Section 1.2 Propositional Logic
25
3 For more puzzles about “knights” and “knaves,” see What Is the Name of This Book? by the logician—and magician—Raymond Smullyan
(Prentice-Hall, 1978).
In Exercises 61–64, you are traveling in a certain country where every inhabitant is either a truthteller who
always tells the truth or a liar who always lies.
3
61. You meet two of the inhabitants of this country, Percival and Llewellyn. Percival says, “At least one of us
is a liar.” Is Percival a liar or a truth teller? What about Llewellyn? Explain your answer.
62. Traveling on, you meet Merlin and Meredith. Merlin says, “If I am a truth teller, then Meredith is a truth
teller.” Is Merlin a liar or a truth teller? What about Meredith? Explain your answer.
63. Next, you meet Rothwold and Grymlin. Rothwold says, “Either I am a liar or Grymlin is a truth teller.” Is
Rothwold a liar or a truth teller? What about Grymlin? Explain your answer.
64. Finally, you meet Gwendolyn and Merrilaine. Gwendolin says, “I am a liar but Merrilaine is not.” Is
Gwendolyn a liar or a truth teller? What about Merrilaine?
S e c t i o n 1 . 2 ProPositional logic
The argument of the defense attorney at the beginning of this chapter made a
number of (supposedly true) statements and then asked the jury to draw a specific
conclusion based on those statements. In Section 1.1, we used the notation of formal logic to represent statements in symbolic form as wffs; because statements
are sometimes called propositions, these wffs are also called propositional wffs.
Now we want to use tools from formal logic to see how to reach logical conclusions based on given statements. The formal system that uses propositional wffs
is called propositional logic, statement logic, or propositional calculus. (The
word calculus is used here in the more general sense of “calculation” or “reasoning,” not “differentiating” or “integrating.”)
Valid Arguments
An argument can be represented in symbolic form as
P 1 ` P 2 ` P 3 ` … ` P n S Q
where P 1 , P 2 , … , P n are the given statements, called the hypotheses, of the argument, and Q is the conclusion of the argument. As usual, the P’s and the Q represent wffs, not merely statement letters. When should this be considered a valid
argument? This question can be stated in several equivalent ways:
• When can Q be logically deduced from P 1 , … , P n ?
• When is Q a logical conclusion from P 1 , … , P n ?
• When does P 1 , … , P n logically imply Q?
• When does Q follow logically from P 1 , … , P n ?
and so forth.
25
3 For more puzzles about “knights” and “knaves,” see What Is the Name of This Book? by the logician—and magician—Raymond Smullyan
(Prentice-Hall, 1978).
In Exercises 61–64, you are traveling in a certain country where every inhabitant is either a truthteller who
always tells the truth or a liar who always lies.
3
61. You meet two of the inhabitants of this country, Percival and Llewellyn. Percival says, “At least one of us
is a liar.” Is Percival a liar or a truth teller? What about Llewellyn? Explain your answer.
62. Traveling on, you meet Merlin and Meredith. Merlin says, “If I am a truth teller, then Meredith is a truth
teller.” Is Merlin a liar or a truth teller? What about Meredith? Explain your answer.
63. Next, you meet Rothwold and Grymlin. Rothwold says, “Either I am a liar or Grymlin is a truth teller.” Is
Rothwold a liar or a truth teller? What about Grymlin? Explain your answer.
64. Finally, you meet Gwendolyn and Merrilaine. Gwendolin says, “I am a liar but Merrilaine is not.” Is
Gwendolyn a liar or a truth teller? What about Merrilaine?
S e c t i o n 1 . 2 ProPositional logic
The argument of the defense attorney at the beginning of this chapter made a
number of (supposedly true) statements and then asked the jury to draw a specific
conclusion based on those statements. In Section 1.1, we used the notation of formal logic to represent statements in symbolic form as wffs; because statements
are sometimes called propositions, these wffs are also called propositional wffs.
Now we want to use tools from formal logic to see how to reach logical conclusions based on given statements. The formal system that uses propositional wffs
is called propositional logic, statement logic, or propositional calculus. (The
word calculus is used here in the more general sense of “calculation” or “reasoning,” not “differentiating” or “integrating.”)
Valid Arguments
An argument can be represented in symbolic form as
P 1 ` P 2 ` P 3 ` … ` P n S Q
where P 1 , P 2 , … , P n are the given statements, called the hypotheses, of the argument, and Q is the conclusion of the argument. As usual, the P’s and the Q represent wffs, not merely statement letters. When should this be considered a valid
argument? This question can be stated in several equivalent ways:
• When can Q be logically deduced from P 1 , … , P n ?
• When is Q a logical conclusion from P 1 , … , P n ?
• When does P 1 , … , P n logically imply Q?
• When does Q follow logically from P 1 , … , P n ?
and so forth.