Validity of Arguments

What is an Argument?

An argument is a set of premises (things you assume) that lead to a conclusion (what you claim follows).

Premises + Conclusion = Argument

A Simple Example

• $p$ = “it rains”

• $q$ = “the ground gets wet”

• Premise 1: $p \to q$ — “If it rains, the ground gets wet”

• Premise 2: $p$ — “It rained”

• Conclusion: $q$ — “The ground is wet”

This feels right. But how do we know for sure?

What Makes an Argument Valid?

An argument is valid if: whenever ALL premises are true, the conclusion is ALSO true.

It’s about the structure, not whether the premises are actually true in real life.

How to Check Validity

Use a truth table:

1. Find all rows where every premise is true
2. Check if the conclusion is also true in those rows
3. If yes in ALL such rows → valid

If there’s even ONE row where all premises are true but the conclusion is false → invalid

Common Valid Forms

These are patterns that always work. Once you recognize them, you don’t need a truth table.

1. Modus Ponens

$p \to q, \; p \; \therefore q$

If p then q. p happened. So q happened.

• $p$ = “it rains”
• $q$ = “I bring an umbrella”

“If it rains, I bring an umbrella. It rained. So I brought an umbrella.”

2. Modus Tollens

$p \to q, \; \neg q \; \therefore \neg p$

If p then q. q didn’t happen. So p didn’t happen.

• $p$ = “it rains”
• $q$ = “I bring an umbrella”

“If it rains, I bring an umbrella. I didn’t bring an umbrella. So it didn’t rain.”

3. Hypothetical Syllogism

$p \to q, \; q \to r \; \therefore p \to r$

If p leads to q, and q leads to r, then p leads to r.

• $p$ = “I oversleep”
• $q$ = “I miss the bus”
• $r$ = “I’m late to work”

“If I oversleep, I miss the bus. If I miss the bus, I’m late to work. So if I oversleep, I’m late to work.”

4. Disjunctive Syllogism

$p \lor q, \; \neg p \; \therefore q$

Either p or q. p is false. So q must be true.

• $p$ = “he’s at home”
• $q$ = “he’s at work”

“He’s either at home or at work. He’s not at home. So he’s at work.”

Common Fallacies

These look valid but aren’t. Don’t fall for them.

1. Affirming the Consequent

$p \to q, \; q \; \therefore p$ ✗ INVALID

If p then q. q happened. So p happened? No — q could have happened for other reasons.

• $p$ = “it rains”
• $q$ = “the ground is wet”

“If it rains, the ground is wet. The ground is wet. So it rained.” ← WRONG

The ground could be wet from a sprinkler.

2. Denying the Antecedent

$p \to q, \; \neg p \; \therefore \neg q$ ✗ INVALID

If p then q. p didn’t happen. So q didn’t happen? No — q could still happen for other reasons.

• $p$ = “it rains”
• $q$ = “the ground is wet”

“If it rains, the ground is wet. It didn’t rain. So the ground isn’t wet.” ← WRONG

Sprinkler again.

Valid vs Sound

Valid = the logic is correct. IF the premises were true, the conclusion would HAVE to follow.

Sound = valid + the premises are actually true in real life.

Example of valid but NOT sound:

• $p$ = “you eat an apple a day”
• $q$ = “you never get sick”

“If you eat an apple a day, you never get sick. You eat an apple a day. So you never get sick.”

The logic is perfect — it’s modus ponens. If the first statement were true, the conclusion would follow.

But the first statement is false in real life. Eating apples doesn’t make you immune to sickness.

So: Valid structure, but not sound because we started with a lie.

Why does this matter?

You can have flawless logic and still be wrong — if you start with bad premises.

Valid = your reasoning is correct. Sound = your reasoning is correct AND your starting points are true.

Summary