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Semantics: Difference between revisions
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Toaq is a loglang, which means that | Toaq is a loglang, which means that given any sentence, we can unambiguously derive its meaning in logic notation. '''Semantics''', the study of meaning, guides us in determining what those results should look like, and how we might use our knowledge of [[syntax]] to derive them. | ||
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], 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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** <math>\forall</math> for "every" | ** <math>\forall</math> for "every" | ||
For example, we might write the interpretation of "indeed, every person is living or dead" as <math>\dagger \forall a | For example, we might write the interpretation of "indeed, every person is living or dead" as <math>\dagger \forall a : \text{poq}(a).\ \text{mie}(a) \lor \text{muaq}(a)</math>. | ||
== Events == | == Events == | ||
One of the most basic jobs of any semantic theory is to define how verbs work. The traditional approach, used widely throughout mathematics, is to represent {{Derani| |Fa jí náomi}} as <math>\text{fa}(\text{j}\mathrm{\acute{i}}, \text{n}\mathrm{\acute{a}}\text{omi})</math>, where the verb is interpreted as a function (here, <math>\left\langle \text{e} \left\langle \text{e}, \text{t} \right\rangle \right\rangle</math>) receiving the subject and any objects as arguments. But sadly, this approach is unable to account for tense, aspect, or adverbs. | |||
Modern semantics research has settled on a single concept to overcome all of these issues: '''events'''. An event is an extra argument passed to a verb representing the action itself; the instance of that verb "happening". For instance, <math>\text{fa}(\text{j}\mathrm{\acute{i}}, \text{n}\mathrm{\acute{a}}\text{omi})(e)</math> represents whether <math>e</math> is an event of the speaker going to the sea. Whereas the first two arguments represent the participants in the action (the goer and the destination), ''e'' stands for the thing that connects them: the going, or the journey. Then, a sentence like {{Derani| |Fa jí náomi}} can be understood as claiming that there ''is'' such an event: <math>\exists e.\ \text{fa}(\text{j}\mathrm{\acute{i}}, \text{n}\mathrm{\acute{a}}\text{omi})(e)</math>. This system is credited to philosopher Donald Davidson, giving it the name '''Davidsonian event semantics'''. | |||
This gives us a systematic way to deal with adverbs: to modify the verb, modify the ''event variable introduced by the verb''. This is intended to reflect the intuition that "I slept briefly" has the same meaning as "My sleep was brief". For example, {{Derani| |Nuo jîm jí}} can be interpreted as <math>\exists e.\ \text{nuo}(\text{j}\mathrm{\acute{i}})(e) \land \text{jim}(e)</math>. And prepositions work similarly: for {{Derani| |Lao jí nîe jío}} we would use <math>\exists e.\ \text{lao}(\text{j}\mathrm{\acute{i}})(e) \land \text{nie}(e, \text{j}\mathrm{\acute{i}}\text{o})</math> — "whether there is some event of me waiting that is inside the building". | |||
With events in our toolbox, tense and aspect also fall into place. If we imagine that every event has a temporal footprint (the points in time at which it takes place), then it seems reasonable that there should be a function to access this information. We call this <math>\tau</math>, the '''temporal trace function''' (type <math>\left\langle \text{v}, \text{i} \right\rangle</math>). Aspect is then understood as making a claim about an event's temporal structure, relative to a reference time determined by the tense. For instance, {{Derani||tam}} makes the claim that the event's temporal trace lies fully within the reference time: <math>\tau(e) \subseteq \text{t}</math>. (This one comes up a lot, because {{Derani||tam}} is the default aspect.) And {{Derani||luı}} makes the claim that the event's temporal trace comes before the reference time: <math>\tau(e) < \text{t}</math>. | |||
So including aspect, the ''complete'' interpretation of {{Derani| |Fa jí náomi}} should be <math>\exists e.\ \tau(e) \subseteq \text{t} \land \text{fa}(\text{j}\mathrm{\acute{i}}, \text{n}\mathrm{\acute{a}}\text{omi})(e)</math>. This is a little cumbersome to read, so you will sometimes see it abbreviated to <math>\text{fa}(\text{j}\mathrm{\acute{i}}, \text{n}\mathrm{\acute{a}}\text{omi})</math> when we're being lazy. | |||
== Presuppositions == | |||
Some statements carry a set of assumptions in addition to their main semantic content. When we say "The current king of France is bald", it is assumed that there ''is'' a current king of France. And likewise, the sentence {{Derani| |Luı nuo sá tıqra nîe náokua}} carries the assumption that {{Derani||náokua}} actually refers to a bathroom. (It would be nonsensical to say such a thing while pointing to, say, a car!) The technical term for an assumption of this kind is a '''presupposition'''. | |||
There's a trick that we can use to write presuppositions alongside a semantic expression: by leveraging the mathematical notion of an expression being '''undefined'''. Just as <math>1 \div x</math> is undefined when <math>x = 0</math>, "the current king of France" should be undefined when France has no king. In semantic notation, we write this as <math>\text{bald}(\text{k})\text{, defined only if king}(\text{k}, \text{France}) </math>. This restricts the possible models to only those that set <math>\text{k}</math> to be a king of France. | |||
