Argument and Validity

Section 1.3 Argument and Validity

Section Objectives.

Define argument (or logical deduction)

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Identify hypothesis and conclusion of a given argument

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Determine the validity of an argument using a truth table

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Determine the validity of an argument using rules of inferences

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Definition 1.3.1.

An argument (logical deduction) is an assertion that a given set of statements \(p_{1},\,p_{2},\dots,p_{n}\text{,}\) called \(\textbf{hypotheses}\) or premises, yield another statement \(Q\text{,}\) called the conclusion. Such a logical deduction is denoted by:

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\(p_{1},\)	\(p_{2},\)	\(\ldots,\)	\(p_{n}\)	\(\vdash\;Q \,\, \text{ or}\)
\(p_{1}\)
\(p_{2}\)
\(\\ \vdots\\\)
\(\frac{p_{n}}{Q}\)

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Example 1.3.2.

Consider the following argument:

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If you study hard, then you will pass the exam.
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You did not pass the exam.
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Therefore, you did not study hard.
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Let \(p\text{:}\) You study hard.
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\(\quad \,\,\, q\text{:}\) You will pass the exam.
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The argument form can be written as:
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\begin{equation*} \begin{array}{r} p \Rightarrow q\\ \neg q\\ \hline \neg p \end{array} \end{equation*}

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When is an argument form accepted to be correct? In normal usage, we use an argument in order to demonstrate that a certain conclusion follows from known premises. Therefore, we shall require that under any assignment of truth values to the statements appearing, if the premises became all true, then the conclusion must also become true. Hence, we state the following definition.

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Definition 1.3.3.

An argument form \(p_1, p_2, p_3, \ldots, p_n \vdash Q\) is said to be valid if \(Q\) is true whenever all the premises \(p_1, p_2, p_3, \ldots, p_n\) are true; otherwise it is invalid.

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Example 1.3.4.

Investigate the validity of the following arguments:

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\(\displaystyle p \Rightarrow q, \, \neg q \vdash \neg p\)
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\(\displaystyle p \Rightarrow q, \, \neg q \Rightarrow r \vdash p\)
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If it rains, crops will be good. It did not rain. Therefore, crops were not good.
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Solution.

First we construct a truth table for the statements appearing in the argument forms.

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\(p\) \(q\) \(\neg p\) \(\neg q\) \(p \Rightarrow q\)

\(T\) \(T\) \(F\) \(F\) \(T\)

\(T\) \(F\) \(F\) \(T\) \(F\)

\(F\) \(T\) \(T\) \(F\) \(T\)

\(F\) \(F\) \(T\) \(T\) \(T\)

The premises \(p \Rightarrow q\) and \(\neg q\) are true simultaneously in row 4 only. Since in this case \(\neg p\) is also true, the argument is valid.
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\(p\)	\(q\)	\(r\)	\(\neg q\)	\(p \Rightarrow q\)	\(\neg q \Rightarrow r\)
\(T\)	\(T\)	\(T\)	\(F\)	\(T\)	\(T\)
\(T\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(T\)	\(F\)	\(T\)	\(T\)	\(F\)	\(T\)
\(T\)	\(F\)	\(F\)	\(T\)	\(F\)	\(F\)
\(F\)	\(T\)	\(T\)	\(F\)	\(T\)	\(T\)
\(F\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(T\)	\(T\)	\(T\)	\(T\)
\(F\)	\(F\)	\(F\)	\(T\)	\(T\)	\(F\)

The \(1^{\text{st}}\text{,}\) \(2^{\text{nd}}\text{,}\) \(5^{\text{th}}\text{,}\) \(6^{\text{th}}\) and \(7^{\text{th}}\) rows are those in which all the premises take value \(T\text{.}\) In the \(5^{\text{th}}\text{,}\) \(6^{\text{th}}\) and \(7^{\text{th}}\) rows however the conclusion takes value \(F\text{.}\) Hence, the argument form is invalid.

