\left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle</math>) to access their semantic content. There could also be a function <math>\text{prop}</math> (type <math>\left\langle \left\langle \text{s}, \text{t} \right\rangle, \text{e} \right\rangle</math>) which lets you convert propositions in the other direction, from functions to individuals. With this approach, quantifying over propositions, as in {{Derani| |Dua jí sía raı}}, looks like this: <math>\neg\exists a.\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{dua}_{\text{w}}(\text{ji}, \text{juna}(a))(e)</math>. Note the use of <math>\text{juna}</math> to convert the variable <math>a</math> into an <math>\left\langle \text{s}, \text{t} \right\rangle</math>, which enables us to reuse the same version of <math>\text{dua}</math> that takes <math>\left\langle \text{s}, \text{t} \right\rangle</math> propositions. | ||
The consequence of this approach is that we now have a layer of abstraction to play with (<math>\text{juna}</math> and <math>\text{sahu}</math>), so that models are free to apply any reasonable resolution to the liar paradox. For example, we can allow the contradiction to exist by setting <math>\text{sahu}(\text{prop}(P))</math> directly equal to <math>\neg P</math>, or we can let <math>\text{juna}</math> and <math>\text{sahu}</math> refer to some more specific notion of truth that holds up to the liar paradox, such as Kripkean truth<ref>Kripke, S., 1975, “Outline of a theory of truth”, ''Journal of Philosophy'', 72: 690–716.</ref> or stable truth<ref>[https://plato.stanford.edu/entries/truth-revision/index.html The Revision Theory of Truth (Stanford Encyclopedia of Philosophy)]</ref>. | The consequence of this approach is that we now have a layer of abstraction to play with (<math>\text{juna}</math> and <math>\text{sahu}</math>), so that models are free to apply any reasonable resolution to the liar paradox. For example, we can allow the contradiction to exist by setting <math>\text{sahu}(\text{prop}(P))</math> directly equal to <math>\neg P</math>, or we can let <math>\text{juna}</math> and <math>\text{sahu}</math> refer to some more specific notion of truth that holds up to the liar paradox, such as Kripkean truth<ref>Kripke, S., 1975, “Outline of a theory of truth”, ''Journal of Philosophy'', 72: 690–716.</ref> or stable/categorical truth<ref>[https://plato.stanford.edu/entries/truth-revision/index.html The Revision Theory of Truth (Stanford Encyclopedia of Philosophy)]</ref>. | ||
== Properties == | == Properties == | ||
A '''property''' is an incomplete proposition; a claim with blanks to be filled. A simple example would be the property "◯ is red", also known as "to be red" or simply "redness". By filling in the blank, you get a proposition: "the apple is red". Properties can be arbitrarily complex, containing nested clauses or even multiple blanks: for example, "◯ can't believe that ◯ is not butter". In Toaq, properties are marked by the complementizer {{Derani||lä}}. | |||
The good news is that once you understand the semantics behind propositions, properties aren't far out of reach. We still have the same concerns: capturing the intension rather than the extension, and enabling variables to refer to properties without creating a paradox. Properties are represented in the same way as propositions, just with an extra parameter or two added for the blanks. This means we get both function-type and individual-type representations. | |||
We use the function representation whenever a property in Toaq is spelled out explicitly with the complementizer {{Derani||lä}}. For example, the property in {{Derani| |Leo jí, lä nuo já}} would be interpreted as <math>\lambda a.\ \lambda w.\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{nuo}_w(a)(e)</math>, a function of type <math>\left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle</math>. And for a property with two blanks, you would use a function of type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle</math>. | |||
But whenever a Toaq variable is used as a property, we need to fall back to the properties as individuals approach, using <math>\text{iq}</math> (type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle</math>) or <math>\text{cuoi}</math> (type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle \right\rangle</math>) to access its semantic content. So, the correct interpretation of {{Derani| |Che nháo sá jua}} would be <math>\exists a\ \text{jua}(a).\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{che}(\text{nh}\mathrm{\acute{a}}\text{o}, \lambda b.\ \text{iq}(b, a))(e) </math>. | |||
TODO: point out that questions are isomorphic to properties | |||
== Notes == | == Notes == | ||
<references /> | <references /> | ||