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[Audio] TAUTOLOGY, CONTRADICTION & CONTINGENCY Tautology A compound statement that is always true, regardless of the truth values of its components, is called a tautology..

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[Audio] TAUTOLOGY, CONTRADICTION & CONTINGENCY Tautology The proposition 𝑝 Λ… ~𝑝 is a tautology as the following table illustrates. p ~𝒑 𝒑 Λ… ~𝒑 T F T F T T Also, the statement 𝑝 Λ… ~𝑝 is a tautology since it is always true..

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[Audio] TAUTOLOGY, CONTRADICTION & CONTINGENCY Contradiction A contradiction is a compound proposition that is always false..

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[Audio] TAUTOLOGY, CONTRADICTION & CONTINGENCY Contradiction The proposition 𝑝 Λ„ ~𝑝 is a contradiction as the following table illustrates. p ~𝒑 𝒑 Λ„~𝒑 T F F F T F Also, the statement 𝑝 Λ„ ~𝑝 is a contradiction since it is always false..

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[Audio] TAUTOLOGY, CONTRADICTION & CONTINGENCY Contingency A contingency is neither a tautology nor a contradiction. Example: The statement ( 𝑝 β†’ π‘ž Λ„ π‘ž) β†’ 𝑝 is a contingency..

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[Audio] CONSTRUCT A TRUTH TABLE AND DETERMINE IF THE FOLLOWING IS A TAUTOLOGY, CONTRADICTION OR CONTINGENCY. 1. 𝑝 Λ… ~ 𝑝 Λ„ π‘ž 2. (𝑝˄~π‘ž)Λ„(π‘Λ„π‘ž) 3. 𝑝 Λ…~𝑝 Λ„ π‘ž.

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[Audio] 1. 𝑝 Λ… ~ 𝑝 Λ„ π‘ž p q T T T F F T F F Therefore, 𝑝 Λ… ~ 𝑝 Λ„ π‘ž is a tautology..

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[Audio] 1. 𝑝 Λ… ~ 𝑝 Λ„ π‘ž p q 𝑝 Λ„ π‘ž ~ 𝑝 Λ„ π‘ž 𝑝 Λ… ~ 𝑝 Λ„ π‘ž T T T F T T F F T T F T F T T F F F T T Therefore, 𝑝 Λ… ~ 𝑝 Λ„ π‘ž is a tautology..

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[Audio] 2. (𝑝˄~π‘ž)Λ„(π‘Λ„π‘ž) p q T T T F F T F F Therefore,(𝑝˄~π‘ž)Λ„(π‘Λ„π‘ž)is a contradiction..

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[Audio] 2. (𝑝˄~π‘ž)Λ„(π‘Λ„π‘ž) p q ~π‘ž (𝑝˄~π‘ž) (π‘Λ„π‘ž) (𝑝˄~π‘ž)Λ„(π‘Λ„π‘ž) T T F F T F T F T T F F F T F F F F F F T F F F Therefore,(𝑝˄~π‘ž)Λ„(π‘Λ„π‘ž)is a contradiction..

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[Audio] 3. (𝑝 Λ…~𝑝) Λ„ π‘ž p q T T T F F T F F Therefore, (𝑝 Λ…~𝑝) Λ„ π‘ž is a contingency.

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[Audio] 3. (𝑝 Λ…~𝑝) Λ„ π‘ž p q ~𝑝 (𝑝 Λ…~𝑝) (𝑝 Λ…~𝑝) Λ„ π‘ž T T F T T T F F T F F T T T T F F T T F Therefore, (𝑝 Λ…~𝑝) Λ„ π‘ž is a contingency.

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[Audio] 4. (~𝑝 Λ„~π‘ž) β†’(π‘ŸΛ„~π‘ž) p q r T T T T T F T F T T F F F T T F T F F F T F F F.

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[Audio] 4. (~𝑝 Λ„~π‘ž) β†’(π‘ŸΛ„~π‘ž) is a contingency p q r ~𝑝 ~π‘ž (~𝑝 Λ„~π‘ž) (π‘ŸΛ„~π‘ž) (~𝑝 Λ„~π‘ž) β†’(π‘ŸΛ„~π‘ž) T T T F F F F T T T F F F F F T T F T F T F T T T F F F T F F T F T T T F F F T F T F T F F F T F F T T T T T T F F F T T T F F.

