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Validity is about the link
An argument is valid when it is impossible, in the sense of there being no describable case, for every premise to be true and the conclusion false. It says nothing about whether any premise is true.
Lesson · Arguments and Validity · What an argument is
Validity and Soundness
Validity asks whether any case has true premises and a false conclusion. Soundness asks, in addition, whether the premises are in fact true. The first can be settled by looking at cases; the second needs the world. Keeping them apart is the first discipline of the whole subject.
valid: no case with premises T, conclusion F
sound: valid, and the premises are true
a lab can check valid, not sound
valid arguments can have false conclusions
Two words, two questions
Core ideas
What this lesson establishes
In everyday speech a valid point is a good point. In logic, validity is a narrow property of the link between premises and conclusion.
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Soundness adds the world
A sound argument is valid and has only true premises. Soundness is the property you want when you are trying to establish the conclusion, and the only part of it a table can check is the first half.
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A bad conclusion is a reason to doubt a premise
If a valid argument leads to a conclusion you reject, you are not entitled to say the argument is invalid. You are committed to rejecting at least one premise. This is the move most philosophical argument turns on.
The material
Valid, invalid, sound, unsound
Four combinations of two properties, and the one combination that is ruled out.
Definition · Validity
An argument is valid when there is no case in which every premise is true and the conclusion is false. Otherwise it is invalid, and any one such case shows it.
Definition · Soundness
An argument is sound when it is valid and every premise is true. Otherwise it is unsound: either invalid, or valid with at least one false premise.
The definition is a ban on one combination: true premises with a false conclusion. Every other combination of true and false can occur in a valid argument. In particular a valid argument can have false premises and a false conclusion, and false premises and a true conclusion.
Example · Valid and unsound
“All whales are fish. All fish fly. So all whales fly.” Suppose the premises were true; then whales would be among the fish and the fish among the fliers, so whales would fly. No case has the premises true and the conclusion false, so the argument is valid. Both premises are false, so it is unsound.
Example · Invalid with true parts
“Paris is in France. So Rome is in Italy.” Every sentence is true. The argument is invalid, because it is easy to describe a case in which the premise is true and the conclusion false: the two sentences are about different things, and nothing ties one to the other.
So the lab, which looks at cases, can settle validity. Take the preset same, with premise p and conclusion p. There are two cases, p true and p false, and in neither is the premise true while the conclusion is false. The lab calls it valid whatever p says. If p is “whales fly”, the argument is valid and unsound; the lab has no way to know that, and no claim about whales enters the check.
The preset other has premise q and conclusion p. Four cases, and in one of them q is true and p false. That single case is all the invalidity there is. The preset both, with premises p and q and conclusion q, has the same four cases and no such one: wherever both premises hold, q holds, because q is one of them.
Theorem · Using a valid argument
If an argument is valid, then accepting all its premises commits you to its conclusion, and rejecting its conclusion commits you to rejecting at least one premise. Which of the two to do is a question about the premises, and the table does not answer it.
The last point is where the lessons ahead begin. A skeptic offers a valid argument whose conclusion is that you know nothing; you may accept the conclusion or reject a premise, and what each costs is the argument. Validity guarantees that you have to choose.
Interactive
Validity by cases
Every row, every counterexample, and the form
Method
Classifying an argument
Two questions, asked separately and in this order.
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Set out the argument
Write the premises and the conclusion in standard form, as in the previous lesson. Classify nothing until you know what is being classified.
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Look for a case
Describe a case in which every premise is true and the conclusion false. Do not ask whether the case actually holds; ask whether it can be described. If one can, the argument is invalid and you are done.
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If there is no such case, call it valid
Validity is a verdict on the absence of a case, so it needs every case checked. Here the lab does what a person does slowly.
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Then, and only then, ask about the premises
A valid argument is sound if every premise is true. That is a question about the world, and an unsound argument has not been refuted so much as turned into a question about which premise to doubt.
Valid and sound
Score 0 / 0Choose the answer you can justify.
Whales and fish
The argument: all whales are fish; all fish fly; so all whales fly.
Valid? Describe a case: premises true, conclusion false.
If whales are fish and fish fly, whales fly.
No such case. The argument is valid.
Sound? Is every premise true?
Whales are mammals. P1 is false.
The argument is unsound.
The lab can do the first half of this. It cannot do the second, because “whales are fish” is a claim about the world and the lab was never told about whales. That division of labour is the reason to separate the two words.
Common mistakes
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Thinking a valid argument has a true conclusionValidity is conditional: if the premises are true, so is the conclusion. “All whales are fish; all fish fly; so all whales fly” is valid, and its conclusion is false; validity allows that only because a premise is false too. A table cannot tell you that the conclusion is true; only that it cannot be false while every premise is true.
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Using “valid” to mean “reasonable”“That is a valid point” praises a claim. In logic the word applies to arguments, never to sentences, and an argument with absurd premises can be perfectly valid. A sentence is true or false; an argument is valid or invalid.
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Rejecting a valid argument's conclusion by calling it invalidIf the argument is valid and you do not like the conclusion, the argument stands and one of the premises must go. Showing that you can find the premise to reject is the work; saying “it cannot be valid” is only a refusal to do it.
Check your understanding
✓
Finish when you can classify an argument twice, with a reason each time.
Given an argument, say whether it is valid by looking for a case with true premises and a false conclusion, and then whether it is sound by checking the premises against what you know. Explain why the lab answers only the first question.
Note: The argument “p, so p” looks trivial, and it is meant to. It is the smallest valid argument, and it shows that validity depends on the form alone: whatever p says, the verdict is the same.