Chapter 7. Propositional and Predicate Logic Comp3710 Artificial Intelligence Computing Science Thompson Rivers University

Propositional/Predicate Logic Course Outline Part I – Introduction to Artificial Intelligence Part II – Classical Artificial Intelligence, and Searching Knowledge Representation Searching Search Methodologies Advanced Search Genetic Algorithms (relatively new study area) Knowledge Represenation and Automated Reasoning Propositinoal and Predicate Logic Inference and Resolution for Problem Solving Rules and Expert Systems Part III – Machine Learning Part IV – Advanced Topics TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Chapter Objectives TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Chapter Outline Propositional logic Introduction What is logic? Why is logic used in Artificial Intelligence? How to use logical operators How to translate an English statement with logic notations Let’s recall complex truth tables Let’s recall tautology and contradictory How to use equivalent propositions How to logically use propositions – propositional logic Introduction to predicate calculus Summary TRU-COMP3710 Propositional/Predicate Logic

1. Introduction [Q] What are expert systems? [Q] What is reasoning? [Wikipedia] Reason is the capacity for consciously making sense of things, applying logic, for establishing and verifying facts, and changing or justifying practices, institutions, and beliefs based on new or existing information (facts and/or observation). To form conclusions, inferences, or judgments [Q] How to automate reasoning? Need to know How to represent information, knowledge, facts and beliefs, and How to apply logic, …

[Q] How to formalize/validate our arguments? Argument = premises (propositions or statements) + conclusion To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true. [Q] Any logical forms for valid arguments?

Examples Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate an error message, then the program syntax is correct and program execution does not result in division by zero. Argument 2: If x is a real number such that x < -2 or x > 2, then x2 > 4. For y, such that y2 /> 4, y /< -2 and y /> 2. The common logical form of both of the above arguments: If p or q, then r. Not r. Therefore not p and not q. Is this logical form valid?

Propositional/Predicate Logic Topics You may recall “Logic and Truth Tables” in MATH 1380. In this unit, Boolean logic Propositional logic Introduction to predicate calculus – first-order predicate logic Propositional logic will be used in the following unit to solve some interesting problems. Based on rules, knowledge, facts, and observation Decide if a given query is valid. TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics 2. What is Logic? Logic: reasoning about the validity of arguments. An argument is valid if its conclusions follow logically from its premises (proposition or statement) – even if the argument doesn’t actually reflect the real world: Mary is a lemon. All lemons are blue. Therefore, Mary is blue. - conclusion Logic is widely used as a representation method of AI, and allows us to easily reason about negatives (i.e., “NOT”) and disjunctions (,i.e., “OR”) However, one of the main weaknesses of traditional logic (i.e., Boolean logic) is its inability to deal with uncertainty and ambiguity. Later probabilistic method and fuzzy logic will be discussed to deal with uncertainty and ambiguity. premises TRU-COMP3710 Propositional/Predicate Logic

3. How to Use Logical Operators Definition of statement A statement (or proposition) is a sentence that is true or false but not both. Examples Two plus two equals four. 2 + 2 = 4 I am a TRU student. x + y > 0 ??? TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Compound Statements Symbols used in complicated logical statements: ~ not ~p negation of p  and p  q conjunction of p and q  or p  q disjunction of p and q  exclusive or p  q Order of operations: ( ) and ~ have the higher precedence. ~p  q = (~p)  q ~(p  q) TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics And (conjunction)  Or (disjunction)  Not (negation)  or ~ Implies (conditional)  (if… then…) Iff (biconditional)  (if and only if) Exclusive OR? TRU-COMP3710 Propositional/Predicate Logic

4. Translating between English and Logic Facts, observations and rules need to be translated into logical notations. For example: It is Raining and it is Thursday: R  T, where R represents “It is Raining”, T represents “it is Thursday”. R and T are called propositional symbols. TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic More complex sentences need predicates. That part of a proposition that is affirmed or denied about the subject. For example, in the proposition We are mortal, mortal is the predicate. E.g., It is raining in New York: R(N) Could also be written N(R), or even just R. [Q] How to express “It is not raining in New York”??? It is important to select the correct level of detail for the concepts you want to reason about. TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics Example It is not hot but it is sunny. It is neither hot nor sunny. -> It is not hot, and it is sunny. It is not hot, and it is not sunny. Let h = “it is hot” and s = “it is sunny.” Then the above statements can be translated as ~h  s ~h  ~s Suppose x is a particular real number. Let p, q, and r symbolize “0 < x,” “x < 3,” and “x = 3.” respectively. Then the following inequalities x  3 0 < x < 3 0 < x  3 can be translated as q  r p  q p  (q  r) TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic 5. Truth Tables Tables that show truth values for all possible inputs to a logical operator. For example: [Q] Truth table for implication () ??? A  B: A is the antecedent, and B is the consequent. A  B  A  B [Q] Can you prove it? How? A  B  B  A [Q] Can you prove it? How? [Q] Truth table for iff () ??? TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics We can produce truth tables for complex logical expressions, which show the overall value of the expression for all possible combinations of variables: TRU-COMP3710 Propositional/Predicate Logic

