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Chapter 1

PHIL 210 Chapter Notes - Chapter 1: Infix Notation, Arity, Binary Relation


Department
Philosophy
Course Code
PHIL 210
Professor
Michael Frank Hallett
Chapter
1

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Simply, symbols that are used to refer to some fixed individual or language.
Max (English) --> max (FOL)
Not capitalized.
Work the same way that names work in the English language.
Individual constants refer to only one object.
Some names in the English language don't refer to anything (ex. Santa Claus fails to refer to
anything concrete).
We allow for nameless objects.
1.1 Individual Constants
Also sometimes called relation symbols.
Symbols used to express some property of objects or a relation between objects.
Subjects/Arguments: max (Max), Claire (claire).
Predicate: Likes (expresses a relation between max and claire).
Max likes Claire.
Example:
These logical subjects are called the arguments of the predicate.
Predicate is whatever is left, when we remove the logical subjects (individual constants).
3 arguments = ternary, so on and so forth.
In FOL, each predicate has a fixed number of arguments.
This quality of being binary or ternary is called the arity of the predicate.
This is a number that tells us how many individual constants the predicate symbol needs
to form a sentence.
In the above example with Max and Claire, the predicate is said to be binary, since it takes 2
arguments.
Claire is taller than Max.
2 arguments: claire and max.
Predicate: Taller.
Formula: Pred(argument 1, argument 2)
Claire is taller than Max. --> Taller(claire, max).
Thus:
Example:
Arity 1: Cube, Tet, Dodec, Small, Medium, Large
Arity 2: Smaller, Larger, LeftOf, RightOf, BackOf, FrontOf, SameSize, SameShape, SameRow,
SameCol, Adjoins, =
Between(b,c,d) --> b is between c and d.
Arity 3: Between
For example, the predicate "young" would be inappropriate, because there isn't a clear
definition/age cut-off that defines the property of young-ness.
Predicates are very specific and defined.
1.2 Predicate Symbols
Formed by a single predicate and the appropriate number of individual constants.
Taller(claire, max) and Cube(a) are atomic sentences.
Infix Notation: Using a "=" symbol, such as a = b.
Prefix Notation: The predicates precedes the arguments, like the arguments above.
Taller(claire, max) --> Claire is taller than Max.
Taller(max, claire) --> Max is taller than Claire.
The order of the arguments plays an important role in the meaning of the atomic sentence.
Atomic sentences make claims about things (i.e. that Claire is taller than Max) that are either
true or false (truth value).
1.3 Atomic Sentences
1.4 General First-Order Languages
Chapter 1 - Atomic Sentences
PHIL 210 Page 1
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