## Operators & Symbols
| **Symbol** | **Operation** | **Example** | **Description** |
|:--------------------------------------:|:-----------------------------------------:|:---------------------------------------:|:-------------------------------------------------------------------------------------------------------- |
| → | Implies | P→Q | A proposition that is only false if Q is false. |
| ↔ | Biconditional | P↔Q | A proposition that denotes logic equivalence, and is true when both predicates are either true or false. |
| ¬ | Negation | ¬P | A proposition that is true if and only if P is false. |
| ∧ | Conjunction | P∧Q | A proposition that is true if and only if both P and Q are true. |
| ∨ | Disjunction | P∨Q | A proposition that is false if and only if both P and Q are false. |
| ⊻ | Exclusive Or | P⊻Q | A proposition that is true if and only if either only P or only Q are true. |
| ∀ | Universal Quantifier | ∀x(Px) | A proposition that states that for all x, P is true. |
| ∃ | Existential Quantifier | ∃x(Px) | A proposition that states that there is at least one X, such that P(x) is true. |
| ∴ | Therefore | ∴Q | The conclusion of a syllogism |
| := | Definition | P:=grass is green | A general symbol used for defining propositions. |
| () | Parentheses | (P∧Q) | A means of grouping propositions together. |
# Logical Necessity
### Tautology
| **P** | **True** |
| --------------------------------- | ------------------------------------ |
| T | T |
| F | T |
# Logical Impossibility
### Contradiction
| **P** | **¬P** | **P∧¬P** |
|:---------------------------------:|:----------------------------------:|:------------------------------------ |
| T | F | F |
| F | T | F |
# Logical Possibility
### Implication (P→Q)
| **P** | **Q** | **P→Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
### Biconditional (P↔Q)
| **P** | **Q** | **P↔Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
### Conjunction (P∧Q)
| **P** | **Q** | **P∧Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F | F | T | F |
### Disjunction (P∨Q)
| **P** | **Q** | **P∨Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
### Exclusive Disjunction (P⊻Q)
| **P** | **Q** | **P⊻Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
# Esoteric Logical Possibility
### Not And (P⊼Q)
| **P** | **Q** | **P⊼Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
### Not Or (P⊽Q)
| **P** | **Q** | **P→Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
### Material Nonimplication (P↛Q)
| **P** | **Q** | **P→Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | F |
| T | F | T |
| F | T | F |
| F | F | F |
### Converse Nonimplication (P↚Q)
| **P** | **Q** | **P→Q** |
|:---------------------------------:|:---------------------------------:|:----------------------------------- |
| T | T | F |
| T | F | F |
| F | T | T |
| F | F | F |
# Hashtags
#coursework
#logic_course
#propositional_logic