First Order Logic Chapter 7

PL is a Weak Representational Language §Propositional Logic (PL) is not a very expressive language because: §Hard to identify objects : mary, 3 §Can't directly talk about properties of individuals or relations between individuals: Employee(Name, SoSec, Salary, Dept) §Generalizations, patterns, regularities can't easily be represented: All persons are mortals

PL is a Weak Representational Language §Consider the problem of representing the following information: Every person is mortal. Bill is a person. Bill is mortal. §How can these sentences be represented so that we can infer the third sentence from the first two?

PL is a Weak Representational Language §In PL we have to create propositional symbols to stand for all or part of each sentence. For example, we might do: Person => Mortal Person-Bill Mortal-Bill §That is, we have used four symbols to represent the three given sentences. But, given this representation, the third sentence is not entailed by the first two.

§A different representation would be to use three symbols to represent the three sentences as Person => Mortal Bill => Person Bill => Mortal §The third sentence is entailed by the first two, but we needed an explicit symbol, Bill, to represent an individual who is a member of the classes "person" and "mortal." §So, to represent other individuals we must introduce separate symbols for each one, with means for representing the fact that all individuals who are ”person" are also "mortal." §FOL is expressive enough to concisely represent this kind of situation.

Representational Advantages §Concise Representations §Recursive Relations §Arbitrary structures (non-fixed size)

Propositional logic v. FOL §PL deals only with propositions that have no internal structure l P : I-love-all-dogs §FOL makes a stronger commitment to what there is (ontology) l objects: things in the world l properties & relations of & between objects §FOL is as an extension of PL §Example in FOL

Alphabet Non-Logical Symbols (meaning given by interpretation) l (infinite) set of object constants - strings of alphanumeric characters beginning with a lowercase letter or a numeral: john, mary l (infinite) set of function constants of all arities - strings of alphanumeric characters beginning with a lowercase letter: fatherOf, distanceBetween, times l (infinite) set of predicates (relation constants) of all arities - strings of alphanumeric characters beginning with an uppercase letter: MarriedTo(john, mary)

Alphabet Logical Symbols (fixed meaning): §Connectives: not (  ) and (  ) or (  ) implies (  ) equivalent (  ) §Punctuation Symbols: (, ) §Truth symbols: TRUE, FALSE

Alphabet Logical Symbols (fixed meaning): §Variable symbols- strings beginning with an uppercase letter near the end of the alphabet: X, Y, Z §Quantifier symbols: Universal:  Existential:  §(Optional) Equality predicate: =

Terms §Terms refer to objects in the world (do not have a truth value assigned to them): l object constants l function constants of arity n, followed by n terms in parentheses separated by commas: fatherof(john, bill) times(4, plus(3, 6)) l variable symbols

Syntax rules: Wffs §atomic formulas are predicates of arity n, followed by n terms in parentheses separated by commas: Loves(john, mary) =(john, john) (= usually in infix form: john = john) (A proposition in PL is an atomic formula of arity 0) §Truth symbols (TRUE, FALSE) are wffs

WFFs Complex wffs (just as in PL, we can make more complicated sentences by combining predicates using connectives): §If A is a wff, so is  A §If A and B are wffs, so are A  B A  B A  B A  B

WFFs §If A is a wff and X is a variable then both  X(A) and  X(A) are wffs;  X (Person(X)  Mortal(X))  X (Student(X)  Smart(X))  X  y Loves(X, Y) §There are no other wffs. §Language: set of all wffs