Propositional Logic

What are Propositions?

Propositions are anything that can be true or false. This could include:

  • Statements like "Birds can fly".

Propositions are not:

  • Statements that aren't clearly true or false, like "I like trains."


Simple propositions can be joined together to make complex statements. There are three basic ways to connect propositions together:

One example where we can see these components in action is in De Morgan's Laws, which state how negation can be distributed across conjunction or disjunction:

"If neither P nor Q are true, then P and Q must both be false."

"If P(x) isn't true for every x, then there exists an x where P(x) is false."

Another example of distribution is this congruence, which works for any combination of and's and or's.


One proposition can imply another, which looks like this:

Roughly, implication in plain English can be stated in the form if P, then Q. However, there are a lot of nuances to what this really means!

Properties of Implication

  • Reversible: Q is true if P is true. However, be careful- this doesn't necessary mean that Q implies P!

  • P is sufficient for Q: Proving P allows us to say that Q is also true.

  • Q is necessary for P: For P to be true, it is necessary that Q is true. (If Q is false, then P is also false.)

Truth Table

(If two propositions have the same truth table, then they are logically equivalent. However, it's still possible for a proposition to imply another even if their truth tables are different!)


Sometimes, we need to define a specific type of variable to work with in a propositional clause. For instance, take the proposition, "There exists a natural number that is equal to the square of itself." We could write this as:


Note: This idea is going to be important for a lot of future sections!


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