User:Silence/Logic


 * Green = True if the statement is true. False if the statement is false.
 * Brown = True if the statement is true.
 * Orange = False if the statement is false.


 * Purple = False if the statement is true. True if the statement is false.
 * Blue = True if the statement is false. (At least one must be true.)
 * Red = False if the statement is true. (At least one must be false.)


 * Black = No correlation.

Statement      GREEN   BROWN  ORANGE   PURPLE    BLUE   RED   BLACK T            T       T      T        F        T      F     T     T             T       T      F        F        F      F     F     F             F       T      F        T        T      T     T     F             F       F      F        T        T      F     F

Statement: (∀x)(Fx ⇒ Px)
 * "All flowers are plants." / "Being a flower implies being a plant."

Antecedent Affirmation: (∃x)(Fx)
 * "There are flowers."

Antecedent Denial: (∃x)(~Fx)
 * "Some things aren't flowers."

Generalized Antecedent Affirmation: (∀x)(Fx)
 * "Everything is a flower."

Generalized Antecedent Denial: (∀x)(~Fx)
 * "Nothing is a flower."

Consequent Affirmation: (∃x)(Px)
 * "There are plants."

Consequent Denial: (∃x)(~Px)
 * "Some things aren't plants."

Generalized Consquent Affirmation: (∀x)(Px)
 * "Everything is a plant."

Generalized Consequent Denial: (∀x)(~Px)
 * "Nothing is a plant."

Negation: ~(∀x)(Fx ⇒ Px)
 * "Not all flowers are plants." / "It is false that all flowers are plants."

Contrary: (∀x)(Fx ⇒ ~Px)
 * "No flowers are plants."
 * Negates the sentence's predicate. ("Contrary" sometimes refers to the simple negation.)

Contradiction: (∃x)(Fx ∧ ~Px)
 * "Some flowers aren't plants." / "There exists a flower that isn't a plant."
 * Reverses whether the sentence is general or particular, and negates the predicate.

Subalternate: (∃x)(Fx ∧ Px)
 * "Some flowers are plants."
 * Reverses whether the sentence is general or particular. (Making this brown is Aristotle's existential fallacy.)

Dual: ~(∃x)(Fx ∧ ~Px)
 * "It is false that some flowers aren't plants."

Opposite: (∀x)(Fx ⇒ Ax)
 * "All flowers are animals."
 * ("Opposite" sometimes refers to the contrary.)

Reverse: (∀x)(Px ⇐ Fx)
 * "Being a plant is implied by being a flower." / "Plants, all flowers are."

Obverse: (∀x)(Fx ⇒ Px)
 * "No flower is not a plant."


 * 

Inverse: (∀x)(~Fx ⇒ ~Px)
 * "No non-flower is a plant." / "All non-flowers are non-plants."
 * Negates a sentence's subject. Equivalent to converse. ("Inverse" sometimes refers to the contrary.)

Converse: (∀x)(Px ⇒ Fx)
 * "All plants are flowers."
 * Transposes a sentence's subject and predicate. Equivalent to inverse, which it is the contrapositive of.

Contrapositive: (∀x)(~Px ⇒ ~Fx)
 * "No non-plant is a flower."

Transpositive: (∀x)(~Px ⇒ ~Fx)
 * "All non-plants are non-flowers."
 * An obverted contrapositive.

'Antecedent Affirmation:
 * "There are flowers."

Antecedent Denial:
 * "There are no flowers."

Simplified logic notation
1. Rule of Assumption.
 * Assume anything, to see what is derivable from it.
 * p ⊢ ⊢

2. Rule of Two Truths. (Conjunction Introduction + Tautology)
 * If two sentences are true, their conjunction is true. (The two sentences can be the same sentence.)
 * p, q / p ∧ q
 * p / p ∧ p

3. Rule of One Truth. (Addition)
 * If a sentence is true, then its negation falsely conjuncts with any statement.
 * p / ~p ⊼ q
 * p / q ⊼ ~p

4. Rule of Derivation. (Conditional Proof)
 * It can't be the case that a statement is true and something derivable from it is false.
 * p ⊢ q / p ⊼ ~q

5. Rule of 'Both' Elimination (Simplification)
 * If a conjunction is true, both conjuncts are true.
 * p ∧ q / p
 * p ∧ q / q

6. Rule of 'Not Both' Elimination (Modus Ponens + Modus Tollens)
 * If a conjunction is false, and one of the conjuncts is true, conclude that the other conjunct is false.
 * p ⊼ q, p / ~q
 * p ⊼ q, q / ~p

7. Rule of Double Negation (Double Negation Introduction + Double Negation Elimination)
 * An atomic or compound sentence is equivalent to the negation of its negation.
 * p / p
 * p / p

Further rules
9. Rule of Instantiation
 * If something is true of all individuals, it is true of any specific individual.
 * (∀x)(Bx) / Ba (for any "a")

10. Rule of Generalization
 * If something is true of an arbitrary individual, it is true of any individual.
 * Ba (for an arbitrary "a") / (∀x)(Bx)

11. Rule of Indiscernibility
 * If all the same properties are predicable of two names, the two names refer to the same individual.
 * (∀B)[(Ba ⊼ ~Bb) ∧ (Bb ⊼ ~Ba)] / =ab

ZFC
Otherwise unspecified particulars: a, b, A, R

1. Extensionality
 * ∀x∀y{∀z[(∈zx ⊼ ∉zy) ∧ (∈zy ⊼ ∉zx)] ⊼ ≢xy}

2. Regularity
 * ∀x{∈ax ⊼ [∈bx ⊼ ∀y(∈yb ⊼ ∈yx)]}

3. Separation
 * ∀z∀w₁

4. Pairing
 * ∀x∀y(∈xa ∧ ∈ya)

5. Union
 * ∀X∀Y∀z[(∈zY ∧ ∈YX) ⊼ ∉zA]

6. Replacement 7. Infinity
 * ∈(∅,A) ∧ ∀x[∈(x,A) ⊼ ∉(x∪{x},A)]

8. Power Set
 * ∀x∀y(⊆yx ⊼ ∉yA)