What is logic? Valid arguments, premises and conclusions

Logic is the study of correct reasoning: of which conclusions follow from which premises, and why. Its basic unit is the argument, a set of statements called premises offered in support of another statement, the conclusion. Logic doesn't tell you whether the premises are true. It tells you whether, if they were true, the conclusion would have to be true as well, and it does that by looking at the argument's form, not its subject.

In short

• An argument is premises plus a conclusion. Logic studies the link between them: whether the conclusion follows.
• An argument is valid if true premises would guarantee a true conclusion; it is sound if it is valid and its premises really are true.
• The connecting words "and", "or", "not" and "if … then" have exact rules, shown in truth tables, and those rules are what computer circuits carry out.

What is an argument?

In logic, an argument isn't a quarrel. It is a set of statements in which some, the premises, are offered as reasons for another, the conclusion. Each statement must be the kind of sentence that is true or false: "Close the door" is not, "The door is closed" is.

The oldest worked-out system is the syllogism, set out by Aristotle in the Prior Analytics in the 4th century BCE, as the Stanford Encyclopedia of Philosophy describes. Here is one:

1. All mammals are warm-blooded. (premise)
2. All whales are mammals. (premise)
3. So all whales are warm-blooded. (conclusion)

Now strip out the subject. The argument has the shape: all M are P; all S are M; so all S are P. Put anything in for the letters, "all violins are instruments; all Stradivarius violins are violins; so all Stradivarius violins are instruments", and the conclusion still follows. Logic studies the shape of reasoning, which is why one rule covers whales, violins and everything else at once.

What is the difference between valid and sound?

An argument is valid when its form makes it impossible for the premises to be true and the conclusion false, in the words of the Internet Encyclopedia of Philosophy. It is sound when it is valid and all its premises actually are true. Validity is about the link; soundness is about the link and the starting points.

	Premises all true	At least one premise false
Valid	Sound. All mammals are warm-blooded; all whales are mammals; so all whales are warm-blooded. The conclusion must be true.	Valid but unsound. All birds can fly; penguins are birds; so penguins can fly. The reasoning works; the first premise is false, and so is the conclusion.
Invalid	Unsound. Some mammals swim; all whales are mammals; so all whales swim. Every statement is true, but the conclusion doesn't follow: the same form would "prove" that all bats swim.	Unsound. All fish can fly; whales can fly; so whales are fish. Bad premises and a bad link.

The bottom-left box is the one that catches people. An argument can have true premises and a true conclusion and still be invalid, because the premises don't support the conclusion; it just happens to be true. The easiest test is the one used there: keep the form, change the subject, and see whether true premises can now lead to a false conclusion. If they can, the form is invalid, and errors of form like this are the formal kind of logical fallacy.

How do "and", "or" and "if … then" work?

Many arguments turn on small connecting words, and logic gives each an exact meaning. Write P and Q for any two statements, T for true and F for false. A truth table lists every combination:

P	Q	not P	P and Q	P or Q	if P then Q
T	T	F	T	T	T
T	F	F	F	T	F
F	T	T	F	T	T
F	F	T	F	F	T

"And" is true only when both parts are. "Or", in logic, is the inclusive or: true when either part is true, or both. "If P then Q" is false in just one case, when P is true and Q is false. In the last two rows, where P is false, it counts as true: "if it rains, the match is off" isn't broken on a day it doesn't rain, whatever happens to the match. The IEP's article on propositional logic gives the same rule. This reading, the material conditional, is the standard one in logic, though philosophers have long argued about how well it fits the everyday "if", which often carries a hint of cause.

How do you test an argument with a truth table?

Truth tables turn checking validity into a mechanical search: an argument is invalid if any row makes all its premises true and its conclusion false. Take this one:

1. If it rains, the match is off.
2. The match is off.
3. So it rained.

Let P be "it rains" and Q be "the match is off". The premises are "if P then Q" and "Q"; the conclusion is "P". Go through the rows. In the third row, P is false and Q is true: "if P then Q" is true, Q is true, so both premises hold, but the conclusion, P, is false. One such row is enough. The argument is invalid; the match could be off because of a frozen pitch. This error is called affirming the consequent.

Now change the second premise to "it rains" and the conclusion to "the match is off". The only row where P and "if P then Q" are both true is the first, and there Q is true. No row breaks it, so the argument is valid. This form is called modus ponens, and the Stoic logician Chrysippus listed it among the basic forms of inference in the 3rd century BCE.

How did logic end up inside computers?

Truth and falsehood can be written as 1 and 0. In 1854 the mathematician George Boole published The Laws of Thought, which treated logic as a kind of algebra on those two values. In 1937 Claude Shannon, a graduate student at MIT, showed in his master's thesis that circuits of switches obey the same algebra: two switches in a row pass current only if both are closed, which is "and"; two side by side pass it if either is, which is "or". The thesis, published in 1938, has been called one of the most important master's theses ever written.

Every modern processor is built from billions of transistors wired into such circuits, called logic gates; how computers work shows them adding numbers. A computer program's "if" is a direct descendant of the truth table above.

What does logic cover, and what doesn't it?

Everything above is deductive logic, the core of the subject, where the conclusion is guaranteed. Much everyday and scientific reasoning is different: it moves from cases to a general rule, or from clues to the best explanation, and makes a conclusion likely rather than certain, which inductive logic and probability study; deductive vs inductive reasoning compares them. Since the mathematician Gottlob Frege's Begriffsschrift of 1879, logic has also become a branch of mathematics, and a mathematical proof is a chain of valid steps from agreed starting points.

The most common misunderstanding is that logic can tell you what's true. It can't, on its own. A perfectly valid argument from a false premise reaches a false conclusion, and the penguin argument shows it. Logic tells you what you are committed to by what you already accept, which makes it as useful for finding a mistaken premise as for proving a conclusion. When an argument that looks valid, from premises that look true, leads somewhere impossible, you have a paradox, and something has to give.

Deciding which premises to trust is the job of evidence. Logic and evidence together are most of what philosophy means by reasoning well.

Check yourself

• Without looking back, explain the difference between a valid argument and a sound one, with an example of an argument that is valid but unsound.
• Why is "if it rains, the match is off; the match is off; so it rained" invalid?
• In what single case is "if P then Q" false?

Sources

• Validity and Soundness — Internet Encyclopedia of Philosophy
• Propositional Logic — Internet Encyclopedia of Philosophy
• Aristotle's Logic — Robin Smith, Stanford Encyclopedia of Philosophy, revised 2022
• Classical Logic — Stewart Shapiro, Stanford Encyclopedia of Philosophy, revised 2026
• A Symbolic Analysis of Relay and Switching Circuits — Wikipedia, on Claude Shannon's 1938 thesis