Propositional Logic
What are Propositions? #
Propositions are anything that can be true or false. This could include:
- Statements like “Birds can fly”.
- Well defined equations with no free variables like $1 + 1 = 3$.
Propositions are not:
- Variables like $x$ or $5$.
- Equations with free variables like $P(x) = y$.
- Statements that aren’t clearly true or false, like “I like trains.”
Connectives #
Simple propositions can be joined together to make complex statements. There are three basic ways to connect propositions together:
- Conjunction is the and operation: for $P \land Q$to be true, $P$and $Q$must both be true.
- Disjunction is the or operation: for $P \lor Q$to be true, either $P$or $Q$must be true.
- Negation is the not operation: if $P$is true, then $\lnot P$is false.
- The law of the excluded middle states that $P$and $\lnot P$ cannot both be true.
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:
$ \lnot(P \lor Q) \iff (\lnot P \land \lnot Q) $
“If neither P nor Q are true, then P and Q must both be false.”
$ \lnot(\forall x)(P(x)) \iff (\exists x)(\lnot P(x)) $
“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.
$ (P \lor Q) \land R \equiv (P \land R) \lor (Q \land R) $
Implication #
One proposition can imply another, which looks like this:
$ P \implies Q $
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.)
- Contrapositive Equivalence: If P implies Q, then $\lnot Q \implies \lnot P$.
- Note that this is different from the converse, which is $Q \implies P$. This statement is not logically equivalent!
Truth Table #
| P | Q | P $\implies$Q | P $\iff$Q |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | T |
Note that the truth table for $P \implies Q$ is equivalent to the one for $\lnot P \lor Q$! That means this formula is logically the same as $P \implies Q$.
(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!)
Quantifiers #
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:
$ (\exists x \in \mathbb{N})(x=x^2) $
You could think about the parentheses almost like defining a scope of variables, like what might happen in programming! Here, the first clause is defining an arbitrary variable $x$to be any natural number.
Exercises #
(Source: Discussion 0 2a)
No, they are not equivalent. We can see this more clearly by converting the implication $Q \implies P$ to $\lnot Q \lor P$ as was demonstrated in the Truth Table section above.
On the left side, this conversion is straightforward, yielding $\forall x \exists y (\lnot Q(x,y) \lor P(x))$.
On the right side, we’ll need to invoke De Morgan’s Laws to convert the ’exists’ into a ‘for all’ since it was negated. This yields $\forall x (\forall y\lnot(Q(x,y)) \lor P(x))$which is not the same thing!
An integer $a$is said to divide another integer $b$ if $a$is a multiple of $b$. Write this idea out using propositional logic (a divides b can be written as $a \mid b$).
Note: This idea is going to be important for a lot of future sections!
$a \mid b \iff (\exists q \in \mathbb{Z})(a = qb)$
In plain English: “There exists an integer $q$such that when we multiply $q$with $b$, we get $a$.”
Resources #
Note 1:
https://www.eecs70.org/assets/pdf/notes/n1.pdf
Discussion 0:
https://www.eecs70.org/assets/pdf/dis00a.pdf