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1 FUNDAMENTALS OF LOGIC NO.2 PROPOSITION AND TRUTH VALUE Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/

Proposition A Proposition is a statement of which truth does not change (i.e. always true or always false). `1 < 2' `There are infinitely many prime numbers.' `A triangle has equal edges.' `Any even number can be expressed as a sum of two prime numbers.' (Gold Bach's Conjecture) `Taro likes Hanako.' `The headquarter of Keio University is at SFC.' `Keio University is a national university.' If a statement contains a variable, it is not a proposition. The truth may change depending on the variable. `x < 5' `Taro likes A.' 2

3 Propositional Variable A propositional variable represents a proposition which cannot be decomposed. It represents an atomic statement. p, q, r, Non atomic statements: A compound statement is not basic because it can be decomposed into smaller statements. `Taro likes Hanako and Hanako likes Taro.' `If the wind blows, the bucket makers make money.' `Taro comes to SFC using bus or bicycle'

4 Compound Proposition Propositions can be combined. A compound statement is composed of some atomic statements. There are four ways of composition: Connective Symbol Name Meaning Other Symbols and conjunction both hold & or disjunction one of them holds imply conditional under some condition, it holds not negation the reverse holds ~

5 Logical Formula A logical formula represents a compound proposition. Definition A propositional variable is a logical formula. If A and B are logical formulae, the followings are also logical formulae: (A B) (A B) (A B) ( A) Example (p q) (p (q ( r))) ( ((p q) (r p)))

6 Omission of Parenthesis Too many parentheses. Omit some of them if there is no confusion. Omit out most parentheses. Give priority to connectives. > > > and are left associative and is right associative. p q r p q r p q r (p q) r p q r p q r Example: ` ' has more priority than ` ', ` ' connects stronger than ` '. Therefore, ` p q' means `(( p) q)'. p q r means p p means p (q p) means

Creating Logical Formulae Let p represent `Taro likes Hanako', q represent `Taro likes Momoko' and r represent `Hanako likes Taro'. Write the following statements as logical formulae. `Taro likes both Hanako and Momoko.' 7 `Taro likes Hanako or Momoko.' `Taro likes Hanako, but Hanako does not like him.' `If Taro likes Hanako, Hanako also likes him.' `If Taro likes Hanako and not Momoko, Hanako likes him.' `Hanako likes Taro who likes Momoko.'

8 Truth Table A proposition has a value of true (T) or false (F). A proposition is either true or false, not both. The negation of true is false, and the negation of false is true. The truth value of a logical formula depends on the true value of propositional variables in the formula. If a formula consists of connecting two formulae with a logical connective, its truth value can be determined from the true value of two sub formulae. The following truth tables show truth value of each logical connective. A B A B A B A A B T F T T F A B T F T T T A B T F T T F A T A F F F F F T F F T T F T

9 Exclusive Or A B is true if A or B is true. When A and B are true, A B is true. exclusive or, xor Exclude the case when both A and B are true. It is true when only one of A or B is true. A B A B T F T T T F T F A B A B T F T F T F T F A B is also used for exclusive or.

Calculating Truth Value of a Formula When a formula A contains propositional variables p 1, p 2,..., p n, the truth value of A can be calculated from the truth value of p 1, p 2,..., p n. Starting form the truth value of p 1, p 2,..., p n, using the truth value table of each connective, the truth value of the formula can be calculated. Example Truth value of p q p 10 p q p q p p q p T T F T T T F F F F F T T T T F F T T T

11 Exercise Calculate the truth value of p (q p). p q q p (q p) p (q p) T T T F F T F F

12 Tautology A is a tautology if A is always true no matter what the truth value of propositional variables p 1, p 2,..., p n inside A. A is also called valid. There are 2 n comination of truth value of p 1, p 2,..., p n. By checking all the cases, we can determine whether A is a tautology or not. Theorem: It is decidable whether a given logical formula is a tautology or not. Example: p (p q) q is a tautology. p q p q p (p q) p (p q) q T T T T T T F F F T F T T F T F F T F T

13 Exercise Show that p p is a tautology. p T F p p Show that (p q) ( q p) is also a tautology. p q p q q p q p (p q) ( q p) T T T F F T F F

14 Summary Mathematical Logic Proposition Propositional variables Logical connectives Logical formulae Truth value Truth table of logical connectives Tautologies

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