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Chapter 2: Problem 12

For statements \(P, Q,\) and \(R\) : (a) Show that \([(P \rightarrow Q) \wedge P] \rightarrow Q\) is a tautology. Note: In symbolic logic, this is an important logical argument form called modus ponens. (b) Show that \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\) is a tautology. Note: In symbolic logic, this is an important logical argument form called Syllogism.

Short Answer

Expert verified
For part (a), the statement \([(P \rightarrow Q) \wedge P] \rightarrow Q\) is a tautology, as proven with a truth table containing only true values in the final column. For part (b), the statement \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\) is also a tautology, which was shown by creating a truth table with only true values in the final column.

Step by step solution

01

Create the truth table for part (a)

We will create a truth table with columns for \(P\), \(Q\), \(P \rightarrow Q\), \((P \rightarrow Q) \wedge P\), and \([(P \rightarrow Q) \wedge P] \rightarrow Q\). | \(P\) | \(Q\) | \(P \rightarrow Q\) | \((P \rightarrow Q) \wedge P\) | \([(P \rightarrow Q) \wedge P] \rightarrow Q\) | |---|---|--------------|---------------------|---------------------------| | T | T | T | T | T | | T | F | F | F | T | | F | T | T | F | T | | F | F | T | F | T | Since the final column contains only true values, the statement \([(P \rightarrow Q) \wedge P] \rightarrow Q\) is a tautology.
02

Create the truth table for part (b)

We will create a truth table with columns for \(P\), \(Q\), \(R\), \(P \rightarrow Q\), \(Q \rightarrow R\), \((P \rightarrow Q) \wedge(Q \rightarrow R)\), and \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\). | \(P\) | \(Q\) | \(R\) | \(P \rightarrow Q\) | \(Q \rightarrow R\) | \((P \rightarrow Q) \wedge(Q \rightarrow R)\) | \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\) | |---|---|---|--------------|--------------|---------------------------|-----------------------------------| | T | T | T | T | T | T | T | | T | T | F | T | F | F | T | | T | F | T | F | T | F | T | | T | F | F | F | T | F | T | | F | T | T | T | T | T | T | | F | T | F | T | F | F | T | | F | F | T | T | T | T | T | | F | F | F | T | T | T | T | Since the final column contains only true values, the statement \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\) is a tautology.

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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 fundamental tools in logic that allow us to visualize how different logical statements behave under various conditions. They display all possible truth values of logical variables and the results of logical operations performed on them.

For instance, if you have a statement like \(P \rightarrow Q\), which means "if \(P\) then \(Q\)," a truth table helps you understand when this statement is true or false.
  • Each row in a truth table represents a unique combination of truth values for the variables involved.
  • The columns represent the individual variables, the expressions to be evaluated, and the final outcome.
  • The last column is of particular interest, as it shows the final result of the logical operation defined by the statement.
By completing a truth table, like those shown for the exercises above, we can determine whether a complex logical expression is always true, that is, a tautology.
Modus Ponens
Modus ponens is a fundamental rule in formal logic, often referred to by its Latin name which means 'the way that affirms by affirming'. It's a basic logical argument that involves a conditional statement and its antecedent. The structure can be stated as:
  • \(P \rightarrow Q\): If \(P\), then \(Q\).
  • \(P\): \(P\) is true.
  • Therefore, \(Q\): \(Q\) is true.
In symbolic logic, modus ponens is essential because it allows us to infer new conclusions from established premises.
For example, if you know that "if it rains, the streets are wet" (\(P \rightarrow Q\)), and "it is raining" (\(P\)), then you can conclude "the streets are wet" (\(Q\)).

