1. Formal Theories SIE 550 Lecture Matt Dube Doctoral Student - Spatial

2. 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 ) predicateconstantvariable

3. Today’s Class Formal Theories Logical Outputs Boolean Operators and Modifiers Truth Tables Mathematical Laws Logical Relations Axioms and Theorems

4. 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

5. 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

6. 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. StatementNegated 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.

7. 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.

8. Truth Table for NOT PNot P TRUEFALSE TRUE opposite Only true if condition is false

9. Truth Table for AND PQP AND Q TRUE FALSE TRUEFALSE Only true if both conditions are true

10. Truth Table for OR PQP OR Q TRUE FALSETRUE FALSETRUE FALSE True if at least one condition is true

11. Truth Table for = (If and only if) PQP = Q TRUE FALSE TRUEFALSE TRUE Only true if both conditions are the same

12. Truth Table for If P, then Q (IMPLIES) PQIf P, then Q TRUE FALSE TRUE FALSE TRUE The uncertainty table: anything can happen

13. Boolean Modifiers Two more relevant terms –∀–∀ For All –∃–∃ There Exists

14. 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

15. 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.

16. 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.

17. 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.

18. Logical Relations Converse –Q implies P Inverse –not P implies not Q Contrapositive –not Q implies not P

19. 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

20. Types of axioms Logical Axioms –Association Rules Non-logical Axioms –All other axioms Ground Axioms –Non-logical Axioms that contain all constants

21. 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