Ready for Proofs?

We have propositions

• p v q, p ^ q
• "If x is divisible by 2, then x * 2 is divisible by 4 for all real numbers"
• "You own a PC and you have a Facebook account"
• "If you own a PC, then you can play DOTA 2"

Chain them together

• "I own a PC and I have Steam installed"
• "If I have Steam installed, I can install DOTA 2"
• "If I can install DOTA 2, then I can play DOTA 2"

Therefore, I can play DOTA 2

Argument

A sequence of statements (premises) that end with a conclusion. With propositional logic, it's a series of propositions ending with a conclusion. In a sense, we're "and-ing" it all together and then inferring something.

An argument form is a sequence of compound propositions involving propositional variables.

Valid Arguments

An argument is considered valid if the conclusion follows based on the truth of the premises. Or, if all the premises are true, then the conclusion is true.

(valid)

p ^ q

--------

∴ p

(not valid)

p v q

--------

∴ p

(valid)

p v q

¬ q

--------

∴ p

Valid Arguments

Note that the premises may not actually be true, but the argument is still valid if the conclusion follows if the premises were true.

* If George is a nice guy, then everyone will get an A.

* George is a nice guy

---------------------

∴ Everyone gets an A

We could make a truth table

Rules of Inference

Existing argument forms we can use to create more complicated argument forms. Or, argument forms we can use to (hopefully) prove our argument is valid. These argument forms are also tautologies.

Combine them with logical equivalences to win at logic.

Example

What can we conclude from the following?

"If I stay up late, then I can get ready for class."

"I can get ready for class if I don't play video games."

"I'm not ready for class. :("

Fallacy

Based on argument forms that look like an inference rule, but is not.

"(q ^ (p → q)) → p" Fallacy of affirming the conclusion

"((p → q) ^ ¬p) → ¬q" Fallacy of denying the hypothesis

Rules of Inference for Quantifiers