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}(\text{L})_\text{w} = \neg \text{L}_\text{w}</math>, which is a problem. Philosophers have studied this paradox extensively, and their responses fall into a few categories:
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)