Session 18

First Order Logic

Rules of Inference

Two common rules for reasoning over predicate logic representations rely on a simple rule from propositional logic involving implication, extended for reasoning with variables:

Modus Ponens (p ^ (p->q) -> q )

man(Marcus) p

x man(x) implies person(x) p -> q

person(Marcus) q

Modus Tolens ( q' ^ (p->q) -> p' )

not( person(Lassie) ) q'

x man(x) implies person(x) p -> q

not( man(Lassie) ) not p

Consider

hoosier(Schafer)

x hoosier(x) -> likes (x,Basketball)

x,y childOf(x,y) AND likes (y, Basketball) -> likes (x,Basketball)

y likes(y,Basketball) -> likes(y,March)

daughterOf(Margaret,Schafer)

x,y daughterOf( x, y )-> childOf( x, y )
hoosier(Wallingford)
hoosier(Letterman)
daughterOf(Sarah,Wallingford)

Universal Instantiation (UI) states that we can infer any sentence obtained by substituting a ground term for the variable

(S1) SUBST({x/Schafer),2) yields hoosier(Schafer) -> likes(Schafer,Basketball)
(S2) SUBST({y/Schafer, x/Margaret),3) yields childOf(Margaret,Schafer) AND likes(Schafer,Basketball -> likes (Margaret,Basketball)
(S3) SUBST({y/Schafer},4) yields likes(Schafer,Basketball -> likes (Schafer,March)
(S4) SUBST({y/Schafer, x/Margaret),5) yields daughterOf(Margaret,Schafer) -> childOf(Margaret,Schafer)

Thus, the reason we can conclude likes(Margaret,March) is not immediately related to the demonstration we on Monday, but instead is related to:

likes( Schafer, Basketball ) [mp 1,S1]

childOf( Margaret, Schafer ) [mp 5,S4]

likes( Margaret, Basketball ) [mp 8,7,S2]

likes( Margaret, March ) [mp 9,S4]

What is the problem with UI? Well, if there are too many ground terms, you end up with too much information...

SUBST({x/Wallingford),2) yields hoosier(Wallingford) -> likes(Wallingford,Basketball)
SUBST({x/Margaret),2) yields hoosier(Margaret) -> likes(Margaret,Basketball)
SUBST({x/March),2) yields hoosier(March) -> likes(March,Basketball)
SUBST({x/Basketball),2) yields hoosier(Basketball) -> likes(Fienup,Basketball)
SUBST({y/Wallingford, x/Margaret),3) yields childOf(Margaret,Wallingford) AND likes(Wallingford,Basketball -> likes (Margaret,Basketball)
SUBST({y/Wallingford, x/Walingford),3) yields childOf(Wallingford,Wallingford) AND likes(Wallingford,Basketball -> likes (Wallingford,Basketball)

The fact is, with the 7 ground terms we have above (Schafer, Margaret, Wallingford, Sarah, Letterman, March, Basketball), 2 unary universal rules, and 2 binary universal rules, we end up with

2x7 + 2x7² = 112 derived rules

An Exercise in Using Logic to Reason Using Forward Chaining

The forward chaining of this is fairly straightforward and is not reproduced here.

An Exercise in Using Logic to Reason Using Backward Chaining

Conclude

likes(Margaret,March)

In 4, bind "y=Margaret"

This is valid if we can prove that likes(Margaret,Basketball) is true.

We could arrive at this by either showing

hoosier(Margaret) (rule #2) or
finding some z where childOf(Margaret,z) AND likes (z, Basketball) (rule #3, remember, variable names are arbitrary.

We are unable to use any rule/fact to show the first part.

We can arrive at childOf(Margaret,z) if we can show DaugherOf(Margaret,z) (rule #7).

We can arrive at likes(z,Basketball) if we can show hoosier(z) (rule #2)

Rule #1 allows us to bind (z=Schafer).

Rule #5 allows us to bind (z=Schafer).

Thus, we can conclude likes(Schafer,Basketball) and childOf(Margaret,Schafer).

Thus, we can conclude likes(Margaret,Basketball).

Thus, we can conclude likes(Margaret,March).

Related to UI is Existential Instantiation (EI). This is more complicated. It states that we can infer any sentence obtained by substituting a constant symbol k that does not appear elsewhere in the knowledge base for the variable:

SUBST({y / K),9) yields hoosier(K) AND NOT likes(K,Indiana)

As long as K does not exist somewhere else in the KB