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[Audio] EQUIVALENT STATEMENTS Equivalent statements, denoted by ≑, are statements whose truth values is always either both true or both false whenever they have identical truth tables..

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[Audio] EQUIVALENT STATEMENTS Example: 𝑝 β†’ π‘ž 𝑖𝑠 π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘‘π‘œ ~π‘ž β†’ ~𝑝 p q 𝑝 β†’ π‘ž ~π‘ž ~𝑝 ~π‘ž β†’ ~𝑝 T T T F F T T F F T F F F T T F T T F F T T T T.

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[Audio] EQUIVALENT STATEMENTS Another example: Determine if each of the following is equivalent by constructing their truth tables. 1. ~ 𝑝 Λ„ π‘ž and ~𝑝 Λ… ~π‘ž 2. ~ 𝑝 Λ… π‘ž and ~𝑝 Λ„~π‘ž 3. 𝑝 β†’ π‘ž and ~𝑝 Λ… π‘ž 4. 𝑝 β†’ π‘ž Λ„ (π‘ž β†’ 𝑝) and 𝑝 π‘ž.

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[Audio] EQUIVALENT STATEMENTS 1. ~ 𝑝 Λ„ π‘ž π‘Žπ‘›π‘‘ ~𝑝 Λ… ~π‘ž p q T T T F F T F F.

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[Audio] EQUIVALENT STATEMENTS 1. ~ 𝑝 Λ„ π‘ž π‘Žπ‘›π‘‘ ~𝑝 Λ… ~π‘ž p q 𝑝 Λ„ π‘ž ~ 𝑝 Λ„ π‘ž ~𝑝 ~π‘ž ~𝑝 Λ… ~π‘ž T T T F F F F T F F T F T T F T F T T F T F F F T T T T Hence, ~ 𝑝 Λ„ π‘ž ≑ ~𝑝 Λ… ~π‘ž.

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[Audio] EQUIVALENT STATEMENTS 2. ~ 𝑝 Λ… π‘ž π‘Žπ‘›π‘‘ ~𝑝 Λ„~π‘ž p q 𝑝 Λ… π‘ž ~ 𝑝 Λ… π‘ž ~𝑝 ~π‘ž ~𝑝 Λ„~π‘ž T T T F F F F T F T F F T F F T T F T F F F F F T T T T Hence, ~ 𝑝 Λ… π‘ž ≑ ~𝑝 Λ„~π‘ž.

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[Audio] EQUIVALENT STATEMENTS 3. 𝑝 β†’ π‘ž π‘Žπ‘›π‘‘ ~𝑝 Λ… π‘ž p q 𝑝 β†’ π‘ž ~𝑝 ~𝑝 Λ… π‘ž T T T F T T F F F F F T T T T F F T T T Hence, 𝑝 β†’ π‘ž ≑ ~𝑝 Λ… π‘ž.

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[Audio] EQUIVALENT STATEMENTS 4. (𝑝 β†’ π‘ž)Λ„ π‘ž β†’ 𝑝 and 𝑝 π‘ž p q (𝑝 β†’ π‘ž) (π‘ž β†’ 𝑝) (𝑝 β†’ π‘ž)Λ„(π‘ž β†’ 𝑝) 𝑝 π‘ž T T T T T T T F F T F F F T T F F F F F T T T T Hence,(𝑝 β†’ π‘ž)Λ„(π‘ž β†’ 𝑝) ≑ 𝑝 π‘ž.

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[Audio] FORMS OF CONDITIONAL PROPOSITIONS Given propositions 𝑝 and π‘ž. There are three propositions that we can derive from the conditional 𝑝 β†’ π‘ž, namely, its 1. converse: π‘ž β†’ 𝑝 2. Contrapositive: ~π‘ž β†’ ~𝑝 3. Inverse: ~𝑝 β†’ ~π‘ž.

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[Audio] FORMS OF CONDITIONAL PROPOSITIONS Truth table:.

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[Audio] FORMS OF CONDITIONAL PROPOSITIONS Consider the following true conditional: 𝑝 β†’ π‘ž: "If today is Saturday, then it is a weekend." State its (a) converse, (b) contrapositive, and (c) inverse, and determine whether each statement is true. Solution: (a) Converse: "If today is a weekend, then it is Saturday." (b) Contrapositive: "If today is not a weekend, then it is not Saturday." (c) Inverse: "If today is not Saturday, then it is not a weekend.".

