which is true unless both ' + ]; This assertion says nothing about the truth of q when p is false, true, no matter what p and q are. Paris is the capital of France. 'return(correct);'; The truth value of an implication is false if and only if its antecedent is true and its consequent is false; otherwise, the truth value is true. the third column gives the corresponding truth values of (p | q). ', '!, | and &. writeTextExercise(30, qCtr++, s); The truth table is ' + The logical operator | is analogous to addition in arithmetic. (∨) or by the word "or." ¬. then q is also true." var optPerm = randPermutation(rawOpt,"inverse"); document.writeln(qStr); and r '(!p) | ' + // -->. The proposition (p → q) is ['If the Sun orbits the Earth, then the Moon is made of cheese; ' + !, |, and &. Problem 2.4: Consider the sentences shown below. The truth assignments i and j defined in the preceding section correspond to the third and sixth rows of this table, respectively. In this case, j does not satisfy (p ∨ q) ∧ (¬q ∨ r). //alert(strArr[0]); + (p & p) and (p | p). the order of operations for evaluating compound propositions, The logical operation &, 2.1 Conjunction. 16 possibilities. 'equivalent to ' + strArr[0] + ' using only ' + The argument asserts that if these two premises are true, then the conclusion is p and q: The first two columns of the table show the truth values of and if p is false whenever q is false, and vice versa. '

' + var optPerm = randPermutation(rawOpt,"inverse"); If p is true, so are ' for (j=0; j'; the conclusion is true. The logical structure of the previous argument can be untangled in the following way. falseProps[whichFalse[1]] + ' & ' + trueProps[whichTrue[3]], if (qArr[which][1]) { Algebraic methods are designed to respect these relationships, independent of what the variables represent. var opt = ['no','yes']; } ['If the Sun orbits the Earth or the Moon is made of cheese, then ' + (p & q) is the intersection var qStr = 'What is the structure of the following argument?' + Here is the truth table for that compound proposition: This compound proposition is always true, no matter the values of p and We say that a truth assignment falsifies a sentence if and only if the sentence is false under that truth assignment. is the union of the set 'of cheese. The operation | is sometimes represented by a vee 'The proposition (' + qTxt[5][0] + ' ) is equivalent to ' + Therefore, this is an invalid argument. Which of the sentences would be true and which would be false? 'Supposedly, a colleague at the University ' + has a logically equivalent proposition that uses only the operations Note that the constituent sentences within any compound sentence can be either simple sentences or compound sentences or a mixture of the two. A logical argument consists of one or more '(truthValues[i],truthValues[j],truthValues[k]) != ' + 28 = 256 possible truth tables involving three basic propositions. proposition (q → p). associations follow from these three. As our first example, consider the English sentence If a person is cool or funny, then he is popular. var optPerm = randPermutation(parts[0][qN],"inverse");