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Semantics: Difference between revisions
Explain presuppositions
(Use a consistent syntax for restricted quantification) |
(Explain presuppositions) |
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== Presuppositions == | == 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 hô}}. 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. | |||
== Propositions == | == Propositions == | ||