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Chapter 1: Problem 36

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \rightarrow q \equiv q \rightarrow p$$

Short Answer

Expert verified
The given statement \(p \rightarrow q \equiv q \rightarrow p\) is true since the expressions on both sides of the equivalence have the same truth values for every possible combination of true or false values for \(p\) and \(q\).

Step by step solution

01

Create Truth Tables for Expressions

First, we create the truth tables for both expressions: \(p \rightarrow q\) and \(q \rightarrow p\). For each expression, list all possible combinations of truth values for \(p\) and \(q\), and then calculate the output for each case. | \(p\) | \(q\) | \(p \rightarrow q\) | |-----|-----|-------------------| | T | T | T | | T | F | F | | F | T | T | | F | F | T | | \(q\) | \(p\) | \(q \rightarrow p\) | |-----|-----|-------------------| | T | T | T | | T | F | F | | F | T | T | | F | F | T |
02

Compare Truth Tables and Determine Equivalence

Now we compare the truth table columns for \(p \rightarrow q\) and \(q \rightarrow p\). If all the values are the same for each combination of \(p\) and \(q\), the two expressions are equivalent. | \(p\) | \(q\) | \(p \rightarrow q\) | \(q \rightarrow p\) | Equivalence? | |-----|-----|-------------------|-------------------|--------------| | T | T | T | T | T | | T | F | F | F | T | | F | T | T | T | T | | F | F | T | T | T | All the values for \(p \rightarrow q\) and \(q \rightarrow p\) are the same in every case.
03

State the Result

Since the expressions \(p \rightarrow q\) and \(q \rightarrow p\) are equivalent based on comparing their truth tables, we can mark the given statement as true: $$p \rightarrow q \equiv q \rightarrow p$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Truth Tables
Truth tables are a fundamental tool in logic and mathematics for determining the validity of logical expressions. They systematically tabulate all possible combinations of truth values that variables within a logical statement can have, and the corresponding output of the logical expression for each combination.

For instance, if we have two variables, such as in the conditional statement \(p \rightarrow q\), there are four possible combinations of truth values for \(p\) and \(q\) (both true, \(p\) true and \(q\) false, \(p\) false and \(q\) true, and both false). A truth table for \(p \rightarrow q\) will display the resulting truth value for each of these combinations.Truth tables are invaluable for understanding the relationships between different logical statements, for proving logical equivalences, and for checking whether a compound statement is a tautology or a contradiction.
Conditional Statements
Conditional statements form the backbone of logical reasoning, often phrased in an 'if-then' format. Such a statement typically reads as 'if \(p\), then \(q\),' symbolically written as \(p \rightarrow q\). The truth value of a conditional depends on the relationship between the premise (the 'if' part, \(p\)) and the conclusion (the 'then' part, \(q\)).

In the context of truth tables, we consider a conditional statement to be true except in the case where the premise is true and the conclusion is false. This is because a promise or a prediction made on false premises can't be considered broken, while one made on a true premise but leading to a false conclusion is.
Tautology
A tautology is a type of statement in logic that is always true, regardless of the truth values of its constituent variables. In other words, it's a statement that is true in every possible situation.

An example of a tautology would be the statement \(p \lor \eg p\), which translates to 'either \(p\) is true, or \(p\) is not true.' By the very definition of 'or' and the law of the excluded middle in classical logic, this statement is always true. Tautologies are essential in logical arguments to demonstrate arguments that are fundamentally valid, and they are considered 'trivially true' because they do not impart any new information about the variables involved.
Contradiction
In contrast to tautologies, a contradiction is a statement that is false under all circumstances. It's an assertion that simultaneously makes two opposing claims, such that it is impossible for the statement to be true in any conceivable scenario.

An example of a contradiction is the statement \(p \land \eg p\), meaning 'it is the case that \(p\) is true, and it is not the case that \(p\) is true at the same time.' Such a proposition defies the principle of non-contradiction, one of the classic laws of thought, which states that contradictory propositions cannot both be true. Recognizing contradictions is crucial for the identification of fallacious reasoning or inconsistent theories.

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Most popular questions from this chapter

Four women, one of whom was known to have committed a serious crime, made the following statements when questioned by the police: (B. Bissinger, Parade Magazine, 1993 ) $$\begin{array}{ll}{\text { Fawn: }} & {\text { "Kitty did it" }} \\ {\text { Kitty: }} & {\text { "Robin did it." }} \\ {\text { Bunny: }} & {\text { "I didn't do it" }} \\ {\text { Robin: }} & {\text { "Kitty lied." }}\end{array}$$ If exactly one of these statements is false, identify the guilty woman.

The Sheffer stroke / is a binary operator" defined by the following truth table.(Note: On page 25 we used the vertical bar \(|\) to mean is a factor of. The actual meaning should be clear from the context. So be careful.) Verify each. (Note: Exercise 58 shows that the logical operators \(|\) and \(\mathrm{NAND}\) are the same. (TABLE CAN'T COPY) $$p \rightarrow q \equiv((p | p)|(p | p))|(q | q)$$

There are seven lots, 1 through \(7,\) to be developed in a certain city. A builder would like to build one bank, two hotels, and two restaurants on these lots, subject to the following restrictions by the city planning board (The Official LSAT PrepBook, 1991): If lot 2 is used, lot 4 cannot be used. If lot 5 is used, lot 6 cannot be used. The bank can be built only on lot \(5,6,\) or \(7 .\) A hotel cannot be built on lot \(5 .\) A restaurant can be built only on lot \(1,2,3,\) or \(5 .\) If a restaurant is built on lot \(5,\) which of the following is not a possible list of locations? A. A hotel on lot 2 and lot 4 is left undeveloped. B. A restaurant on lot 2 and lot 4 is left undeveloped. C. A hotel on lot 2 and lot 3 is left undeveloped.

Construct a truth table for each proposition. $$(p \wedge q) \rightarrow(p \vee q)$$

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: x
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