Note that this <math>\text{defined only if}</math> clause can appear anywhere within an expression, not just at the top level. One example where it ''needs'' to be embedded in a sub-expression is in {{Derani| |Gaq tú deo ké pao âq}}. This becomes: <math>\forall a : \text{deo}(a).\ \exists e.\ \tau(e) \subseteq \text{t}\ \land\ \text{gaq}(a, [\text{P}(a)\text{, defined only if }\text{pao}(\text{P}(a), a)])(e)</math>. Moving the <math>\text{defined only if}</math> clause to the top level wouldn't work, because it uses the variable <math>a</math>, which is only available inside the scope of the <math>\forall</math> function. | |||
In lambda expressions, you might also come across the syntax <math>\lambda a : \text{naokua}(a).\ \text{ti}(\text{t}\mathrm{\acute{i}}\text{qra}, a)</math>, where <math>\lambda</math> is imagined to be a quantifier restricted by <math>\text{naokua}(a)</math>. This is the same thing as writing <math>\lambda a .\ (\text{ti}(\text{t}\mathrm{\acute{i}}\text{qra}, a)\text{, defined only if naokua}(a))</math>. | |||
== 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 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 this framework, we can understand modals as making claims about alternate worlds. For instance, {{Derani| |Shê, ꝡä tao sı súq fáfuaq, nä cho súq hóq}} means "in all possible worlds, minimally different from the real world, in which you go to see the movie, you like it". In semantic notation, that looks like: <math>\forall w: (\text{SHE}(\text{w}, w)\ \land\ \exists e.\ \tau(e) \subseteq \text{t}\ \land\ \text{si}_w(\text{s}\mathrm{\acute{u}}\text{q}, \text{f}\mathrm{\acute{a}}\text{fuaq})(e))).\ \exists e.\ \tau(e) \subseteq \text{t'}\ \land\ \text{cho}_w(\text{s}\mathrm{\acute{u}}\text{q}, \text{f}\mathrm{\acute{a}}\text{fuaq})(e)</math>. The function <math>\text{SHE}(\text{w}, w)</math> is the part that stands for "<math>w</math> is a possible world minimally different from the real world". The technical term for this function is the '''accessibility relation''', because it defines which worlds we can "access" and talk about using the modal {{Derani||she}}. | |||
Some modals, such as {{Derani||daı}}, use the quantifier <math>\exists</math> instead of <math>\forall</math>, because for something to be possible, it only needs to be true in one possible world. Other modals, such as {{Derani||dı}}, use a completely different accessibility relation (<math>\text{DUAI}</math>) to talk about ''acceptable worlds'' rather than possible worlds. And other modals, such as {{Derani||ao}}, use an accessibility relation that presupposes that the complement is not true in the reference world, to achieve a counterfactual effect. This world metaphor really is flexible enough to account for all modals! | |||
Note that similarly to events, we sometimes get lazy and neglect to write the world arguments on verbs. | |||
== Propositions == | == Propositions == | ||
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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{prop}(P))</math> directly equal to <math>\neg | 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 == | ||
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We use the function representation whenever a property in Toaq is spelled out explicitly with the complementizer {{Derani||lä}}. For example, the property in {{Derani| |Leo jí, lä nuo já}} would be interpreted as <math>\lambda a.\ \lambda w.\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{nuo}_w(a)(e)</math>, a function of type <math>\left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle</math>. And for a property with two blanks, you would use a function of type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle</math>. | We use the function representation whenever a property in Toaq is spelled out explicitly with the complementizer {{Derani||lä}}. For example, the property in {{Derani| |Leo jí, lä nuo já}} would be interpreted as <math>\lambda a.\ \lambda w.\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{nuo}_w(a)(e)</math>, a function of type <math>\left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle</math>. And for a property with two blanks, you would use a function of type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle</math>. | ||
But whenever a Toaq variable is used as a property, we need to fall back to the properties as individuals approach, using <math>\text{iq}</math> (type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle</math>) or <math>\text{cuoi}</math> (type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle \right\rangle</math>) to access its semantic content. So, the correct interpretation of {{Derani| |Che nháo sá jua}} would be <math>\exists a\ \text{ | But whenever a Toaq variable is used as a property, we need to fall back to the properties as individuals approach, using <math>\text{iq}</math> (type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle</math>) or <math>\text{cuoi}</math> (type <math>\left\langle \text{e}, \left\langle \text{e}, \left\langle \text{e}, \left\langle \text{s}, \text{t} \right\rangle \right\rangle \right\rangle \right\rangle</math>) to access its semantic content. So, the correct interpretation of {{Derani| |Che nháo sá jua}} would be <math>\exists a : \text{jua}_\text{w}(a).\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{che}_\text{w}(\text{nh}\mathrm{\acute{a}}\text{o}, \lambda b.\ \text{iq}(b, a))(e) </math>. | ||
== Notes == | == Notes == | ||
<references /> | <references /> | ||