Computing Truth Values SYMBOLIC LOGIC Definition An assertion is a statement. A proposition is a statement or a declarative sentence which is either true or false, but not both. If a proposition is true we assign the truth value “TRUE” to it. If a proposition is false, we assign the truth value “FALSE” to it. We will denote by “T” or “1”, for the truth value TRUE and by “F" or “0” for the truth value FALSE. SYMBOLIC LOGIC Examples The following are examples of propositions: 1. 2 > 4 2. The billionth prime, when written in base 10, ends in a 3. 3. All men are mortals. 4. 9 is a prime number. SYMBOLIC LOGIC Non-Examples The following are non-examples of propositions: 1. x < y 2. Factor ๐ฅ 2 + 2๐ฅ + 1. 3. ๐ฅ = 10 4. How old are you? SYMBOLIC LOGIC Exercises *Decide whether the following are propositions or not: 1. 2. 3. 4. 5. 23 = z 10 – 7 = 3 5 < 27 All women are mammals. Where do you live? SYMBOLIC LOGIC Definition A propositional variable, denoted by ๐, ๐, ๐ … denotes an arbitrary proposition with an unspecified truth value. A propositional variable is a variable that represents a proposition. SYMBOLIC LOGIC Definition Given two propositional variables ๐ and ๐. These two propositional variables maybe combined to form a new one. These are combined using the logical operators or logical connectives : “and”, “or” or “not”. SYMBOLIC LOGIC These new proposition are: 1. (Conjunction of P and Q) P and Q, denoted by ๐ ๐; 2. (Disjunction of P and Q) P or Q, denoted by ๐ ๐; 3. (Negation of P) not P, denoted by ¬ ๐. Truth Tables ๐ ¬๐ 1 0 0 1 ๐ 1 1 0 0 ๐ 1 0 1 0 ๐ ๐ 1 0 0 0 ๐ ๐ 1 1 1 0 Note: Other books represent the negation of P as ~P. SYMBOLIC LOGIC Inclusive and exclusive disjunction: 1. Inclusive disjunction denoted by ๐ ๐ is True when either or both of the disjuncts are True. 2. Exclusive disjuction denoted by P โ Q is True when only one of the disjuncts is true and the other is false. Truth Table: ๐ 1 1 0 0 ๐ 1 0 1 0 ๐ ๐ PโQ 1 0 1 1 1 1 0 0 SYMBOLIC LOGIC Definition The proposition “P implies Q”, denoted by ๐โนQ is called an implication. The operand P is called the hypothesis, premise or antecedent while the operand Q is called the conclusion or the consequence ๐ 0 0 1 ๐ 0 1 0 ๐โนQ 1 1 0 1 1 1 SYMBOLIC LOGIC Definition Given the implication ๐โนQ , its converse is ๐โนP , its inverse is¬๐โน¬Q, and its contrapositive is ¬๐โน¬P . The operand P is called the hypothesis, premise or antecedent and the operand Q is called the conclusion or the consequence ๐ ๐ ๐โนQ ¬๐โน¬Q ๐โนP ¬๐โน¬P 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 1 SYMBOLIC LOGIC Definition A biconditional proposition is expressed linguistically by preceding either component by ‘if and only if’. The truth table for a biconditional propositional form, symbolised by P โท Q is shown below. ๐ ๐ 0 0 1 1 0 1 0 1 PโทQ 1 0 0 1 SYMBOLIC LOGIC Exercises I. 1. q = “You miss the final exam.” r = “You pass the course.” Express q ๏ ~r in English. 2. Construct a truth table for ~p ๏ ~q. Exercises II. Let p and q be propositions p: 4 is a rational number. q: √3 is an irrational number. Express each of these propositions as an English sentence: 3) ~p 6) p ๏ q 4) p v q 7) p ๏ซ q 5) p แดง q Lecture 1 14 Exercises III. Given: p: 4 is an even integer. q: 5 is an odd integer. Write each of the ff. in terms of p, q and logical connectives: 8) 4 is an even integer and 5 is an odd integer. 9) 4 is not an even integer or 5 is an odd integer. 10) If 4 is an even integer then 5 is an odd integer. 11) 4 is an even integer whenever 5 is an odd integer. 12) 4 is not an even integer if and only if 5 is not an odd integer. Lecture 1 15 Do Worksheets 2 & 3 Definition A propositional form is an assertion which contains at least one propositional variable and maybe generated by the following rules: 1. A propositional variable standing alone is a propositional form; 2. If P is a propositional form, then Q is also a propositional form; 3. If P and Q are propositional forms, then ๐ ๐, ๐ ๐, ๐ โบ ๐ are propositional forms; 4. A string of symbols containing propositional variables, connectives and parentheses is a propositional form if and only if it can be obtained by infinitely many applications of rules (1.); (2.) or (3.) above. SYMBOLIC LOGIC Definition Let X be a set of propositions. A truth assignment (to X) is a function ๏ด : X ๏ฎ {true, false} that assigns to each propositional variable a truth value. (A truth assignment corresponds to one row of the truth table. If a truth value of a compound proposition under truth assignment ๏ด is true, we say that ๏ด satisfies P, otherwise we say that ๏ด falsifies P. A tautology is a propositional form where every truth assignment satisfies P, i.e. All entries of its truth table are true. A contradiction or absurdity is a propositional form