Valid and Invalid Arguments

MATH1061

Whether Arguments Are Valid or Invalid

An Argument can be considered like this:

If today is Tuesday, then I am wearing a pink shirt. Today is Tuesday. Therefore, I am wearing a pink shirt.

Valid argument: The statement (Premise 1 $\wedge$ Premise 2 $\wedge$ $\dots$ $\wedge$ Last Premise) $⟹$ Conclusion is a Tautology/Redundant

Let $p$ represent the statement “today is Monday”, and $q$ represent the statement “I am wearing a pink shirt”.

Then this argument can be written as:

$$\begin{array}{rl}p⟹q& \\ p& \\ \therefore q& \end{array}$$

This is a valid argument, but it has a false conclusion. Note that the premise “If today is Tuesday, then I am wearing a pink shirt” is false.

Activity 1:

Consider the following argument:

• If wages are raised, buying increases.
• If there is a depression, buying does not decrease.
• Therefore, there is not a depression or wages are not raised.

Let $w$ represent the statement “wages are raised”, $b$ represent the statement “buying increases”, and $d$ represent the statement “there is a depression”.

If written in symbolic form:

$$\begin{array}{r}w⟹b\\ d⟹\sim b\\ \therefore \sim d\vee \sim w\end{array}$$

Proving an Argument is Valid (using Rules of inference)

Premise 1: $p⟹q$ Premise 2: $p\vee r$ Premise 3: $p\vee \sim r$ Conclusion: $q$

Use the laws of logical equivalence and the rules of inference to show that this argument is valid.

1. $p⟹q$
2. $p\vee r$
3. $p\vee \sim r$

$$\{\begin{array}{llcc}(p\vee r)\wedge (p\vee \sim r)& & & \text{From 2. and 3 by conjunction. }\end{array}$$

Activity 5:

Proving an argument is valid (by checking if the argument is invalid):

Premise 1: $p⟹q$ Premise 2: $q⟹r$ Premise 3: $\sim p\vee \sim q$ Conclusion: $r$

$$\begin{array}{r}\text{Suppose all premises are true and the conclusion is false.}\\ \{\begin{array}{llc}\text{Since the conclusion is false,}& q& \text{is false.}\\ \text{Since 2. is true and}& r& \text{is false, }\end{array}\end{array}$$

Activity 6:

Premise 1: $p⟹q$ Premise 2: $p\vee r$ Premise 3: $p\vee \sim r$ Conclusion: $q$

$$\begin{array}{r}\text{Suppose all premises are true and the conclusion is false.}\\ \{\begin{array}{l}\text{Since conclusion is false},q\text{ is false.}\\ \text{Since 1. is true and}q\text{ is false,}p\text{is false.}\end{array}\end{array}$$