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Let \(p\text{:}\) It rains.
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\(\qquad q\text{:}\) Crops are good.
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\(\qquad \neg p\text{:}\) It did not rain.
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\(\qquad \neg q\text{:}\) Crops were not good.
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The argument form is \(p \Rightarrow q, -p \vdash -q\)
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Now we can use truth table to test validity as follows:
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\(p\) \(q\) \(\neg p\) \(\neg q\) \(p \Rightarrow q\)

\(T\) \(T\) \(F\) \(F\) \(T\)

\(T\) \(F\) \(F\) \(T\) \(F\)

\(F\) \(T\) \(T\) \(F\) \(T\)

\(F\) \(F\) \(T\) \(T\) \(T\)

Examining the table, we can see that the premises \(p \Rightarrow q\) and \(\neg p\) are true in both rows 3 and 4. However, in row 3, \(\neg q\) is false, which means the argument is invalid.
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The premises \(p \Longrightarrow q\) and \(\neg p\) are true simultaneously in row 4 only. Since in this case \(\neg q\) is also true, the argument is valid.
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Remark 1.3.5.

What is important in validity is the form of the argument rather than the meaning or content of the statements involved.
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The argument form \(p_{1},\ p_{2},p_{3},\ldots,p_{n}\ ├\ \ Q\) is valid iff the statement
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\((p_{1} \land p_{2} \land p_{3} \land \ldots \land p_{n}\ ) \Longrightarrow Q\) is a tautology.
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Rules of Inference.

Below we list certain valid deductions called rules of inferences.

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Modus Ponens
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\(\begin{array}{c} p \\ p \implies q \\ \hline q \end{array}\)
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Modus Tollens
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\(\begin{array}{c} p \implies q \\ \neg q \\ \hline \neg p \end{array}\)
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Principle of Syllogism
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\(\begin{array}{c} p \Rightarrow q \\ q \Rightarrow r \\ \hline p \Rightarrow r \end{array}\)
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Principle of Adjunction
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1. \(\displaystyle \begin{array}{c} p \\ q \\ \hline p \land q \end{array}\)
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2. \(\displaystyle \frac{q}{p \lor q}\)
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Principle of Detachment
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\(\begin{array}{c} p \land q \\ \hline p, q \end{array}\)
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Modus Tollendo Ponens
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\(\begin{array}{c} \neg p \\ p \lor q \\ \hline q \end{array}\)
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Modus Ponendo Tollens
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\(\begin{array}{c} \neg(p \land q) \\ p \\ \hline \neg q \end{array}\)
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Constructive Dilemma
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\(\begin{array}{c} (p \Rightarrow q) \land (r \Rightarrow s) \\ p \lor r \\ \hline q \lor s \end{array}\)
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Principle of Equivalence
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\(\begin{array}{c} p \Leftrightarrow q \\ p \\ \hline q \end{array}\)
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Principle of Conditionalization
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\(\begin{array}{c} p \\ \hline q \Rightarrow p \end{array}\)
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Formal proofs of validity of an argument.

Definition 1.3.6.

A formal proof of a conclusion \(Q\) given hypotheses \(p_{1},\ p_{2},p_{3},\ldots,p_{n}\) is a sequence of steps, each of which applies some inference rule to hypotheses or previously proven statements (antecedent) to yield a new true statement (the consequent).

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A formal proof of validity is given by writing on the premises and the statements which follows from them in a single column, and setting off in another column, to the right of each statement, its justification. It is convenient to list all the premises first.

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Example 1.3.7.

Show that \(p \Longrightarrow \neg q,q \vdash \neg p\) is valid.

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Solution.

1. \(q\) is true	premise
2. \(p \Longrightarrow \neg q\) is true	premise
3. \(q \Longrightarrow \neg p\) is true	contrapositive of (2)
4. \(\neg p\) is true	Modus Ponens using (1) and (3)

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Example 1.3.8.

Show that the hypotheses

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It is not sunny this afternoon and it is colder than yesterday.
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If we go swimming, then it is sunny.
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If we do not go swimming, then we will take a canoe trip.
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If we take a canoe trip, then we will be home by sunset.
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Lead to the conclusion:

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\(\qquad\) We will be home by sunset.

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Solution.

Let \(p\text{:}\) It is sunny this afternoon.

