Every way to show an argument is invalid

An argument is invalid exactly when there is at least one possible situation in which every premise is true and the conclusion is false. Such a situation is called a counterexample to the argument. An argument is valid exactly when no counterexample exists.

What is this saying? To show an argument is invalid, you don't need to show its conclusion is false. You only need to describe one situation, real or imagined, where the premises all hold and the conclusion still fails. One such situation settles it.

Why should I care? "It doesn't follow" is a claim people make and dispute all the time. This test turns it into something you can demonstrate: produce the counterexample and the argument is finished, however true its conclusion happens to be.

Intuition. Validity is a promise: whenever the premises are true, the conclusion is true too. A promise about every situation is broken by a single situation where it fails, the same way one black swan breaks "all swans are white."

Ingredients.

• Possible situation: any consistent way the world could be, not only the way it happens to be.
• Every premise true: all of them at once, in that same situation.
• Conclusion false: in that same situation, not in some other one.

Example. "Every spider has eight legs. Charlotte has eight legs. So Charlotte is a spider." Counterexample: Charlotte is a scorpion. Both premises are true in that situation and the conclusion is false, so the argument is invalid, even if the real Charlotte turns out to be a spider.

What goes wrong without "every premise true." A situation where the conclusion is false but some premise is false too proves nothing. A valid argument can have a false conclusion when a premise is false as well.

Call a form valid when none of its substitution instances can have all true premises and a false conclusion. Then:

1. Every substitution instance of a valid form is a valid argument.
2. If even one substitution instance of a form has all true premises and a false conclusion, the form is invalid.

What is this saying? The letters in a form are blanks, and an argument you get by filling the blanks is an instance of the form. If a pattern of reasoning is good, it is good whatever you fill it with. If one filling of the pattern leads from truths to a falsehood, the pattern itself is broken.

Why should I care? This is why logic can study patterns instead of topics. Show a form is valid once, and every argument with that form is valid, about weather, medicine, or anything else. The reverse doesn't hold: an argument with an invalid form can still be valid because of what its words mean. It also gives a practical way to attack a bad argument: keep its form, swap in content where the premises are plainly true and the conclusion plainly false.

Intuition. A valid form gets its guarantee from its connecting words, such as if … then, or, and, and not, and those are the same in every instance. The topic words only fill the blanks, so they can't switch the guarantee on or off.

Ingredients.

• Form: an argument with each distinct claim replaced by a letter, with the same letter used every time the same claim appears.
• Substitution instance: what you get by putting a claim in for each letter, the same claim for every copy of that letter.
• All true premises, false conclusion: the counterexample combination, here found by choosing the fillings.

Example. "If Paris is in Germany, then Paris is in Europe. Paris is in Europe. So Paris is in Germany." The premises are true and the conclusion is false. By part 2, the form behind it (if P then Q; Q; so P) is invalid, and so no argument is made valid just by having that form.

Where the rule stops. Part 2 condemns the form, not every argument that has it. "If Ann is taller than Ben, then Ben is shorter than Ann. Ben is shorter than Ann. So Ann is taller than Ben." This has the invalid form above, yet it is valid, because of what taller and shorter mean. An invalid form can't prove an argument valid, but some other feature of the argument might.

The form

1. If P, then Q.
2. Q.

\therefore P.

is invalid. It has instances with true premises and a false conclusion.

What is this saying? Knowing that P leads to Q, and that Q happened, does not tell you P happened. Something else may have caused Q.

Why should I care? This is one of the most common reasoning mistakes there is, because it looks almost exactly like modus ponens. "If the engine is flooded, the car won't start. The car won't start. So the engine is flooded." A dead battery also stops the car.

Intuition. "If P, then Q" says P is one way to get Q. It never says P is the only way. The second premise lands on the result, and a result can have many causes.

Ingredients.

• Antecedent: the "if" part, P.
• Consequent: the "then" part, Q.
• Affirming the consequent: the second premise asserts the consequent and the conclusion asserts the antecedent. Modus ponens runs the other way: it asserts the antecedent and concludes the consequent.

Example. "If Paris is in Germany, then Paris is in Europe. Paris is in Europe. So Paris is in Germany." Both premises are true. The conclusion is false.

Non-example. "If Paris is in Germany, then Paris is in Europe. Paris is in Germany. So Paris is in Europe." This is modus ponens, and it is valid. Its first premise is still true, its second is false, and validity only promises a true conclusion when every premise is true.

The form

1. If P, then Q.
2. Not P.

\therefore Not Q.

is invalid. It has instances with true premises and a false conclusion.

What is this saying? Knowing that P leads to Q, and that P did not happen, does not tell you Q didn't happen. Q could still come about some other way.

Why should I care? It sounds like careful reasoning and often slips past unnoticed. "If you study, you'll pass. You didn't study. So you won't pass." Some people pass without studying, and the premises never ruled that out.

