Symbolic Logic

[Fringe Pioneer]

In symbolic logic, you can have simple truth functional statements (statements that can be true or false). Arbitrary constants that represent a simple statement are represented by a single capital letter.

To make more complex truth functional statements, you can apply any number of any of the various logical operators.

Variables can represent a truth functional statement of any complexity, and are represented by a single lowercase letter.

Operator	Operation
~	Negation (NOT)
·	Conjunction (AND)
∨	Disjunction (OR)
⊃	Material Implication (IF...THEN) (IMPLIES)
≡	Material Equivalence (IF AND ONLY IF)

SPOILER: Click to show

x	~x
False	True
True	False

SPOILER: Click to show

x	y	x · y
False	False	False
False	True	False
True	False	False
True	True	True

SPOILER: Click to show

x	y	x ∨ y
False	False	False
False	True	True
True	False	True
True	True	True

SPOILER: Click to show

x	y	x ⊃ y
False	False	True
False	True	True
True	False	False
True	True	True

SPOILER: Click to show

x	y	x ≡ y
False	False	True
False	True	False
True	False	False
True	True	True

An argument is a set of premises (given truth functional statements) and the conclusion that they are supposed to derive, also a truth functional statement.

A person likes math or likes literature.
If a person likes math and literature, then a person likes philosophy.
A person neither not likes math nor not likes literature.
/∴ A person likes philosophy.

A person can perform calculus.
A person cannot make good poetry.
/∴ A person likes to eat cake.

An argument can be valid or invalid. An argument is invalid if there is an instance where all premises can be simultaneously true and the conclusion false; otherwise, an argument is valid.

The validity of an argument can be tested either by creating truth tables or by proofs. When using proofs, various rules of inference must be used to derive the conclusion from the premises.

[Elmach]

I use ⇔ and = for if and only if/ material biconditional.

Speaking of which, this gives a good list of all possible binary logical operators. (That is what you call it, right?)

[Fringe Pioneer]

Well, the tilde is an unary logical operator, not a binary one, but yes, logical operators.

The author of my philosophy book seems to have a different set of symbols than what other authors have, unless this is just what Wikipedia has...

[endy123]

yeah, I've noticed different authors use different symbols somewhat randomly.

Didn't know that they taught symbolic logic in Philosophy. I learned it in Discrete Mathematics myself. Pretty handy though wherever you go.