where every truth assignment is false; A contingency is a propositional form that is neither tautology nor contradiction. Examples: P V ๏P is a tautology. P ๏ ๏P is a contradiction. For each of the following compound propositions determine if it is a tautology, contradiction or contingency: 1. (p v q) ๏ ๏p ๏ ๏q 2. P v q v r v (๏p ๏ ๏q ๏ ๏r) 3. (p ๏ฎ q) ๏ซ (๏p v q) Definition A logically equivalent propositional form have identical values for each assignment of the truth values to their component propositional variables. We can denote the logical equivalent variables P and Q as: ๐ โบ๐ or ๐ ≡๐ (๐ฅ + 2)2 and ๐ฅ 2 + 4๐ฅ + 4 are regarded as equivalent algebraic expressions. SYMBOLIC LOGIC Example Show that P โน Q and ¬P โ Q are logically equivalent propositional forms. ๐ 0 0 1 1 ๐ PโนQ ¬P ¬P โ Q 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 P โน Q and ¬P โ Q are logically equivalent propositional forms. SYMBOLIC LOGIC Example Given the propositional forms Q โ ¬P, ¬Q โน ¬P and ¬P โ ¬Q, between which pairs of these forms does the relation logical equivalence exist? ๐ ๐ ¬P ¬Q Q โ ¬P ¬Q โน ¬P ¬P โ ¬Q, 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 1 1 1 ¬Q โน ¬P and ¬P โ ¬Q are logically equivalent SYMBOLIC LOGIC The following are logical identities or rules of replacement. SYMBOLIC LOGIC SYMBOLIC LOGIC 12. Exportation ((P ส Q) โน R) โบ (P โน (Q โน R)) SYMBOLIC LOGIC SYMBOLIC LOGIC Example Show that ¬(Q โ P) โบ P โน¬Q. Solution ¬(Q โ P) โบ โบ โบ ¬Q โ¬P ¬P โ¬Q P โน¬Q (De Morgan’s) (Commutativity) (MI) SYMBOLIC LOGIC Example Show that P โ [(P โ Q) โ R] โบ P โ (Q โ R). Solution P โ [(P โ Q) โ R] โบ โบ โบ โบ [P โ (P โ Q)] โ (P โ R) (Dist) [(P โ P) โ Q] โ (P โ R) (Assoc) (P โ Q) โ (P โ R) (Indempotence) P โ (Q โ R) (Dist) SYMBOLIC LOGIC Do Worksheet 4 SYMBOLIC LOGIC Definition An argument is a collection of propositions wherein it is claimed that one of the propositions, called the conclusion, follows from the other propositions, called the premise of the argument. the conclusion is usually preceded by such words as therefore, hence, then, consequently. Classification of Arguments: 1. Inductive argument is an argument where it is claimed that within a certain probability of error, the conclusion follows from a premise; and 2. Deductive argument is an argument where is it claimed that the conclusion absolutely follows from the premise. SYMBOLIC LOGIC A deductive argument is said to be valid if whenever the premises are all true, then the conclusion is also true. In other words if ๐1 , ๐2 , … ๐๐ are premises and Q is the conclusion of the argument ๐1 ๐๐๐ ๐2 , ๐๐๐ … ๐๐ therefore Q is valid if and only if the corresponding prepositional form (๐1 ๐2 … ๐๐ ) โน ๐, is a tautology. Otherwise, the argument is invalid. SYMBOLIC LOGIC To show that an argument is invalid, we have to show an instance where the conclusion is false and the premises are all true. Show that the following argument is invalid using Truth Table. SYMBOLIC LOGIC To show the validity of arguments, we may use the truth table. However, this method is impractical specially if the argument contains several propositional variables. A more convenient method is by deducing the conclusion from the premises by a sequence of shorter, more elementary arguments known to be valid. SYMBOLIC LOGIC Rules of Inference These are known valid argument forms. SYMBOLIC LOGIC SYMBOLIC LOGIC SYMBOLIC LOGIC SYMBOLIC LOGIC Construct a formal proof of validity of the following arguments: a) Jack is in Paris only if Mary is in New York. Jack is in Paris and Fred is in Rome. Therefore, Mary is in New York. b) If Mark is correct then unemployment will rise and if Ann is correct then there will be a hard winter. Ann is correct. Therefore unemployment will rise or there will be a hard winter or both. SYMBOLIC LOGIC Solution for (a): J: M: F: Jack is in Paris. Mary is in New York. Fred Is in Rome. The premises of the argument are J โน M and J โ F. The conclusion is M. 1. J โน M 2. J โ F 3. J 4. M (premise) (premise) (2. Simp) (1, 3. MP) SYMBOLIC LOGIC Solution for (b): M: U: A: H: Mark is correct. Unemployment will rise. Ann is correct. There will be a hard winter. The premises of the argument are: (M โน U) โ (A โน H) and A. The conclusion is: U โ H. 1. (M โน U) โ (A โน H) 2. A 3. (A โน H) โ (M โน U) 4. A โน H 5. H 6. H โ U 7. U โ H (premise) (premise) (1. Comm) (3. Simp) (4, 2. MP) (5. Add) (6. Comm) SYMBOLIC LOGIC Alternative Solution for (b): M: U: A: H: Mark is correct. Unemployment will rise. Ann is correct. There will be a hard winter. The premises of the argument are: (M โน U) โ (A โน H) and A. The conclusion is: U โ H. 1. (M โน U) โ (A โน H) 2. A 3. A โ M 4. M โ A 5. U โ H (premise) (premise) (2. Add) (3. Comm) (1, 4. CD) SYMBOLIC LOGIC Do Worksheet 5 SYMBOLIC LOGIC