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Chapter 1

PHIL 210 Chapter Notes - Chapter 1: Arity, Atomic Sentence, Set Theory

Course Code
PHIL 210
Michael Hallett

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Chapter 1, LPL
LPL language: Cube(b) means b is a cube; Larger(c, f) means c > f; Between(b, c, d)
means c is between b and d. Note: this must mean that Cube/Larger/Between are
predicates/relations, whereas b/c/d/f are simple terms
Names are called individual constants (ex a/b/c/d/e/f/n1/n2…)
Individual constants must name an existing object (no empty)
Individual constants must name one object (no multivalued)
An object can have any non-negative integer number of names associated with the
object (0, 1, 2, …)
Each predicate has a fixed number of arguments, arity
Every predicate is interpreted by a determinate property or relation of the same arity as
the predicate (?)
So ex. RightOf(a, b) (the sentence) is interpreted by the property of (a, b) (in
particular, orientation with respect to each other)? Makes sense I guess.
= uses ‘infix’ notation: = trapped within 2 arguments. Other predicates use ‘prefix
notation; they come before arguments.
Predicates designate properties, names designate objects, sentences make claims/express
Claims have a truth value which is either true or false.
Note: Are claims objects? So can you have names for sentences in this language? Ex.
“Cube(a)” = b… this way, the example used in the powerpoint i.e. Dodec(Dodec(a)) not
being a properly formed sentence would be false. It is properly formed… just false.
Dodec(a) is a claim, and therefore, Dodec(Dodec(a)) takes on the truth value false. I can
also frame this question as: are there different classes of objects that aren’t really inter-
replaceable? That would mean that predicates can only take in some objects and not
others. This also means that some objects have properties which other objects don’t
have (for example, namely, the property of its interaction with the predicate, for which
other objects have NA interaction with the predicate).
25 Atomic sentences consist of a predicate of arity n, and n names
Order is crucial! (non-commutative)
FOL languages differ in the names and predicates they contain, and thus the sentences
that can be formed
Ex. Arithmetic, set theory
They have in common connectives and quantifiers
Sometimes when trying to translate from English to FOL, there is no predefined
language… so you need to decide on names and predicates
Objects are anything we can make claims about.
31 Note: Ex. 1.11.3: is LessThan(Contagion(AIDS), Contagion(HIV)) atomic? It only
contains 1 predicate and 2 objects. EDIT: By the looks of it, it is indeed an atomic
Function symbols help express complex claims in atomic sentences.
Individual constants are simplest terms, more complex terms are built up with function
Convention in textbook: predicates will be capitalized, functions will be in lower case
33 When designing a language in FOL< you should make sure that complex terms refer to
unique, existing individuals
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