Laqme: cheo higher-orderness example

2021-11-11T23:23:23Z

cheo higher-orderness example

← Older revision		Revision as of 23:23, 11 November 2021
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	<blockquote>		<blockquote>
	{{t\|cheo}}: ___ all satisfy binary relation ___ among each other.		{{t\|cheo}}: ___ all satisfy binary relation ___ among each other.<br>
			<math>\text{Cheo}(x, R) \iff \forall a \forall b ( \text{Among}(a, x) \wedge \text{Among}(b, x) \wedge (a \neq b) \to R(a, b))</math>
	</blockquote>		</blockquote>

Laqme at 21:44, 3 November 2021

2021-11-03T21:44:52Z

← Older revision		Revision as of 21:44, 3 November 2021
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	This connection between predicate logic and natural language has been explored since the 1970s by Montague ([https://en.wikipedia.org/wiki/Montague_grammar ''Montague grammar'' on Wikipedia]).		This connection between predicate logic and natural language has been explored since the 1970s by Montague ([https://en.wikipedia.org/wiki/Montague_grammar ''Montague grammar'' on Wikipedia]).

			== See also ==
			* [https://en.wikipedia.org/wiki/Predicate_logic Predicate logic] on Wikipedia.
			* [[Plural logic]], another flavor of the logic Toaq is based on.

Laqme: Created page with "'''Predicate logic''' is formal logic where the alphabet includes quantifiers ( $\forall$ , $\exists$ …), variables (''x'', ''y'', ''z''…), and '''predi..."

2021-11-03T21:10:16Z

Created page with "'''Predicate logic''' is formal logic where the alphabet includes quantifiers (<math>\forall</math>, <math>\exists</math>…), variables (''x'', ''y'', ''z''…), and '''predi..."

New page

'''Predicate logic''' is formal logic where the alphabet includes quantifiers (<math>\forall</math>, <math>\exists</math>…), variables (''x'', ''y'', ''z''…), and '''predicate symbols''' (''P'', ''Q''…) that can relate variables or constant terms.

For example, <math>\forall x \exists y ( P(x, y) \vee \neg Q(y, x) )</math> is a sentence in predicate logic, meaning: “for all ''x'', there exists some ''y'', such that either P(''x'', ''y'') holds, or Q(''y'', ''x'') does not hold.”

In this case, P and Q are “non-logical symbols” standing for some relations, and x and y are variables ranging over some implicit domain. The precise relations and domain are opaque to the ''rules'' of predicate logic — the point of formal logic is arguably to state inferences like <math>\forall x P(x) \iff \neg \exists x \neg P(x)</math> without worrying about what ''x'' and ''P'' stand for — but can be assigned meaning when combined with some '''interpretation''' or '''model''' for the logic.

Even in its simplest form, predicate logic can be used to model many statements in natural language, where the predicate symbols and constants/terms are taken to model verbs relating nouns:

<blockquote>
Every person who lives in Perth lives in Australia.<br>
<math>\forall p(\text{Person}(p) \wedge \text{LivesIn}(p, \text{Perth}) \to \text{LivesIn}(p, \text{Australia}))</math>
</blockquote>

Toaq is based on this formalism. Its content words are explicitly ''n''-ary predicates:

<blockquote>
{{t|poq}}: ___ is a person.<br>
{{t|bua}}: ___ lives in ___.
</blockquote>

It allows for slots that themselves accept other predicates/relations, meaning Toaq is really based on '''higher-order logic''':

<blockquote>
{{t|cheo}}: ___ all satisfy binary relation ___ among each other.
</blockquote>

This connection between predicate logic and natural language has been explored since the 1970s by Montague ([https://en.wikipedia.org/wiki/Montague_grammar ''Montague grammar'' on Wikipedia]).

Predicate logic - Revision history

Laqme: cheo higher-orderness example

Laqme at 21:44, 3 November 2021

Laqme: Created page with "'''Predicate logic''' is formal logic where the alphabet includes quantifiers (\forall, \exists…), variables (''x'', ''y'', ''z''…), and '''predi..."

Laqme: Created page with "'''Predicate logic''' is formal logic where the alphabet includes quantifiers ( $\forall$ , $\exists$ …), variables (''x'', ''y'', ''z''…), and '''predi..."