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Semantics: Difference between revisions
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Say that you have an idea of what the world is like—maybe you have a mental model in your head, or maybe you have a database to look things up in. If your knowledge is complete enough, then that model lets you answer a question, or tell whether what someone said is true, by interpreting their words and then "looking up" the answer. But more often than not, people are working with incomplete knowledge. In this case, if someone tells you something, a model lets you interpret their words and then ''work backwards'' from the meaning to figure out what must be true about the world. | Say that you have an idea of what the world is like—maybe you have a mental model in your head, or maybe you have a database to look things up in. If your knowledge is complete enough, then that model lets you answer a question, or tell whether what someone said is true, by interpreting their words and then "looking up" the answer. But more often than not, people are working with incomplete knowledge. In this case, if someone tells you something, a model lets you interpret their words and then ''work backwards'' from the meaning to figure out what must be true about the world. | ||
A note for the adventurous: There are alternative approaches to semantics that don't involve models, such as [https://plato.stanford.edu/entries/proof-theoretic-semantics/#InfIntAntRea proof-theoretic semantics], in which the meaning of a statement is determined purely by its relationships to other statements in a formal proof system. There have been some attempts to apply this approach to Lojban and Toaq semantics<ref>[https://mostawesomedude.github.io/brismu/ brismu], a sketch of an inferential approach to Lojban semantics</ref><ref>[ | A note for the adventurous: There are alternative approaches to semantics that don't involve models, such as [https://plato.stanford.edu/entries/proof-theoretic-semantics/#InfIntAntRea proof-theoretic semantics], in which the meaning of a statement is determined purely by its relationships to other statements in a formal proof system. There have been some attempts to apply this approach to Lojban and Toaq semantics<ref>[https://mostawesomedude.github.io/brismu/ brismu], a sketch of an inferential approach to Lojban semantics</ref><ref>[[:File:Hoemuı.pdf|Hoemuı]], the beginnings of a sketch of an inferential approach to Toaq semantics (super outdated)</ref>, but when it comes to natural language semantics, the model-based approach described here is far more common. | ||
== Basic notation == | == Basic notation == | ||
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== Worlds == | == Worlds == | ||
Another important concept for any semantic theory to cover is '''modality''': the treatment of words such as {{Derani||she}}, {{Derani||daı}}, {{Derani||ao}}, and {{Derani||dı}}. We use these words to make claims not about the actual state of the world, but about possibilities, obligations, or beliefs. The tried and true | Another important concept for any semantic theory to cover is '''modality''': the treatment of words such as {{Derani||she}}, {{Derani||daı}}, {{Derani||ao}}, and {{Derani||dı}}. We use these words to make claims not about the actual state of the world, but about possibilities, obligations, or beliefs. The tried and true system for reasoning about modality, named after philosopher Saul Kripke, is known as '''Kripke semantics'''. | ||
In Kripke semantics, we imagine that there are a multitude of '''worlds''': one world, <math>\text{w}</math>, represents the real world, while others represent alternate timelines. Then, every verb is extended to take a world argument: for example, <math>\exists e.\ \text{saqsu}_\text{w}(\text{j}\mathrm{\acute{i}})(e)</math> computes whether there is an event of the speaker whispering ''in the real world'', with the world variable being written in a subscript for readability. | In Kripke semantics, we imagine that there are a multitude of '''worlds''': one world, <math>\text{w}</math>, represents the real world, while others represent alternate timelines. Then, every verb is extended to take a world argument: for example, <math>\exists e.\ \text{saqsu}_\text{w}(\text{j}\mathrm{\acute{i}})(e)</math> computes whether there is an event of the speaker whispering ''in the real world'', with the world variable being written in a subscript for readability. | ||
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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 == | ||