Intuition. "If P, then Q" tells you what happens when P is true. It says nothing about what happens when P is false. Denying P walks into territory the conditional never covered.

Ingredients.

• Antecedent: the "if" part, P.
• Consequent: the "then" part, Q.
• Denying the antecedent: the second premise denies the antecedent and the conclusion denies the consequent. Modus tollens runs the other way: it denies the consequent and concludes that the antecedent is false.

Example. "If Paris is in Germany, then Paris is in Europe. Paris is not in Germany. So Paris is not in Europe." Both premises are true. The conclusion is false.

Non-example. "If Paris is in Germany, then Paris is in Europe. Paris is not in Europe. So Paris is not in Germany." This is modus tollens, and it is valid.

A standard-form categorical syllogism is valid on the Boolean reading if and only if it satisfies all five rules:

1. The middle term is distributed in at least one premise.
2. Any term distributed in the conclusion is distributed in the premise that contains it.
3. The premises are not both negative.
4. A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.
5. If both premises are universal, the conclusion is not particular.

Breaking rule 1 is the undistributed middle; breaking rule 2 is illicit major or illicit minor; rule 3 is exclusive premises; rule 5 is the existential fallacy. Breaking rule 4 is an affirmative conclusion from a negative premise (or a negative conclusion from affirmative premises).

What is this saying? You can check a syllogism without drawing anything: look at which terms each statement is about "all of," count the negatives, and check whether the conclusion claims existence that the premises don't. Pass all five checks and the syllogism is valid; fail any one and it is invalid.

Why should I care? The rules are faster than a Venn diagram and they name exactly what is wrong with a bad syllogism, which is what you need to explain or repair it.

Intuition. The middle term is the bridge between the other two terms; if neither premise talks about the whole middle class, the two premises may be talking about different parts of it and never meet. And a conclusion can't say more about a class, or claim more existence, than the premises gave it.

Ingredients.

• Distributed term: one the statement says something about every member of. A distributes its subject; E distributes both terms; I distributes neither; O distributes its predicate.
• Middle term: in both premises, not in the conclusion.
• Negative: an E or O statement. Universal: A or E. Particular: I or O.

Example. "All dogs are mammals. All cats are mammals. So all cats are dogs." The middle term "mammals" is the predicate of two A statements, so it is never distributed. Rule 1 fails: undistributed middle.

What breaks without rule 5. "All unicorns are horses. All horses are animals. So some unicorns are animals." Both premises are true (there are no unicorns to break the first), but the conclusion claims a unicorn exists.

An argument in sentential logic with premises A_1, \dots, A_n and conclusion C is valid if and only if no row of their joint truth table makes every premise true and C false. When this holds we write A_1, \dots, A_n \models C. Because a table has finitely many rows, this test always gives an answer.

What is this saying? To test an argument, build one table with a column for each premise and for the conclusion, and look only at the rows where every premise is T. If the conclusion is T on all of those rows, the argument is valid. If it is F on even one, the argument is invalid, and that row is a countermodel.

Why should I care? Chapter 3 defined validity in terms of possible situations, which you had to imagine. For sentential arguments this turns the definition into a mechanical check that is guaranteed to finish.

Intuition. In sentential logic, all that matters about a situation is which sentence letters it makes true. So "every possible situation" comes down to "every row."

Ingredients.

• A_1, \dots, A_n: the premises. C: the conclusion.
• \models: "entails." A_1, \dots, A_n \models C says every row making all the premises true makes C true.
• Countermodel: a row with every premise T and C F.

Example. For P \to Q, Q, therefore P: the row P F, Q T makes both premises T and the conclusion F. So the argument (affirming the consequent) is invalid.

What does not count. A row where every premise AND the conclusion is T shows nothing. Valid and invalid arguments both usually have such rows.

1. If the premises A_1, \dots, A_n are inconsistent, then A_1, \dots, A_n \models C for every sentence C.
2. If C is a tautology, then A_1, \dots, A_n \models C for any premises at all, including none.

What is this saying? An argument whose premises can never all be true is valid whatever its conclusion. An argument whose conclusion can never be false is valid whatever its premises. In both cases the premises and conclusion may have nothing to do with each other.

Why should I care? These cases look like mistakes, and a common first reaction is to call them invalid. They follow directly from the definition, and seeing why makes the definition clearer. They are also why validity alone isn't enough: an argument with inconsistent premises is valid but can never be sound.

Intuition. Validity only forbids one kind of row: premises all T, conclusion F. If the premises are never all T, or the conclusion is never F, that kind of row can't exist.

Ingredients.

• Inconsistent: no row makes all the premises true together.
• Tautology: true on every row.
• \models: entails.

Example. P, \neg P, therefore Q is valid. So is R, therefore Q \vee \neg Q.

What this does not mean. Valid is not the same as good. Neither kind of argument could persuade anyone: one has premises that can't all be true, and the other proves something that needed no premises.