Computing Truth Values SYMBOLIC LOGIC Definition An assertion
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
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
A propositional variable is a variable that represents a proposition. SYMBOLIC LOGIC
SYMBOLIC LOGIC
1 0 0 1 Note: Other books represent the negation of P as ~P. SYMBOLIC LOGIC
P⋁Q 0 1 1 0 SYMBOLIC LOGIC
SYMBOLIC LOGIC
1 0 1 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 ⟷ Q is shown below. P⟷Q SYMBOLIC LOGIC
Do Activities 3 & 4
SYMBOLIC LOGIC
Definition Let X be a set of propositions. A truth assignment (to X): X {true, false} that assigns to each propositional variable a truth value. (A truth assignment corresponds to one row of the truth table). A tautology is a propositional form where every truth assignment is true. 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 q) ( p v q)
SYMBOLIC LOGIC
Example Show that P ⟹ Q and ¬P ⋁ Q are logically equivalent propositional forms. P⟹Q 1 1 0 1 P ⟹ Q and ¬P ⋁ Q are logically equivalent propositional forms. SYMBOLIC LOGIC
Example Given the propositional forms Q ⋁ ¬P, ¬Q ⟹ ¬P and ¬P ⋀ ¬Q, between which pairs of these forms does the relation logical equivalence exist? Q ⋁ ¬P ¬P ⋀ ¬Q, 1 1 1 0 0 0 0 0 1 1 0 ¬Q ⟹ ¬P and Q V ¬P are logically equivalent SYMBOLIC LOGIC
The following are logical identities or rules of replacement. SYMBOLIC LOGIC
SYMBOLIC LOGIC
12. Exportation ((P ʌ Q) �R) �(P �(Q �R)) SYMBOLIC LOGIC
SYMBOLIC LOGIC
Example Show that ¬(Q ⋀ P) ⟺ P ⟹¬Q. Solution ¬(Q ⋀ P) ⟺ ⟺ ⟺ ¬Q ⋁¬P ¬P ⋁¬Q P ⟹¬Q (De Morgan’s) (Commutativity) (MI) SYMBOLIC LOGIC
Example Show that P ⋀ [(P ⋀ Q) ⋁ R] ⟺ P ⋀ (Q ⋁ R). Solution P ⋀ [(P ⋀ Q) ⋁ R] ⟺ ⟺ [P ⋀ (P ⋀ Q)] ⋁ (P ⋀ R) (Dist) [(P ⋀ P) ⋀ Q] ⋁ (P ⋀ R) (Assoc) (P ⋀ Q) ⋁ (P ⋀ R) (Idempotence) P ⋀ (Q ⋁ R) (Dist) SYMBOLIC LOGIC
Do Activity 5 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
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 ⟹ M and J ⋀ F. The conclusion is M. 1. J ⟹ M 2. J ⋀ F 3. J 4. M (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 ⟹ U) ⋀ (A ⟹ H) and A. The conclusion is: U ⋁ H. 1. (M ⟹ U) ⋀ (A ⟹ H) 2. A 3. (A ⟹ H) ⋀ (M ⟹ U) 4. A ⟹ H 5. H 6. H ⋁ U 7. U ⋁ H (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 ⟹ U) ⋀ (A ⟹ H) and A. The conclusion is: U ⋁ H. 1. (M ⟹ U) ⋀ (A ⟹ H) 2. A 3. A ⋁ M 4. M ⋁ A 5. U ⋁ H (premise) (2. Add) (3. Comm) (1, 4. CD) SYMBOLIC LOGIC
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