Logical reasoning

Arguments

An argument is made up of propositions. A series of propositions called hypotheses lead up to a final proposition called the conclusion.

A valid argument is a guarantee that the conclusion is true whenever all of the hypotheses are true.

The form of an argument can be written out like:

p1
p2
p3
...
__
∴ c (the tripple dot means "therefore")

The order of the propositions does not matter.

Truth Value

The truth value of an argument is determined by it's form. So even this argument:

1+1 != 3
if 1+1 = 3, then dogs can fly
_____________________
∴ dogs cannot fly

even though it seems valid, this argument is false because it's form is invalid

¬p
p -> q
______
∴ q

this form is invalid if p = false and q = true

Rules of inference with propositions

These arguments are known to be valid, and can be used to prove other arguments as valid:

modus ponens
p
p -> q
∴ q

modes tollens
¬q
p -> q
∴ ¬p

addition
p
∴ p ∨ q

simplification
p ∧ q
∴ p

conjunction
p
q
∴ p ∧ q

hypothetical syllogism
p -> q
q -> r
∴ p -> r

disjunctive syllogism
p ∨ q
¬p
∴ q

resolution
p ∨ q
¬p ∨ r
∴ q ∨ r

Inference

When using arguments in proofs, you should label the thing you are doing on each line:

3 is an integer. Element definition.
c is an arbitrary integer above zero. Hypothesis.
t Hypothisis
t ^ c Conjunction of lines 2 and 3

Inference for quantified statements

Universal Instantiation
c is an element (arbitrary or particular)
∀x P(x)
∴ P(c)

Universal Generalization
c is an arbitrary element
P(c)
∴ ∀x P(x)

Existential Instantiation
∃x P(x)
∴ (c is a particular element) ∧ P(c)

Existential Generalization
c is an element (arbitrary or particular)
P(c)
∴ ∃x P(x)

When using existential instantiation in proofs make sure to use a new particular for each hypothesis because one hypothesis does not imply the other.