CITS2211 Discrete Structures Propositional Logic Unit coordinator: Rachel Cardell-Oliver July 31, 2017
Highlights This lecture will address the following questions: 1 What is a proposition? 2 How are propositions combined? 3 How are propositions related to natural language and things in the real world such as computer programs? 4 How is the truth or falsity of a proposition decided?
Reading Mathematics for Computer Science Introduction Chapter 1, Section 1.1 Chapter 3, Section 3.1 Chapter 3, Section 3.2
Why study propositional logic English language has many ambiguities. This is tolerable in normal conversation but not if we wish to formulate precise ideas. Propositions and logical connectives arise all the time in computer programs. For example, in conditional statements or loops. Computer programs themselves can be reasoned about using logic. It is possible (although difficult) to prove a program is correct. Leading CPU chip manufacturers now routinely prove chip correctness to avoid mistakes like the notorious Intel division bug in the 1990s.
The first topic in cits2211 is logic. Definition (from www.oxforddictionaries.com/definition/english/logic): Reasoning conducted or assessed according to strict principles of validity.
Example: Warm up: You have 4 cards. Each card has a number on one side and a letter on the other. If the cards are dealt so that you can see K 5 2 J Which cards would you have to turn over, to test the rule that if there s a J on one side then there s a 5 on the other side? Give reasons for your answer.
Definition: A proposition is a statement that is either true or false. Example: All men are mortal Example: The earth is flat Example: 1+1 = 2 Example: The evergreen forests of Canada comprise spruce, pine and fir trees.
In English, we combine simple propositions into more complex ones using connectives. Example: The earth is not flat (not F) Example: The earth is flat or I am mad (F or M) Example: The earth is flat unless I am mad (F or M) Example: Your money or your life (M or L, but be aware this really means M xor L) Example: If all men are mortal and all Greeks are men then all Greeks are mortal (if MX and GM then GX) Example: All babies are illogical (B implies not L)
Example: It is raining today and today is winter. (R and W) Example: It is raining despite the season being winter. (R and W) Example: If I can afford it then I will buy it (A implies B) Example: I will buy it if I can afford it (A implies B) Example: I will buy it when I can afford it (A implies B) Example: I can afford it only if I buy it (A implies B)
The logic operators are: not and or implies (if-then) iff (if and only if) The meaning of these operators can be defined using a truth table.
P P T F F T P Q P Q T T T F F T F F P Q P Q T T T F F T F F
P P T F F T P Q P Q T T T T F F F T F F F F P Q P Q T T T T F T F T T F F F
P Q P Q T T T F F T F F P Q P Q T T T F F T F F
P Q P Q T T T T F F F T T F F T P Q P Q T T T T F F F T F F F T
Implication is important, remember an implication is true exactly when the if-part is false or the then-part is true. False implies anything, True can not imply false. Example: True: If pigs can oink then the sky is blue Example: False: If the sky is blue then the sky is red Example: True: If pigs can fly then the sky is red Example: True: If pigs can fly then the sky is blue
Propositional logic is a formal language A formal language consists of 1 A set of permitted symbols; its alphabet 2 Rules defining the correct order of symbol in sentences; its syntax 3 Assignment of meaning to correctly written sentences; its semantics
Here is the formal language of propositional logic: 1 The alphabet of propositional logic consists of symbols for denoting propositions called identifiers. For example, P, Q and R, punctuation symbols, that is brackets ( ), and propositional connectives,,,,. 2 The syntax of propositional logic is defined by the following rules for construction of well formed formulae (wff), which represent propositions: An identifier is a proposition. If P and Q are propositions then so are P, P Q, P Q, P Q and P Q. Note that arbitrarily complex propositions can be built recursively using these rules. 3 The semantics of a proposition (its truth value) is derived from the possible truth values of its sub-parts and the truth tables for connectives.
Which of the following are wff of propositional logic? Why? P Q Q P Q R (S P) P P P Q
Brackets can be used to resolve ambiguity. But over-using brackets can be irritating. Definition: Precedence of Symbols: B rackets N egation C onjunction D isjunction I mplication E quivalence Example: Add (redundant!) brackets to the following formula to show how it is interpreted under the operator precedence rules. P Q S T V W
Today s lecture will address the following questions: 1 How to use truth tables to evaluate compound propositions 2 Introduce some common rules of propositional logic 3 How to identify tautologies and equivalences
Definition: Two propositions are equivalent if they have the same truth value for all possible values of their identifiers. Here are some commonly used propositional equivalences. Each can be derived from a truth table (exercise). Expression Equivalent To Name of Rule P Q Q P Commutativity P Q Q P (P Q) R P (Q R) Associativity (P Q) R P (Q R) (P Q) P Q De Morgan s laws (P Q) P Q P Q P Q Implication P Q Q P Contrapositive P ( P) Double negation
Example: Derive De Morgan s law (P Q) equivalent to P Q using a truth table. LHS RHS P Q P Q (P Q) P Q P Q LHS RHS T T T F F T F F
Example: Derive the contrapositive law using a truth table. LHS RHS P Q P Q Q P Q P LHS RHS T T T F F T F F
Propositional Logic in Programs Propositional logic is used in computer programs. If ( x > 0 ( x <= 0 && y > 100 ) ) return true; else return false; Example: Use propositional logic to simplify this statement.
Definition: A tautology is a compound proposition which is true under all possible assignments of truth values to its prime propositions. Example: Show that P P is a tautology. Definition: A contradiction is a compound proposition which is false under all possible assignments of truth values to its prime propositions. Example: Show that P P is a contradiction.
Definition: An contingent proposition is one which is neither a tautology nor a contradiction. Example: Show that P P is a contingent proposition. Definition: Two propositions are logically equivalent if and only if their equivalence is a tautology. Example: Show that P Q is logically equivalent to P Q
Axioms of Propositional Logic (1) Expression Equivalent To Name of Rule P Q Q P Commutativity P Q Q P (P Q) R P (Q R) Associativity (P Q) R P (Q R) P (Q R) (P Q) (P R) Distributivity P (Q R) (P Q) (P R) (P Q) P Q De Morgan (P Q) P Q P P Double Negation P P T Excluded Middle P P F Contradiction
Axioms of Propositional Logic (2) Expression Equivalent To Name of Rule P Q P Q Implication P Q (P Q) (Q P) Equivalence P (P Q) P Or Absorption P P P P T T P F P P (P Q) P And Absorption P P P P T P P F F P P Identity
Axioms of Propositional Logic (3) The table of rules on the previous two pages (axioms) can be used to derive all tautologies. An even more elegant axiomatisation was discovered by Edward Vermilye Huntington in 1933. Expression Equivalent To Name of Rule P Q Q P Commutativity (P Q) R P (Q R) Associativity ( P Q) ( P Q) P Huntington axiom But note that these axioms are not very convenient for writing proofs. So you are recommended to use a wider range of laws.