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Chapter 7 Notes
Sawyer Watts
Predicates are descriptions and relations for individuals. They are what allow us to go in depth
and see how the statements from last chapter relate to each other, by better describing
relations and the individuals linked by those relations. This isn’t possible with mere statement
logic. For example, there could be an individual “x” who is a human, and who likes cooking. The
predicates H (for human) and C (for liking cooking) would both be predicate that individual “x”
would belong to, in the form of H (x) and C (x). These are one place predicates, and there are
multiple-placed ones as well.
Multiple-place predicates involve one individual acting on something else, like an individual “a”
liking an individual “b”. The predicate here would be L(<a,b>), with an ordered pair of “a” and
“b” being contained as a member of the two-place predicate L. As a generality, one-place
predicates are known as properties, and multiple-place ones as relations.
Additionally, the predicates ∃ and ∀ have special meanings in predicate logic. ∃ means,
“There is at least one x” and ∀ which means “for all x”, where “x” is an individual, or group of
individuals. These predicates are used on other, “normal” predicates to provide additional
restrictions or descriptions to the reader.
An interesting thing to note for these predicates would be the relation between their negations.
The phrase, “There is not at least one individual x” (~∃x) is logically equivalent to the phrase
“For all x, there is not an individual” (∀x ~). The reverse of this is also true. The phrase,
“There is at least one individual x who is not” is equivalent to ”There is not, for all x, an
individual” (~∀x). To summarize this, the negation of the exceptional is equivalent to the
universal negation, and likewise the negation of the universal is equivalent to the
exceptional negation.
All this being said, predicates are still incapable of describing concepts such as time, as well
as implicatures like sarcasm. They require even more detailed languages of logic.