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[Audio] Symbolic Arguments An argument is an assertion that a given series of 𝑠 𝑃1,𝑃2, … , π‘ƒπ‘š called premises yields (has a consequence) another statement Q, called the conclusion. The premises are intended to demonstrate or at least provide some evidences for the conclusion..

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[Audio] Symbolic Arguments Example: Some of the following are arguments. Identify their premises and conclusions. 1. He's a Libra, since he was born in the last week of September. Answer: Premise: He was born in the last week of September. Conclusion: He's a Libra..

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[Audio] Symbolic Arguments 2. He was breathing and therefore alive. Answer: Premise: He was breathing. Conclusion: He was alive..

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[Audio] Symbolic Arguments 3. Can I go now? Answer: Not an argument 4. Nikki is my niece because her mother is my younger sister. Answer: Premise: Her mother is my younger sister. Conclusion: Nikki is my niece..

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[Audio] Symbolic Arguments The conclusion indicators, an expression prefixed to a sentence to indicate that it states a conclusion. The premise indicators, an expression prefixed to a sentence to indicate that it states a premise. therefore consequently then for since It is a fact that As shown by the thus hence implies given that fact that so For the reason that Granted that.

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[Audio] VALID OR NOT? If I will study for the examination, then I will get a passing score. I studied for the examination. Therefore, I got a passing score. If I will study for the examination, then I will get a passing score. I got a passing score. Therefore, I studied for the examination..

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[Audio] Valid Arguments An argument is said to be valid when all the premises are true it forces the conclusion to be true. An argument which is not valid is called an invalid argument or fallacy..

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[Audio] Valid Arguments Theorem: The argument consisting of the premises 𝑃1,𝑃2, … , 𝑃𝑛 and conclusion Q is valid if and only if the proposition 𝑃1 Λ„ 𝑃2 Λ„ … Λ„ 𝑃𝑛 β†’ 𝑄 is a tautology..

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[Audio] List of SomeValid Arguments 1. Law of Detachment (also called modus ponens) Symbolically, the argument is written: Premise 1: 𝑝 β†’ π‘ž Premise 2: 𝑝____ ∴ π‘ž.

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[Audio] 1. Law of Detachment (also called modus ponens) [(𝑝 β†’ π‘ž) Λ„ 𝑝] β†’ π‘ž p q 𝑝 β†’ π‘ž (𝑝 β†’ π‘ž) Λ„ 𝑝 [(𝑝 β†’ π‘ž) Λ„ 𝑝] β†’ π‘ž T T T T T T F F F T F T T F T F F T F T The truth table above shows that we have a valid argument, since the compound statement is a tautology..

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[Audio] List of SomeValid Arguments 2. Law of Contraposition (also called modus tollens) Symbolically, the argument is written: Premise 1: 𝑝 β†’ π‘ž Premise 2: ~π‘ž____ ∴ ~𝑝.

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[Audio] 2. Law of Contraposition (also called modus tollens) [(𝑝 β†’ π‘ž) Λ„ ~π‘ž] β†’ ~𝑝 𝑝 π‘ž 𝑝 β†’ π‘ž ~π‘ž (𝑝 β†’ π‘ž) Λ„~π‘ž ~𝑝 [(𝑝 β†’ π‘ž) Λ„ ~π‘ž] β†’ ~𝑝 T T T F F F T T F F T F F T F T T F F T T F F T T T T T The truth table above shows that we have a valid argument, since the compound statement is a tautology..

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[Audio] List of SomeValid Arguments 3. Law of Syllogism Symbolically, the argument is written: Premise 1: 𝑝 β†’ π‘ž Premise 2: π‘ž β†’ π‘Ÿ____ ∴ 𝑝 β†’ π‘Ÿ.