6. Tautology and Contradictory Topics 6. Tautology and Contradictory A tautology is true under any interpretation. The expression A ˅ ¬A is a tautology. This means it is always true, regardless of the value of A. P is a tautology: this is written ╞ P An expression which is false under any interpretation is contradictory (or unsatisfiable). A  ¬A Some expressions are satisfiable, but not always valid. This means that they are true under some interpretation, but not under all interpretations. A  B TRU-COMP3710 Propositional/Predicate Logic

7. How to Use Equivalent Propositions Two expressions are equivalent if they always have the same logical value under any interpretation: A ˄ B  B ˄ A [Q] How to prove the above equivalence? Equivalences can be proven by examining truth tables. [Q] Are there more equivalences? TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic A ˅ A  ??? A ˄ A  ??? A ˄ (B ˄ C)  (A ˄ B) ˄ C A ˅ (B ˅ C)  (A ˅ B) ˅ C A ˄ (B ˅ C)  (A ˄ B) ˅ (A ˄ C) A ˅ (B ˄ C)  ??? A ˄ (A ˅ B)  ??? A ˅ (A ˄ B)  ??? A ˄ true  ??? A ˄ false  ??? A ˅ true  ??? A ˅ false  ??? [Q] DeMorgans’ Laws ??? [Q] Why do we need these equivalences? By using the above and other equivalences, logical expressions can be simplified. TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics Some more TRU-COMP3710 Propositional/Predicate Logic

8. How to Use Propositions Propositional logic in this chapter is a logical system. It deals with propositions. Propositional calculus is the language we use to reason about propositional logic. A sentence in propositional logic is called a well-formed formula (wff) (or sentence). TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Propositional calculus: BNF (Backus-Naur Form) – The following are wff’s: P, Q, R… – propositional symbols true, false (A) ¬A A ˄ B A ˅ B A → B A ↔ B Any combination of wff’s is a wff. [Q] Is P  Q  (B  C)  A  B  D  (E) a wff ??? TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Deduction: the process of deriving a conclusion from a set of assumptions. Will be discussed in the following unit again to solve some problems. If we deduce a conclusion C from a set of assumptions, we write: {A1, A2, …, An} ├ C If C can be concluded without any assumption ├ C The inference rule A ├ B is expressed as A B Given A, B is deduced (or concluded). The meaning is that if A is true (or valid), then B is true (or valid). [Q] What if A is false? B can be true or false. TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic  introduction Given A and B, we can deduce A  B. A, B {A, B} ├ A ˄ B A ˄ B  introduction A_ A  B  elimination A  B A  B A B  elimination (called Modus Ponens) A, A  B [Q] Can you prove? B   elimination   A A Some valid inference rules TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Reduction to absurdity (proof by contradiction)  A . _ Contradiction, i.e., false A  Induction (called deduction theorem) C__ A  C Some valid inference rules TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics  Introduction A__ B  A Modus Tollens ~B, A  B ~A Some valid inference rules TRU-COMP3710 Propositional/Predicate Logic

9. Introduction to Predicate Calculus Predicate Calculus extends the syntax of propositional calculus with predicates and quantifiers: P(X) – P is a predicate. First Order Predicate Calculus (FOPC) allows predicates to apply to objects or terms, but not functions or predicates. Just introduction in this unit. Predicate calculus is used in solving more complex problems. TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Quantifiers  and   - For all: x P(x) is read “For all x’es, P (x) is true”. E.g., for all TRU COMP students, they are smart.  - There Exists: x P(x) is read “there exists an x such that P(x) is true”. E.g., there is a TRU COMP student who is not smart. Relationship between the quantifiers: x P(x)  ¬ ((x)¬P(x)) “If There exists an x for which P holds, then it is not true that for all x P does not hold”. TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics Properties of Logical Systems Four factors to consider of: Soundness: Is every theorem valid? Completeness: Is every tautology a theorem? Decidability: Does an algorithm exist that will determine if a wff is valid? Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions? TRU-COMP3710 Propositional/Predicate Logic

Propositional/Predicate Logic Topics 10. Summary Propositional logic Propositions Boolean logic Deduction; inference rules TRU-COMP3710 Propositional/Predicate Logic