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 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
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