Logic and Truth Tables
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1 Logic and ruth ables What is a ruth able? A truth table is a tool that helps you analyze statements or arguments in order to verify whether or not they are logical, or true. here are five basic operations that you will utilize when creating a truth table. hese operations are the conjunction, disjunction, negation, conditional, and bi-conditional. hese operations are also referred to as and, or, not, if-then, and if and only if. he rules for these operations are as follows: Logic Operations AND (conjunction) And Statements hese statements are true only when both p and q are true (as the rigorous definition of and implies.) p q p q Example: I will bring both a pen AND a pencil to the tutoring session. Only if I bring both is this true. OR Or Statements hese statements are false only when both (disjunction) p and q are false (follows the definition of or.) p q p q Example: I will bring a pen OR a pencil to the tutoring appointment. Only if I don t do either is this false. NO ~ (negation) Not Statements he not is simply the opposite or complement of its original value. p ~p Example: p = It is raining ~p = It is NO raining I-HEN If hen Statements hese statements are false only when p is (conditional) true and q is false (because anything can follow from a false premise.) p q p q Example: I I am elected HEN taxes will go down. Only if I am elected and taxes don t go down is this false. Provided by the Academic Center for Excellence 1 Logic and ruth ables Reviewed August 2017
2 I AND ONLY I (bi-conditional) p q p q If and Only If Statements hese statements are true only when both p and q have the same truth values. Example: axes will go down I AND ONLY I I am elected. Only if I am elected and taxes go down, or I am not elected and taxes do not go down is this true. Constructing ruth ables o create a truth table, follow these steps: 1. Determine the number of variables; for n variables, create a table with 2 n rows. If there are two variables (p, q), then you will need 2 2 or 4 rows. If there are three variables (p, q, and r), you will need 2 3 or 8 rows. 2. List the variable and every combination of and for the given variables. WO VARIABLE HREE VARIABLE p q p q r 3. hen start with negations ( nots or ~ ), create a new column for each piece of the statement or argument, filling in the truth values as you go. Work from simple pieces to more difficult pieces until you have the truth values for the whole problem. 4. If two statements have the same truth values, then they are equivalent. Equivalents here are a number of equivalents in logic. his means that these statements have been proven true, and you can use these statements without having to prove them. he symbol for equivalent is. he following are the most commonly used equivalents: p p p p p p p q ~p q p q ~q ~p p p p p p p p ~p p ~p p (q r) (p q) (p r) ~(p q) ~p ~q DeMorgan s Law #1 ~(p q) ~p ~q DeMorgan s Law #2 p (q r) (p q) (p r) Provided by the Academic Center for Excellence 2 Logic and ruth ables
3 ruth able Example Statement: (p q) (~p q) p q Step 1: Determine the number of variables and rows needed, then write down all possible combinations of p and q. p q Step ~p 2: p q ~p q Write down values of ~p (or not p ) p Step q 3: ~p p q p q ~p q Left Side: Is (p q) true? Use values from p and q columns. p Step q 4: ~p p q ~p q p q ~p q Right Side: Is (~p q) true? Use values from ~p and q columns. p Step q 5: ~p p q ~p q p q ~p q p q ~p q Is the entire statement true? No. he entire statement is true only when the last column s truth values are all rue. In this case, (p q) is not equivalent to (~p q) because they do not have the same truth values. If every cell in the last column is rue, then the entire statement is true. his is called a tautology. Provided by the Academic Center for Excellence 3 Logic and ruth ables
4 ruth ables for Arguments A logical argument is made up of two parts: the premises and the conclusion. Arguments are usually written in the following form: Arguments of this form can be turned into a logical statement. irst, assign variables to each premise and conclusion. It is cold = p It is not cold = ~p My motorcycle will start = q My motorcycle will not start = ~q You can now re-write the argument using logical operators. Look for keywords like if and not. p ~q q ~p inally, the argument can be rewritten as a logical statement. Arguments can always be written in the following form: [Premise One Premise wo] Conclusion Our example would be rewritten as: [ (p ~q) q ] ~p his statement can then be proven valid or invalid using a truth table, as shown on the following page. Provided by the Academic Center for Excellence 4 Logic and ruth ables
5 ruth able Example Statement: [ (p ~q) q ] ~p p q Step 1: Determine the number of variables and rows needed, then write down all possible combinations of p and q. p q ~p Step 2: [ (p ~q) q ] ~p Write down values of ~p (or not p ) p Step q 3: ~p ~q [ (p ~q) q ] ~p Write down values of ~q (or not q ) Step 4: (p ~q) ~q) q ] ~p Left Side: Start within the inner most parentheses. Is (p ~q) true? Use values from p and ~q columns. p q ~p ~q [ (p Step 5: p q ~p ~q [ (p ~q) (p q ] ~q) ~p q Left Side: Work outwards towards the outer most parentheses. Is [ (p ~q) q ] true? Use values from (p ~q) and q columns. Step p q 6: ~p ~q (p ~q) (p ~q) q [ (p ~q) q ] [ (p ~q) q ] ~p ~p Is the entire statement true? Yes. his is a tautology Provided by the Academic Center for Excellence 5 Logic and ruth ables
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