Arguments and Validity: When Conclusions Follow

Modus ponens never fails.

If P then Q. P. Therefore Q.

It doesn't matter what P and Q mean. "If it's raining, the ground is wet. It's raining. Therefore the ground is wet." Valid. "If snorgs are blibble, the moon is cheese. Snorgs are blibble. Therefore the moon is cheese." Equally valid.

The pattern guarantees the conclusion follows from the premises — regardless of content.

Learn a handful of these patterns, and you can recognize valid reasoning wherever you encounter it. You can also spot when someone's argument doesn't fit any valid pattern — which means it might be a fallacy.

Modus Ponens

The most fundamental valid form.

P → Q (If P then Q) P (P is true) ∴ Q (Therefore Q)

The conditional asserts that P being true guarantees Q. Once you establish P is true, Q must follow.

Example: If you study, you'll pass. You studied. Therefore, you'll pass.

This is sometimes called "affirming the antecedent" — you affirm P, the "if" part.

Modus Tollens

The contrapositive argument.

P → Q (If P then Q) ¬Q (Q is false) ∴ ¬P (Therefore P is false)

If P would make Q true, but Q is false, then P can't be true.

Example: If it rained, the ground would be wet. The ground is dry. Therefore, it didn't rain.

This is "denying the consequent" — you deny Q, the "then" part.

Hypothetical Syllogism

Chaining conditionals.

P → Q (If P then Q) Q → R (If Q then R) ∴ P → R (Therefore if P then R)

If P leads to Q, and Q leads to R, then P leads to R.

Example: If it rains, the ground gets wet. If the ground gets wet, the flowers grow. Therefore, if it rains, the flowers grow.

This lets you chain long sequences of implications.

Disjunctive Syllogism

Eliminating alternatives.

P ∨ Q (P or Q) ¬P (P is false) ∴ Q (Therefore Q)

If one of two options must be true, and we eliminate one, the other must hold.

Example: Either the battery is dead or the bulb is broken. The battery isn't dead. Therefore, the bulb is broken.

Constructive Dilemma

Two paths to a conclusion.

P → Q (If P then Q) R → S (If R then S) P ∨ R (P or R) ∴ Q ∨ S (Therefore Q or S)

If we have two conditionals and know one antecedent holds, one consequent must hold.

Example: If I take the train, I'll be late. If I drive, I'll be stressed. I'm either taking the train or driving. Therefore, I'll either be late or stressed.

Conjunction and Simplification

Conjunction: Combining truths.

P Q ∴ P ∧ Q

If both are true, their conjunction is true.

Simplification: Extracting truths.

P ∧ Q ∴ P (or ∴ Q)

If a conjunction is true, each conjunct is true.

Addition

Weakening to a disjunction.

P ∴ P ∨ Q

If P is true, then "P or anything" is true.

This seems odd — you're making a weaker claim. But it's logically valid and sometimes useful in proofs.

The Fallacies: Invalid Forms

Affirming the Consequent (INVALID)

P → Q Q ∴ P ???

If it rained, the ground is wet. The ground is wet. Therefore it rained? No! The sprinklers could be on.

The conditional only says P guarantees Q. It doesn't say Q requires P.

Denying the Antecedent (INVALID)

P → Q ¬P ∴ ¬Q ???

If it rains, the ground gets wet. It's not raining. Therefore the ground isn't wet? No! The sprinklers again.

The conditional says nothing about what happens when P is false.

Comparing Valid and Invalid

The valid forms work with the structure of the conditional correctly. The invalid forms confuse necessary and sufficient conditions.

Recognizing Patterns in the Wild

Real arguments rarely announce their form. You have to extract it.

"The murder weapon was either the knife or the candlestick. But the knife has no fingerprints, so the candlestick must be the weapon."

Structure: Either K or C. Not K. Therefore C. — Disjunctive syllogism. Valid.

"If the economy improves, the president gets credit. The president is getting credit. So the economy must have improved."

Structure: If E then C. C. Therefore E. — Affirming the consequent. Invalid.

Using Valid Forms in Proofs

These forms become proof rules:

1. Given: P → Q, Q → R
2. By hypothetical syllogism: P → R
3. Given: P
4. By modus ponens: R

Each step applies a valid form. The chain of valid steps guarantees the conclusion.

The Complete Toolkit

These patterns are the building blocks of deductive reasoning. Every valid argument is ultimately a combination of these fundamental forms.

Part 7 of the Logic series.

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