Propositional Logic Flashcards
A proposition is a
declaration of fact that is either true or false, but not both
¬
Negation
not
∧
Conjunction
and
∨
Disjunction
(inclusive) or
⨁
Exclusive Disjunction
exclusive or, xor
→
Conditional
If .. then, ..implies..
↔
Biconditional
if and only if, iff
negation is defined by the following truth table:

Objects cannot be negated, only statements
Incorrect negation:
¬JackistallerthanJoe ≡(¬Jackistallerthan¬Joe)
The correct negation is:
¬JackistallerthanJoe ≡JackisatmostastallasJoe.
Negation Rules for Inequalities
the negation of a strict inequality is non-strict inequality and vice versa.
¬𝑥>𝑦 ≡(𝑥≤𝑦)
¬𝑥≥𝑦 ≡(𝑥<𝑦)
You must exercise special care when negating a double inequality.
The correct negation of 1 ≤ 𝑥 ≤ 2 is
𝑥<1∨𝑥>2
Conjunction

Disjunction

Exclusive Disjunction (aka Exclusive Or)

Conditional

Biconditional
𝑝 ↔ 𝑞 ≡ (𝑝 → 𝑞) ∧ (𝑞 → 𝑝)

Observe that 𝑝 ↔ 𝑞 is the negation of
𝑝⨁𝑞.
Also observe that the biconditional is true exactly when p and q have the same truth value, whether that is true or false.
It is not a very good idea to attempt to reduce the understanding of “necessary” and “sufficient” typestatements to the mindlessly symbolical level.

Just like “sufficient” type phrases in English defy simplistic word order analysis, so do “necessary” type statements.

It is helpful to change “necessary” to
“precondition”.For example, we rephrase
“A cloudless sky is necessary to see the stars” as
“A cloudless sky is a precondition for seeing the stars.”
The word precondition makes it clear that the cloudless sky does not guarantee that we can see the stars
“p only if q” means
𝑝 → 𝑞
If you find this hard to remember, just translate the conditional as if the word only was not there, and then switch premise and conclusion.
“p unless q” means
¬𝑞 → 𝑝
This rule shows that unless means if not. “You will get sick unless you take the medicine” means “You will get sick if you don’t take the medicine.”
𝑞 → 𝑝 is called the converse of
𝑝 → 𝑞
¬𝑝 → ¬𝑞 is called the inverse of
𝑝 → 𝑞


