Propositional Logic

When we talk about logic, we're also talking about arguments - how to make sure that the conclusion of an argumemt follows from the premises. Arguments written as a block of text can be vague, or prone to misinterpretation. Propositional logic is the process of converting the premises of an argument ("propositions") into symbolic form using operators and symbols.

Operators

operator	name	logical function	used to translate
˜	tilde	negation	not, it is not the case that
·	dot	conjunction	and, also, moreover, but
∨	wedge	disjunction	or, unless, either... or
⊃	horseshoe	conditional (material implication)	if... then, only if
≡	triple bar	biconditional (material equivalence)	if and only if

Truth Functions

Conjunction

p	q	p · q
T	T	T
T	F	F
F	T	F
F	F	F

Disjunction

p	q	p ∨ q
T	T	T
T	F	Y
F	T	T
F	F	F

Conditional

p	q	p ⊃ q
T	T	T
T	F	F
F	T	T
F	F	T

Bionditional

p	q	p ≡ q
T	T	T
T	F	F
F	T	F
F	F	T

Negation

p	~p
T	F
F	T

Negation
NOT p, the opposite of the given truth value.

Conjunction
p AND q, both statements need to be true for a conjuction to be true.

Disjunction
EITHER p OR q, at least one statement needs to be true for a disjunction to be true.

Conditional
IF p THEN q ... p ONLY IF q, conditional statements are false if the antecedent is true and the consequent is false, yet true in any other case. while this may not seem intuitive at first, you can illustrate this concept clearly with an example (or, just memorize it). Want to check your knowledge? Do the homework now, or wait until you have finished reading this section... ^_^

Homework checkpoint!

Write the conditional statement(s) for the following:

If Twilight is the Element of Friendship (F), then Fluttershy is the Element of Kindness (K).
Twilight is the Element of Friendship (F) if Fluttershy is the Element of Kindness (K).
Twilight is the Element of Friendship (F) only if Fluttershy is the Element of Kindness (K).

At first, conditionals may be confusing. Some statements may look to be in a certain order, but mean something different logically. A statement may be written like "then ... if", yet omitting the "then", like #2 on the homework. On the other hand, "only if" is a good indication that the second statement must be true for the statement to be true, no matter what. The difference between "if" and "only if" is that in p "only if" q, statement q is the necessary condition that must be true in order to have a true conditional; but, if the statement is p "if" q, statement p is the necessary condition (there is no modification preposition before the first statement in #2 - it is simply stated as necessary).

Note: When I say first statement and second statement, I am talking about the order of the un-symbolized statements (or, how they appear before they are translated). Antecedent and consequent refer to the logical order - the "first" and "second" statements respectively when an argument is translated into symbolic form.

Let's say you enter an art competition, and your friend promises you that "If you win the art competition, then I will take you out for dinner". On what conditions would you say that your friend broke their promise?

T ⊃ T is true. If both statements are true, the conditional statement is true.
T ⊃ F is false. If you won the competition yet your friend did not take you out for dinner, the statement is false. Winning the competition implies with certainty that your friend takes you out for dinner - it does not take into account that your friend can change their mind, or the other nuances that are present in reality. Symbolizing arguments takes away lots of uncertainty, which is why it is applicable in computer science - programs only do what you tell them to do.
F ⊃ T is true. If you do not win the competition, yet your friend takes you out to dinner, it is still true because conditional statements don't go two ways - i.e. your friend taking you out to dinner does not imply with certainty that you won the competition. Your friend could have taken you out for dinner for another reason!
F ⊃ F is true. If you do not win the competition and your friend does not take you out to dinner, the statement is true. There is no contradiction taking place - if you did not win, you cannot say that your friend would have not taken you to dinner if you had won. This logic applies to F ⊃ T as well.

Bionditional
p IF AND ONLY IF> q, material equivalence - more tba.

Natural Deduction

hiiii

Rules of Inference

1. Rules of Implication 2. Rules of Replacement 3. Assumption Rules

Most information on this page is adapted from A Concise Introduction to Logic, 12th ed.