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Semantics: Difference between revisions
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The first interpretation is '''propositions as functions'''. The idea is to interpret a complementizer phrase as a function which takes a world as an input, and outputs the truth value of the proposition in that world (type <math>\left\langle \text{s}, \text{t} \right\rangle</math>). So for example, in {{Derani| |Chı jí, ꝡä za ruqshua}}, we would interpret the complementizer phrase as <math>\lambda w.\ \exists e.\ \tau(e) > \text{t} \land \text{ruqshua}_w(e)</math>, and pass this as an argument to the main verb, giving <math>\exists e'.\ \tau(e') \subseteq \text{t'} \land \text{chi}_{\text{w}}(\text{ji}, \lambda w.\ \exists e.\ \tau(e) > \text{t} \land \text{ruqshua}_w(e))(e')</math>. Note that it would be wrong to interpret the complementizer phrase as <math>\exists e.\ \tau(e) > \text{t} \land \text{ruqshua}_{\text{w}}(e)</math>, because this evaluates to a simple truth value, which fails to capture the statement's semantic content. No one goes around saying "I believe [TRUE]" or "I believe [FALSE]". By using a function, we capture the statement's '''intension''' (its abstract connotation) rather than its '''extension''' (the concrete truth value held by the statement in the real world). | The first interpretation is '''propositions as functions'''. The idea is to interpret a complementizer phrase as a function which takes a world as an input, and outputs the truth value of the proposition in that world (type <math>\left\langle \text{s}, \text{t} \right\rangle</math>). So for example, in {{Derani| |Chı jí, ꝡä za ruqshua}}, we would interpret the complementizer phrase as <math>\lambda w.\ \exists e.\ \tau(e) > \text{t} \land \text{ruqshua}_w(e)</math>, and pass this as an argument to the main verb, giving <math>\exists e'.\ \tau(e') \subseteq \text{t'} \land \text{chi}_{\text{w}}(\text{ji}, \lambda w.\ \exists e.\ \tau(e) > \text{t} \land \text{ruqshua}_w(e))(e')</math>. Note that it would be wrong to interpret the complementizer phrase as <math>\exists e.\ \tau(e) > \text{t} \land \text{ruqshua}_{\text{w}}(e)</math>, because this evaluates to a simple truth value, which fails to capture the statement's semantic content. No one goes around saying "I believe [TRUE]" or "I believe [FALSE]". By using a function, we capture the statement's '''intension''' (its abstract connotation) rather than its '''extension''' (the concrete truth value held by the statement in the real world). | ||
This approach is nice and simple, but it does have limitations. In Toaq, we can not only reference propositions with {{Derani||ꝡä}}, but we can also assign them to variables, or even quantify over them, as in the sentence {{Derani| |Dua jí sía raı}}. A naive approach to interpreting this sentence would be <math>\neg\exists P.\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{dua}_{\text{w}}(\text{ji}, P)(e)</math>, where the variable {{Derani||ráı}} is taken to range over functions of type <math>\left\langle \text{s}, \text{t} \right\rangle</math>. But if you let Toaq variables range over functions from the model, this now lets you construct the '''liar paradox''', a sentence which contradicts itself: {{Derani| |Sahu ní ruaqse}}. Interpreting this sentence, we get <math>\text{L}_\text{w} = \text{sahu} | This approach is nice and simple, but it does have limitations. In Toaq, we can not only reference propositions with {{Derani||ꝡä}}, but we can also assign them to variables, or even quantify over them, as in the sentence {{Derani| |Dua jí sía raı}}. A naive approach to interpreting this sentence would be <math>\neg\exists P.\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{dua}_{\text{w}}(\text{ji}, P)(e)</math>, where the variable {{Derani||ráı}} is taken to range over functions of type <math>\left\langle \text{s}, \text{t} \right\rangle</math>. But if you let Toaq variables range over functions from the model, this now lets you construct the '''liar paradox''', a sentence which contradicts itself: {{Derani| |Sahu ní ruaqse}}. Interpreting this sentence, we get <math>\text{L}_\text{w} = \text{sahu}_\text{w}(\text{L}) = \neg \text{L}_\text{w}</math>, which is problematic. Philosophers have studied this paradox extensively, and come up with a few different possible responses: | ||
* Restrict the language's syntax so that it can't even express the liar paradox (not an option for a human language like Toaq) | * Restrict the language's syntax so that it can't even express the liar paradox (not an option for a human language like Toaq) | ||