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Semantics: Difference between revisions
go down the proposition rabbit hole
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Toaq is a loglang, which means that from any sentence, we can unambiguously derive its meaning in logic notation. [[Syntax]] describes how this process works; '''semantics''' | Toaq is a loglang, which means that from any sentence, we can unambiguously derive its meaning in logic notation. [[Syntax]] describes how this process works; '''semantics''' tells us how to interpret the result. | ||
The refgram tells you that {{Derani| |Luı nuo sá tıqra nîe náokua}} translates to <math>\exists x: \text{tıqra}_\text{w}(x).\ \exists e.\ \text{τ}(e)<\text{t}\land \text{nuo}_\text{w}(x)(e)\land \text{nıe}_\text{w}(e, \text{n}\mathrm{\acute{a}}\text{okua})</math>. The reality is that this isn't "just" logic notation: it's a very specific notation that has been purpose-built for describing natural language semantics, and this article will help you understand the core concepts behind it. | The refgram tells you that {{Derani| |Luı nuo sá tıqra nîe náokua}} translates to <math>\exists x: \text{tıqra}_\text{w}(x).\ \exists e.\ \text{τ}(e)<\text{t}\land \text{nuo}_\text{w}(x)(e)\land \text{nıe}_\text{w}(e, \text{n}\mathrm{\acute{a}}\text{okua})</math> . The reality is that this isn't "just" logic notation: it's a very specific notation that has been purpose-built for describing natural language semantics, and this article will help you understand the core concepts behind it. | ||
== Models == | == Models == | ||
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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], which | 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://github.com/MostAwesomeDude/brismu brismu], a sketch of an inferential approach to Lojban semantics</ref><ref>[https://cdn.discordapp.com/attachments/311223912044167168/850159530011918357/hoemui.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. | ||
== Semantic calculus == | == Semantic calculus == | ||
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One interesting thing about this notation is that every expression has a '''type''', like some programming languages do. These include: | One interesting thing about this notation is that every expression has a '''type''', like some programming languages do. These include: | ||
* <math>\text{e}</math>, the type of individuals, which encompasses everything you can refer in Toaq. This is a rather broad category, so to help us get more specific when we need it, it includes a couple of subtypes: | * <math>\text{e}</math>, the type of individuals, which encompasses everything you can refer to in Toaq. This is a rather broad category, so to help us get more specific when we need it, it includes a couple of subtypes: | ||
** <math>\text{v}</math>, the type of [[Event|events]] (things that can happen). More on them later. | ** <math>\text{v}</math>, the type of [[Event|events]] (things that can happen). More on them later. | ||
** <math>\text{i}</math>, the type of time intervals | ** <math>\text{i}</math>, the type of time intervals | ||
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== Propositions == | == Propositions == | ||
A '''proposition''' is, in the broadest sense, anything that bears a truth value, such as a fact, a belief, or the meaning of a sentence. We can say that the sentence " | A '''proposition''' is, in the broadest sense, anything that bears a truth value, such as a fact, a belief, or the meaning of a sentence. We can say that the sentence "Die Erde ist ein Planet" expresses the proposition that the Earth is a planet, and likewise, in the sentence "I believe that I saw a ghost", we can identify "that I saw a ghost" as referring to the proposition that the speaker saw a ghost. | ||
In Toaq, we | In Toaq, we use the complementizer {{Derani||ꝡä}} to create a reference to a proposition, which can then become the complement of another verb. So, our semantic theory needs to account for this construct, and it turns out that it's best to use two different "interpretations" of propositions for this purpose. | ||
The first | 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: | |||
* Restrict the language's syntax so that it can't even express the liar paradox (not an option for a human language like Toaq) | |||
* Allow models to contain contradictions, by departing from classical logic in some way (for example, by adding a 3rd truth value, or otherwise weakening the logic to prevent [[wikipedia:Principle_of_explosion|explosion]]) | |||
* Use a more specific notion of truth for propositions, so that the language doesn't literally contain its own truth predicate | |||
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>. | |||
== Properties == | == Properties == | ||