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Understanding Names, Predicates, and Truth Values in Symbolic Logic

Explore the basics of names, predicates, sentences, and truth values in first-order logic. Learn about atomic sentences, predicates' arities, and making claims with precise meanings. Get ready for assignments and homework problems.

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Understanding Names, Predicates, and Truth Values in Symbolic Logic

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  1. PHIL012 Class Notes 1/12/2001

  2. Outline • Announcements, web page • Names and Predicates • Sentences, claims, and truth value • Atomic sentences • Homework Problems • Assignment for Monday (1/15)

  3. Names • A name in a formal language refers to an object in the world. • In FOL the object must exist. That is, every name must refer to an actual object.

  4. Language World “Name1” Object1 “Name2” Object2 “Pegasus”

  5. Names Continued • No Name can name more than one object. • In other words, every name must have a unique reference. • Or, every name in FOL must be a “rigid designator”. • An object can have more than one name or even be unnamed.

  6. Good Language World “The Morning Star” “The Evening Star” “Venus”

  7. Bad Language World “John”

  8. Predicates • Predicates in FOL refer to propertiesof objects in the world or relations between objects. • Examples of Predicates: • “IsBlue” or simply “Blue” • “IsMarriedTo” or simply “Married” • “Can’tStandBrittanySpears” or simply “BrittanyHater”

  9. “Arity” • Each predicate refers to a fixed number of objects. • In other words, each predicate will take a fixed number of names as arguments* or parameters. • We refer to the number of arguments a predicate takes as its “arity” *Note that this is not the same thing as an argument made up of sentences that makes a claim.

  10. “Arities”of Predicates in Tarski’s World • Arity 1: Cube, Tet, Dodec, Small, Medium, Large • Example: Cube(a) - “a” is the name of an object that has the property of being a cube. • Arity 2: Smaller, Larger, LeftOf, RightOf, BackOf, FrontOf • Example: Smaller(a,b) - This says that an object named by “a” is smaller than an object named by “b”

  11. “Arities”of Predicates in Tarski’s World • Arity 3: Between • Example: Between(a,b,c) - An object named by “a” is between objects named by “b” and “c”. • Remember: Predicates must have a precise meaning in FOL.

  12. Names and Predicates Language World Name Object Predicate Property

  13. Sentences and Claims • A sentence in FOL, like in English, makes a claim. • A claim is a statement about a state of affairs in the world. • Claims have truth value. • That is, claims are either true or false.

  14. Claims and Truth Value • A claim is true if it represents an actual state of affairs with regard to objects in the world. It is false if it does not. • Consider the sentence expressing the claim “Mary like Jim”: Likes(Mary,Jim). • This sentence and this claim will be true just in case Mary actually does like Jim and false if she does not.

  15. Atomic Sentences • An atomic sentence is a predicate followed by the correct number of names, such as Tet(a), Between(a,b,c), and Large(b). • An atomic sentence is the simplest type of sentence that makes a claim. • In other words, an atomic sentence is the smallest unit of FOL that can have a truth value.

  16. Homework Problems

  17. Assignment for Monday (1/15) • Meet with your groups at least once. • Read LFOL 2.4-2.7 if you haven’t already. • Continue working LFOL probs. 2.1-2.10 • Work LFOL problems 2.11-2.17 • For practice, sample the exercises in chapter 1 of Tarski’s World

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