This form of reasoning is critical in proofs and in building complex logical arguments. In the exercise, showing that \([(P \rightarrow Q) \wedge P] \rightarrow Q\) is a tautology illustrates the correctness and universality of modus ponens.
Logical Syllogism
A logical syllogism is another important form of argument that takes two premises to arrive at a conclusion. In classical logic, it usually takes the form of:
  • \(P \rightarrow Q\): If \(P\), then \(Q\).
  • \(Q \rightarrow R\): If \(Q\), then \(R\).
  • Therefore, \(P \rightarrow R\): If \(P\), then \(R\).
This form, known as hypothetical syllogism, is powerful in demonstrating how initial premises can lead to a logical conclusion through intermediate steps.
In the exercise provided, a truth table is used to derive that \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\) is a tautology. This signifies that if both given premises are true, then the conclusion necessarily follows.

Syllogism is widely used in logical arguments, proofs, and even in everyday reasoning. It lays the foundation for understanding more complex logical structures.
Symbolic Logic
Symbolic logic, or formal logic, is the study of symbols and the rules for manipulating them to express logical forms and syllogisms. Its primary aim is to enable precise representations of logical expressions.

By using symbols like \(\wedge\) (and), \(\vee\) (or), and \(\rightarrow\) (implies), complex ideas can be compactly and clearly communicated. The symbolic form carries advantages:
  • It reduces ambiguity found in everyday language.
  • It provides a clear framework to analyze logical arguments using mathematical methods such as truth tables.
  • It enables automation for logical calculation by algorithms or computer programs.
For example, by translating "if it rains, then the streets are wet" into symbolic logic as \(P \rightarrow Q\), logical operations become simpler to perform.

The discipline underlies many areas, including mathematics, computer science, and philosophy, offering tools to evaluate the consistency, validity, and soundness of arguments.

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Most popular questions from this chapter

Use set builder notation to specify the following sets: (a) The set of all integers greater than or equal to 5. (b) The set of all even integers. \(\star(c)\) The set of all positive rational numbers. (d) The set of all real numbers greater than 1 and less than 7 . \star (e) The set of all real numbers whose square is greater than 10 .

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of "I will play golf and I will mow the lawn" as "I will not play golf or I will not mow the lawn." (a) We will win the first game and we will win the second game. (b) They will lose the first game or they will lose the second game. (c) If you mow the lawn, then I will pay you \$20. (d) If we do not win the first game, then we will not play a second game. (e) I will wash the car or I will mow the lawn. (f) If you graduate from college, then you will get a job or you will go to graduate school. (g) If I play tennis, then I will wash the car or I will do the dishes. (h) If you clean your room or do the dishes, then you can go to see a movie. (i) It is warm outside and if it does not rain, then I will play golf.

Prime Numbers. The following definition of a prime number is very important in many areas of mathematics. We will use this definition at various places in the text. It is introduced now as an example of how to work with a definition in mathematics. Definition. A natural number \(p\) is a prime number provided that it is greater than 1 and the only natural numbers that are factors of \(p\) are 1 and \(p .\) A natural number other than 1 that is not a prime number is a composite number. The number 1 is neither prime nor composite. Using the definition of a prime number, we see that \(2,3,5,\) and 7 are prime numbers. Also, 4 is a composite number since \(4=2 \cdot 2 ; 10\) is a composite number since \(10=2 \cdot 5 ;\) and 60 is a composite number since \(60=4 \cdot 15\). (a) Give examples of four natural numbers other than \(2,3,5,\) and 7 that are prime numbers. (b) Explain why a natural number \(p\) that is greater than 1 is a prime number provided that For all \(d \in \mathbb{N},\) if \(d\) is a factor of \(p,\) then \(d=1\) or \(d=p\). (c) Give examples of four natural numbers that are composite numbers and explain why they are composite numbers. (d) Write a useful description of what it means to say that a natural number is a composite number (other than saying that it is not prime).

Suppose that Daisy says, "If it does not rain, then I will play golf." Later in the day you come to know that it did rain but Daisy still played golf. Was Daisy's statement true or false? Support your conclusion.

In calculus, we define a function \(f\) with domain \(\mathbb{R}\) to be strictly increasing provided that for all real numbers \(x\) and \(y, f(x)
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