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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{w}(\text{prop}(P))</math> directly equal to <math>\neg P_\text{w}</math>, or we can let <math>\text{juna}</math> and <math>\text{sahu}</math> refer to some more specific notion of truth that | 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{w}(\text{prop}(P))</math> directly equal to <math>\neg P_\text{w}</math>, or we can let <math>\text{juna}</math> and <math>\text{sahu}</math> refer to some more specific notion of truth that is resistant 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 == | ||