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[Audio] 3. Law of Syllogism 𝑝 π‘ž π‘Ÿ 𝑝 β†’ π‘ž π‘ž β†’ π‘Ÿ 𝑝 β†’ π‘ž Λ„ (π‘ž β†’ π‘Ÿ) 𝑝 β†’ π‘Ÿ [ 𝑝 β†’ π‘ž Λ„ (π‘ž β†’ π‘Ÿ)] β†’ (𝑝 β†’ π‘Ÿ) T T T T T T T T T T F T F F F T T F T F T F T T T F F F T F F T F T T T T T T T F T F T F F T T F F T T T T T T F F F T T T T T The truth table above shows that we have a valid argument, since the compound statement is a tautology..

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[Audio] List of SomeValid Arguments 4. Rule of Disjunctive Syllogism Symbolically, the argument is written: Premise 1: 𝑝 Λ… π‘ž Premise 2: ~𝑝____ ∴ π‘ž.

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[Audio] 4. Rule of Disjunctive Syllogism [(𝑝 Λ… π‘ž) Λ„ ~𝑝] β†’ π‘ž 𝑝 π‘ž 𝑝 Λ… π‘ž ~𝒑 (𝑝 Λ… π‘ž) Λ„~𝒑 [(𝑝 Λ… π‘ž) Λ„ ~𝑝] β†’ π‘ž T T T F F T T F T F F T F T T T T T F F F T F T The truth table above shows that we have a valid argument, since the compound statement is a tautology..

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[Audio] List of SomeValid Arguments Example: Consider the following arguments. Verify if it is valid or invalid. Identify the valid argument used. a. If Mark finishes his homework, then he can watch a movie. Mark finishes his homework. Therefore, Mark can watch a movie. Let p be "Mark finishes his homework", and q be " Mark can watch a movie.' Clearly, by Law of Detachment, this argument is valid..

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[Audio] List of SomeValid Arguments Example: Consider the following arguments. Verify if it is valid or invalid. Identify the valid argument used. b. If Mark finishes his homework, then he can watch a movie. Mark cannot watch a movie. Therefore, Mark did not finish his homework. Let p be "Mark finishes his homework", and q be " Mark can watch a movie.' Clearly, by Law of Contraposition, this argument is valid..

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[Audio] List of SomeValid Arguments Example: Consider the following arguments. Verify if it is valid or invalid. Identify the valid argument used. c. If it rains today, I will wear my rain jacket. If I will wear my rain jacket, I will keep dry. Therefore, if it rains today, I will keep dry. Let p be "It rains today", and q be " I will wear my rain jacket." and r be "I will keep dry". Clearly, by Law of Syllogism, this argument is valid..

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[Audio] List of SomeValid Arguments d. Gregorio's pencil is in his bag or it is on his table. Gregorio's pencil is not in his bag. Therefore, Gregorio's pencil is on his table. Let p be "Gregorio's pencil is in his bag", and q be "It is on his table." Clearly, by Rule of Disjunctive Syllogism, this argument is valid..

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[Audio] List of SomeValid Arguments e. If Mark finishes his homework, then he can watch a movie. Mark watches a movie Therefore, he finishes his homework. Let p be "Mark finishes his homework", and q be " Mark can watch a movie.".

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[Audio] List of SomeValid Arguments Truth Table of [(𝑝 β†’ π‘ž) Λ„ π‘ž] β†’ 𝑝: p q 𝑝 β†’ π‘ž (𝑝 β†’ π‘ž) Λ„ π‘ž [(𝑝 β†’ π‘ž) Λ„ π‘ž] β†’ 𝑝 T T T F F T F F Using the truth table the argument is invalid..

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[Audio] List of SomeValid Arguments Truth Table: p q 𝒑 β†’ 𝒒 (𝒑 β†’ 𝒒) Λ„ 𝒒 [(𝒑 β†’ 𝒒) Λ„ 𝒒] β†’ 𝒑 T T T T T T F F F T F T T T F F F T F T Using the truth table the argument is invalid..

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[Audio] List of SomeValid Arguments f. If Mark finishes his homework, then he can watch a movie. If he watches a movie, then he will buy pack of popcorn. If Mark finishes his homework, then he will buy a pack of popcorn. Let p be "Mark finishes his homework", and q be " Mark can watch a movie." and r be "Mark will buy a pack of popcorn." Clearly, by Law of Syllogism, this argument is valid..

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[Audio] Rules of Inferences Law of Detachment (Modus Ponens) Law of Contraposition (Modus Tollens) Law of Syllogism Rule of Disjunctive Syllogism.