Syntax: Propositional Logic

Propositions are denoted with capital letters P, Q, R...

For example:

P = "I cheated"

Q = "I wrote exam"

Lowercase letters (p, q, r...) are used for general propositions that have no meaning.

(❕ Use lowercase letters for general proofs)

Connectives

• p is well-formed formula (we will just use wff to save space)
• ¬p is a wff (It means not p)
• p ∧ q is a wff (It means p and q)
• p ∨ q is wff (It means p or q)
• p → q is wff (It means if p then q)

Additionally, we have if and only if p then q and we write it as p ↔ q.

Practice

Let's do a little practice to strengthen our newly introduced connectives.

Translate into English:

We know that

P = I cheat; Q = I will get caught; R = I write an exam; S = I will fail.

Translate ( R ∧ P) → (Q ∧ S) in to English.

Answer:

So the proper translation will be: If I write an exam and I cheat then I will get caught and I will fail.

Translate into Propositional logic:

Given:

"If James not die then Mary will not get any money and Jame's family will be happy."

Let's create a formula for these set of propositions. First, we should figure out what are the propositions. There are three propositions here:

1. James dies
2. Marry will get money
3. Jame's family will be happy

Of course, just propositions seem a little bit weird😜.

Then, assign them with letters:

• P = James dies
• Q = Marry will get money
• R = Jame's family will be happy

Now, we should change some words with connective symbols:

• If .., then => →
• not => ¬
• and => ∧

Finally, we put everything together:

¬P → ( ¬Q ∧ R )

This was the common syntax of Propositional Logic. Hope it was helpful.

Written by Asadulloh Rakhimov