Logic PDF

SH1902 Logic Propositions A proposition is a declarative statement that can be evaluated either true or false but cannot be both. A simple proposition consists of one (1) declarative sentence or statement. A compound proposition consists of two (2) or...

SH1902 Logic Propositions A proposition is a declarative statement that can be evaluated either true or false but cannot be both. A simple proposition consists of one (1) declarative sentence or statement. A compound proposition consists of two (2) or more simple propositions joined together by logical operators. Logical Operators Logical operators are words that either: combine two (2) or more simple propositions to form a new compound proposition; or modify the meaning of a proposition. Propositions can be denoted by variables (usually uppercase). Consider two (2) simple propositions below denoted 𝑃𝑃 and 𝑄𝑄. 𝑃𝑃: The animal barks. 𝑄𝑄: The animal is a dog. A. Conjunction: Uses the word “and” to join together two (2) propositions Symbol: ∧ Example: 𝑃𝑃 ∧ 𝑄𝑄 Read as: 𝑃𝑃 AND 𝑄𝑄 (The animal barks AND it is a dog.) TRUE: When ALL its components are TRUE FALSE: When AT LEAST ONE (1) of its components is FALSE B. Disjunction: Uses the word “or” to join together two (2) propositions Symbol: ∨ Example: 𝑃𝑃 ∨ 𝑄𝑄 Read as: 𝑃𝑃 OR 𝑄𝑄 (The animal barks or it is a dog.) TRUE: When AT LEAST ONE (1) of its components is TRUE FALSE: When ALL of its components are FALSE C. Implication: Uses “if-then” to construct a new proposition from two (2) propositions In an implication, the first proposition is called the premise while the second is called the conclusion. Symbol: → Example: 𝑃𝑃 → 𝑄𝑄 Read as: IF 𝑃𝑃, THEN 𝑄𝑄 (IF the animal barks, THEN it is a dog.) 09 Handout 1 *Property of STI  [email protected] Page 1 of 4  SH1902 (Premise: The animal barks; Conclusion: The animal is a dog.) TRUE: When the premise is FALSE, or when the premise and the conclusion are BOTH TRUE FALSE: When the premise is TRUE but the conclusion is FALSE D. Biconditional: Uses “if and only if” or “is equivalent to” to construct a new proposition from two (2) propositions A biconditional statement simply states that two propositions are equivalent, that is, if the first one is true, then the second must also true, and if the second is true, then the first must also be true. Symbol: ↔ Example: 𝑃𝑃 ↔ 𝑄𝑄 Read as: 𝑃𝑃 IF AND ONLY IF 𝑄𝑄 (The animal barks IF AND ONLY IF it is a dog.) TRUE: When BOTH propositions have the same truth value, that is, when BOTH are TRUE, or when BOTH are FALSE FALSE: When the propositions have opposite truth value, that is, one is TRUE but the other is FALSE E. Negation: Precedes a proposition with the word “not” Symbol: ∼ Example: ∼ 𝑃𝑃 Read as: The animal does NOT bark. TRUE: When the proposition is FALSE FALSE: When the proposition is TRUE TRUTH TABLES 𝑃𝑃 𝑄𝑄 𝑃𝑃 ∧ 𝑄𝑄 𝑃𝑃 ∨ 𝑄𝑄 𝑃𝑃 → 𝑄𝑄 𝑃𝑃 ↔ 𝑄𝑄 ∼ 𝑃𝑃 ∼ 𝑄𝑄 T T T T T T F F T F F T F F F T F T F T T F T F F F F F T T T T Tautologies and Contradictions A tautology is a proposition that is always true under any circumstance. On the other hand, a contradiction is always false under any circumstance. In determining whether a proposition is a tautology, a contradiction, or neither, truth tables are used. Example: Let 𝑃𝑃 and 𝑄𝑄 be simple propositions. Determine which of these compound propositions are tautology/ies, contradiction/s, or neither. a. 