1 Chapter 7 Propositional and Predicate Logic

2 Chapter 7 Contents (1) l What is Logic? l Logical Operators l Translating between English and Logic l Truth Tables l Complex Truth Tables l Tautology l Equivalence l Propositional Logic

3 Chapter 7 Contents (2) l Deduction l Predicate Calculus Quantifiers  and  l Properties of logical systems l Abduction and inductive reasoning l Modal logic

4 What is Logic? l Reasoning about the validity of arguments. l An argument is valid if its conclusions follow logically from its premises – even if the argument doesn’t actually reflect the real world: nAll lemons are blue nMary is a lemon nTherefore, Mary is blue.

5 Logical Operators l AndΛ l OrV l Not ¬ l Implies → (if… then…) l Iff ↔ (if and only if)

6 Translating between English and Logic l Facts and rules need to be translated into logical notation. l For example: nIt is Raining and it is Thursday: R Λ T nR means “It is Raining”, T means “it is Thursday”.

7 Translating between English and Logic l More complex sentences need predicates. E.g.: nIt is raining in New York: nR(N) nCould also be written N(R), or even just R. l It is important to select the correct level of detail for the concepts you want to reason about.

8 Truth Tables l Tables that show truth values for all possible inputs to a logical operator. l For example: l A truth table shows the semantics of a logical operator.

9 Complex Truth Tables l We can produce truth tables for complex logical expressions, which show the overall value of the expression for all possible combinations of variables:

10 Tautology The expression A v ¬A is a tautology. l This means it is always true, regardless of the value of A. A is a tautology: this is written ╞ A l A tautology is true under any interpretation. l An expression which is false under any interpretation is contradictory.

11 Equivalence l Two expressions are equivalent if they always have the same logical value under any interpretation: A Λ B  B Λ A l Equivalences can be proven by examining truth tables.

12 Some Useful Equivalences l A v A  A l A Λ A  A l A Λ (B Λ C)  (A Λ B) Λ C l A v (B v C)  (A v B) v C l A Λ (B v C)  (A Λ B) v (A Λ C) l A Λ (A v B)  A l A v (A Λ B)  A l A Λ true  AA Λ false  false l A v true  trueA v false  A

13 Propositional Logic l Propsitional logic is a logical system. l It deals with propositions. l Propositional Calculus is the language we use to reason about propositional logic. l A sentence in propositional logic is called a well-formed formula (wff).

14 Propositional Logic l The following are wff’s: l P, Q, R… l true, false l (A) l ¬A l A Λ B l A v B l A → B l A ↔ B

15 Deduction l The process of deriving a conclusion from a set of assumptions. l Use a set of rules, such as: AA → B B l (Modus Ponens) l If we deduce a conclusion C from a set of assumptions, we write: l {A 1, A 2, …, A n } ├ C

16 Deduction - Example

17 Predicate Calculus l Predicate Calculus extends the syntax of propositional calculus with predicates and quantifiers: nP(X) – P is a predicate. l First Order Predicate Calculus (FOPC) allows predicates to apply to objects or terms, but not functions or predicates.

18 Quantifiers  and  l  - For all: n xP(x) is read “For all x’es, P (x) is true”. l  - There Exists: n x P(x) is read “there exists an x such that P(x) is true”. l Relationship between the quantifiers: n xP(x)  ¬(x)¬P(x) n“If There exists an x for which P holds, then it is not true that for all x P does not hold”.

19 Properties of Logical Systems l Soundness: Is every theorem valid? l Completeness: Is every tautology a theorem? l Decidability: Does an algorithm exist that will determine if a wff is valid? l Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions?

20 Abduction and Inductive Reasoning l Abduction: BA → B A l Not logically valid, BUT can still be useful. l In fact, it models the way humans reason all the time: nEvery non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. l Not valid reasoning, but likely to work in many situations.

21 Modal logic l Modal logic is a higher order logic. l Allows us to reason about certainties, and possible worlds. l If a statement A is contingent then we say that A is possibly true, which is written: ◊A l If A is non-contingent, then it is necessarily true, which is written:  A