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\(\qquad q\text{:}\) It is colder than yesterday.

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\(\qquad r\text{:}\) We go swimming.

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\(\qquad s\text{:}\) We take a canoe trip.

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\(\qquad t\text{:}\) We will be home by sunset.

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Then

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1. \(\neg p \land q\)	hypothesis
2. \(\neg p\)	simplification using (1)
3. \(r \implies p\)	hypothesis
4. \(\neg r\)	Modus Tollens using (2) and (3)
5. \(\neg r \implies s\)	hypothesis
6. \(s\)	Modus Ponens using (4) and (5)
7. \(s \implies t\)	hypothesis
8. \(t\)	Modus Ponens using (6) and (7)

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Exercises Exercises

1.

Use the truth table method to show that the following argument forms are valid.

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\(\neg p \Longrightarrow \neg q,\ q\ \vdash\ p\text{.}\)
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\(p \Longrightarrow \neg p,\ p,\ r \Longrightarrow q\ \vdash\ \neg r\text{.}\)
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\(p \Longrightarrow q,\ \neg r \Longrightarrow \neg q\ \vdash \neg r \Longrightarrow \neg p\text{.}\)
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\(\neg r \vee \neg s,\ (\neg s \Longrightarrow p) \Longrightarrow r\ \vdash\ \neg p\text{.}\)
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\(p \Longrightarrow q,\ \neg p \Longrightarrow r,\ r \Longrightarrow s \vdash\ \neg q \Longrightarrow s\text{.}\)
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2.

For the following arguments given a, b and c below:

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Identify the premises.
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Write argument forms.
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Check the validity.
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1. If he studies medicine, he will get a good job. If he gets a good job, he will get a good wage. He did not get a good wage. Therefore, he did not study medicine.
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2. If the team is late, then it cannot play the game. If the referee is here, then the team can play the game. The team is late. Therefore, the referee is not here.
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3. If the professor offers chocolate for an answer, you answer the professor’s question. The professor offers chocolate for an answer. Therefore, you answer the professor’s question.
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3.

Give formal proof to show that the following argument forms are valid.

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\(\neg p \Longrightarrow \neg q,\ q\ \vdash\ p\text{.}\)
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\(p \Longrightarrow \neg q,\ p,\ r \Longrightarrow q\ \vdash\ \neg r\text{.}\)
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\(p \Longrightarrow q,\ \neg r \Longrightarrow \neg q\ \vdash\ \neg r \Longrightarrow \neg p\text{.}\)
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\(\neg r \land \neg s,\ (\neg s \Longrightarrow p) \Longrightarrow r\ \vdash\ \neg p\text{.}\)
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\(p \Longrightarrow q,\ \neg p \Longrightarrow r,\ r \Longrightarrow s\ \vdash\ \neg q \Longrightarrow s\text{.}\)
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\(\neg p \vee q,\ r \Longrightarrow p,\ r\ \vdash\ q\text{.}\)
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\(\neg p \land \neg q,\ (q \vee r) \Longrightarrow p\ \vdash\ \neg r\text{.}\)
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\(p \Longrightarrow (q \vee r),\ \neg r,\ p\ \vdash\ q\text{.}\)
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\(\neg q \Longrightarrow \neg p,\ r \Longrightarrow p,\ \neg q\ \vdash\ r\text{.}\)
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4.

Prove the following are valid arguments by giving formal proof.

If the rain does not come, the crops are ruined and the people will starve. The crops are not ruined or the people will not starve. Therefore, the rain comes.
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If the team is late, then it cannot play the game. If the referee is here then the team can play the game. The team is late. Therefore, the referee is not here.
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Checkpoint 1.3.9.

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Checkpoint 1.3.10.

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\(p\)	\(q\)	\(\neg p\)	\(\neg q\)	\(p \Rightarrow q\)
\(T\)	\(T\)	\(F\)	\(F\)	\(T\)
\(T\)	\(F\)	\(F\)	\(T\)	\(F\)
\(F\)	\(T\)	\(T\)	\(F\)	\(T\)
\(F\)	\(F\)	\(T\)	\(T\)	\(T\)