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Semantics: Difference between revisions
Write the worlds section
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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. | 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> | 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> computes 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> — "there is some event of me waiting that is inside the building". | 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>. | 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. | 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 == | == Presuppositions == | ||
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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>. | 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 == | |||
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 approach to 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 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/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{jua}(a).\ \exists e.\ \tau(e) \subseteq \text{t} \land \text{che}(\text{nh}\mathrm{\acute{a}}\text{o}, \lambda b.\ \text{iq}(b, a))(e) </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{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 /> | ||