Formal Theories

SIE 550 Lecture

Matt Dube

Doctoral Student - Spatial

Recap of Friday and Monday

• Formal Languages– Terminal and Non-terminal Symbols– Well Formed Formulas

• First Order Languages– Pathway to Computer Logic– Backus-Naur Operators– Predicates, Constants, Variables

mother( linda , X )

predicate constant variable

Today’s Class

• Formal Theories

• Logical Outputs

• Boolean Operators and Modifiers

• Truth Tables

• Mathematical Laws

• Logical Relations

• Axioms and Theorems

Formal Theory

• Language for association• Built on a foundation of primary assertions

– Assumed to be true– father(henry,susan)

• Rules imposed to infer information– Mechanisms for inference– pgrandfather(X,A)=father(X,Z)father(Z,A)

• Statements to prove– pgrandfather(steve,susan)?

Translation: Henry is Susan’s father

Logical Outputs

• Binary Output– TRUE– FALSE

• Multi-valued Output– TRUE– FALSE– MAYBE

• Fuzzy Logic– % of truth

Black and white form

More human form

Statistical form

Boolean Algebra

• AND– All terms are true

• OR– At least one term is true

• NOT– Term is false

• =– Both terms have the same truth value

• IMPLIES– Both true, or first statement false

Today is Monday and I am in class.

I am here or I am not here.

Statement Negated

I am an open set = My complement is a closed set.

If I am in Orono then I am in Maine.

If I am in Ohio then I am in Maine.

Truth Tables

• Status of terms• Status under the operators• Can be simple or complex• Equivalent logical results are equivalent

statements• If all values in a truth column are TRUE,

this is a tautology• If all values are FALSE, this is a

contradiction.

Truth Table for NOT

P Not P

TRUE FALSE

FALSE TRUE

opposite

Only true if condition is false

Truth Table for AND

P Q P AND Q

TRUE TRUE TRUE

TRUE FALSE FALSE

FALSE TRUE FALSE

FALSE FALSE FALSE

Only true if both conditions are true

Truth Table for OR

P Q P OR Q

TRUE TRUE TRUE

TRUE FALSE TRUE

FALSE TRUE TRUE

FALSE FALSE FALSE

True if at least one condition is true

Truth Table for = (If and only if)

P Q P = Q

TRUE TRUE TRUE

TRUE FALSE FALSE

FALSE TRUE FALSE

FALSE FALSE TRUE

Only true if both conditions are the same

Truth Table for If P, then Q (IMPLIES)

P Q If P, then Q

TRUE TRUE TRUE

TRUE FALSE FALSE

FALSE TRUE TRUE

FALSE FALSE TRUE

The uncertainty table: anything can happen

Boolean Modifiers

• Two more relevant terms– ∀

• For All

– ∃• There Exists

Laws

• Idempotent Laws – Intersection and Union• Identity Laws – Equality• Complement Laws – Opposites• Commutative Laws – Reversal• Associative Laws – Arbitrary Grouping• Distributive Laws – Multiplication• Absorption Laws• DeMorgan’s Rules – Distributing Not• Modus Ponens• Modus Tollens • Modus Barbara

Modus Ponens

• Latin– Mode that affirms by affirming

• Affirming the Antecedent

• Law of Detachment

• Example:– If today is Monday, then I have class.– Today is Monday– Therefore I have class.

Modus Tollens

• Latin– The way that denies by denying

• Denying the consequent

• Example:– If I am an archer, then I own a bow.– I don’t own a bow.– Therefore I am not an archer.

Modus Barbara

• Latin– To measure barbarously

• Coming to a conclusion based on successive implications and then strip the middle information

• Example:– If I learn more, then I know more.– If I know more, then I forget more.– If I forget more, then I know less.– Therefore:

• if I learn more, then I know less.

Logical Relations

• Converse– Q implies P

• Inverse– not P implies not Q

• Contrapositive– not Q implies not P

Components of Formal Theories

• Axioms– Base level facts – needs no proof– father(henry,susan)

• Association Rules– grandfather(X,Y)=father(X,Z)father(Z,Y)

• Theorems– father(X,susan) -> who is susan’s father– father(X,A) -> all fathers of all children

Types of axioms

• Logical Axioms– Association Rules

• Non-logical Axioms– All other axioms

• Ground Axioms– Non-logical Axioms that contain all constants

Example

• father(john,suzy)

• father(geoff,larry)

• father(robert,john)

• father(geoff,robert)

• grandfather(X,Y)::=father(X,Z)father(Z,Y)• ggrandfather(X,Y)::=father(X,W)father(W,Z)father(Z,Y)

grandfather(geoff,john)?

THEOREM