Three classical foundational laws of logic
These three laws, sometimes called the "laws of thought," have been considered the bedrock of classical logic since Aristotle.
Law of Identity
A is A. Everything is identical to itself. Each thing is what it is and not something else. An apple is an apple, not an orange.
Law of Non-Contradiction
A cannot be both A and not-A at the same time and in the same respect.
Something cannot simultaneously be and not be. A door cannot be both open and closed at the same moment.
Law of Excluded Middle
Either A or not-A. There is no middle ground.
Every proposition is either true or false. The light is either on or off—there is no third option between being and non-being.
Other Major Laws of Logic
Law of Sufficient Reason
Nothing exists or is true without a sufficient reason why it is so and not otherwise.
Everything has an explanation or cause. If something is true, there must be a reason it is true rather than false.
Leibniz's Law (Identity of Indiscernibles)
If A and B are identical, they share all the same properties.
If two things are truly the same thing, whatever is true of one must be true of the other. You can substitute identical things without changing truth.
Law of Bivalence
Every proposition is either true or false, not both, not neither.
This strengthens the Law of Excluded Middle by insisting propositions must have exactly one of two truth values.
Modus Ponens
If P implies Q, and P is true, then Q is true.
If the rule holds and the condition is met, the consequence follows. If "rain makes streets wet" and it is raining, then streets are wet.
Modus Tollens
If P implies Q, and Q is false, then P is false.
If the consequence did not occur, the condition did not happen. If streets are not wet, then it did not rain.
Law of Syllogism (Hypothetical Syllogism)
If P implies Q, and Q implies R, then P implies R.
Logical chains connect. If A leads to B, and B leads to C, then A leads to C.
De Morgan's Laws
The negation of "A and B" equals "not-A or not-B." The negation of "A or B" equals "not-A and not-B."
These laws govern how negation distributes over conjunction and disjunction, showing the relationship between AND and OR.
Principle of Explosion
From a contradiction, anything follows.
If you accept both A and not-A as true, you can derive any proposition whatsoever. Contradiction destroys logical coherence.