2024-06-28 02:13:12 -05:00
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## Operators & Symbols
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| <font color="CC6600">**Symbol**</font> | <font color="CC6600">**Operation**</font> | <font color="CC6600">**Example**</font> | <font color="CC6600">**Description**</font> |
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|:--------------------------------------:|:-----------------------------------------:|:---------------------------------------:|:-------------------------------------------------------------------------------------------------------- |
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| → | Implies | P→Q | A proposition that is only false if Q is false. |
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| ↔ | Biconditional | P↔Q | A proposition that denotes logic equivalence, and is true when both predicates are either true or false. |
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| ¬ | Negation | ¬P | A proposition that is true if and only if P is false. |
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| ∧ | Conjunction | P∧Q | A proposition that is true if and only if both P and Q are true. |
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| ∨ | Disjunction | P∨Q | A proposition that is false if and only if both P and Q are false. |
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| ⊻ | Exclusive Or | P⊻Q | A proposition that is true if and only if either only P or only Q are true. |
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| ∀ | Universal Quantifier | ∀x(Px) | A proposition that states that for all x, P is true. |
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| ∃ | Existential Quantifier | ∃x(Px) | A proposition that states that there is at least one X, such that P(x) is true. |
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| ∴ | Therefore | ∴Q | The conclusion of a syllogism |
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| := | Definition | P:=grass is green | A general symbol used for defining propositions. |
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| () | Parentheses | (P∧Q) | A means of grouping propositions together. |
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# Logical Necessity
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### Tautology
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| <font color="CC6600">**P**</font> | <font color="CC6600">**True**</font> |
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| --------------------------------- | ------------------------------------ |
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| T | T |
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| F | T |
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# Logical Impossibility
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### Contradiction
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| <font color="CC6600">**P**</font> | <font color="CC6600">**¬P**</font> | <font color="CC6600">**P∧¬P**</font> |
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|:---------------------------------:|:----------------------------------:|:------------------------------------ |
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| T | F | F |
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| F | T | F |
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# Logical Possibility
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### Implication (P→Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P→Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | T |
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| T | F | F |
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| F | T | T |
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| F | F | T |
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### Biconditional (P↔Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P↔Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | T |
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| T | F | F |
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| F | T | F |
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| F | F | T |
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### Conjunction (P∧Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P∧Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | T |
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| T | F | F |
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| F | T | F |
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| F | F | F | F | T | F |
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### Disjunction (P∨Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P∨Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | T |
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| T | F | T |
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| F | T | T |
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| F | F | F |
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### Exclusive Disjunction (P⊻Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P⊻Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | F |
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| T | F | T |
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| F | T | T |
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| F | F | F |
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# Esoteric Logical Possibility
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### Not And (P⊼Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P⊼Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | F |
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| T | F | T |
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| F | T | T |
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| F | F | T |
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### Not Or (P⊽Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P→Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | F |
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| T | F | F |
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| F | T | F |
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| F | F | T |
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### Material Nonimplication (P↛Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P→Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | F |
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| T | F | T |
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| F | T | F |
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| F | F | F |
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### Converse Nonimplication (P↚Q)
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| <font color="CC6600">**P**</font> | <font color="CC6600">**Q**</font> | <font color="CC6600">**P→Q**</font> |
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|:---------------------------------:|:---------------------------------:|:----------------------------------- |
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| T | T | F |
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| T | F | F |
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| F | T | T |
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| F | F | F |
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# Hashtags
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#coursework
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#logic_course
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#propositional_logic
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