𝑃𝑃 ∨ (∼ 𝑃𝑃) c. 𝑃𝑃 ∨ (∼ 𝑄𝑄) b. 𝑄𝑄 ∧ (∼ 𝑄𝑄) d. (𝑃𝑃 ∨ 𝑄𝑄) ∧ [(∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄)] 09 Handout 1 *Property of STI  [email protected] Page 2 of 4  SH1902 Solutions: a. 𝑃𝑃 ∨ (∼ 𝑃𝑃) 𝑃𝑃 ∼ 𝑃𝑃 𝑃𝑃 ∨ (∼ 𝑃𝑃) T F T F T T Conclusion: It is a tautology. b. 𝑄𝑄 ∧ (∼ 𝑄𝑄) 𝑄𝑄 ∼ 𝑄𝑄 𝑄𝑄 ∧ (∼ 𝑄𝑄) T F F F T F Conclusion: It is a contradiction. c. 𝑃𝑃 ∨ (∼ 𝑄𝑄) 𝑃𝑃 𝑄𝑄 ∼ 𝑄𝑄 𝑃𝑃 ∨ (∼ 𝑄𝑄) T T F T T F T T F T F F F F T T Conclusion: It is NEITHER a tautology nor a contradiction. d. (𝑃𝑃 ∨ 𝑄𝑄) ∧ [(∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄)] 𝑃𝑃 𝑄𝑄 ∼ 𝑃𝑃 ∼ 𝑄𝑄 𝑃𝑃 ∨ 𝑄𝑄 (∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄) (𝑃𝑃 ∨ 𝑄𝑄) ∧ [(∼ 𝑃𝑃) ∧ (∼ 𝑄𝑄)] T T F F T F F T F F T T F F F T T F T F F F F T T F T F Conclusion: It is a contradiction. Rules of inference are rules that provide the way of drawing a correct conclusion from a given premise. With these rules are syllogisms that draw correct conclusion from two (2) or more premises. Examples: If 𝑄𝑄 is the consequence of 𝑃𝑃, and 𝑃𝑃 happens, then 𝑄𝑄 also happens. (This rule of inference is known as Modus Ponens.) 09 Handout 1 *Property of STI  [email protected] Page 3 of 4  SH1902 If 𝑄𝑄 is the consequence of 𝑃𝑃, and 𝑄𝑄 did not happen, then 𝑃𝑃 does not happen. (This rule of inference is known as Modus Tollens.) The classmate I have without a pen is Mike. The classmate who borrowed my pen is my only classmate without a pen. Therefore, the person who borrowed my pen is Mike. All dolphins are mammals. All mammals have kidneys. Therefore, all dolphins have kidneys. (The last two (2) arguments in which the conclusion absolutely follows from the given premises is called syllogism.) A fallacy is a kind of reasoning in which the conclusion does not necessarily or logically follow from the premise. Hence, it is also known as faulty, invalid, or erroneous reasoning. Examples: In a class of 50 students, 35 receive daily allowance worth above P100. Hence, all 50 students receive allowance above P100. (The error of attributing to the whole what is observed to some is known as the fallacy of composition.) If you do anything you want, then you will find joy in life. (The error of failing to give logical connection between the premise and the conclusion, but rather the arguments appeal to one’s emotion is known as the fallacy of relevance.) References: Chua, R., Ubarro, A., & Wu, Z. (2016). Soaring 21st century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Fernando, O. (2016) Next century mathematics (general mathematics). Quezon City: Phoenix Publishing House. Lim, Y., Nocon E., Nocon, R., & Ruivivar L. (2016). Math for engaged learning (general mathematics). Quezon City: Sibs Publishing House. Melosantos, L. (2016). Math connections in the digital age (general mathematics). Quezon City: Sibs Publishing House. Regacho, C., Benjamin, M., & Oryan, S. (2017). Mathematics skills for life. Quezon City: Abiva Publishing House, Inc. Zorilla, R. (2016). General mathematics for senior high school. Malabon City: Mutya Publishing House. 09 Handout 1 *Property of STI  [email protected] Page 4 of 4