User:Silence/Logic
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'Antecedent Affirmation:
- "There are flowers."
- 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 Denial:
- "There are no flowers."
Simplified logic notation
| Constants | Variables | |
|---|---|---|
| Subjects | { }, ∅, | *x, *y, *z, *B, *C, *D, B∪C, B∩C, {x}, {x,y} |
| Predicates | Tx, Fx, =xy, =(x,y), ∈xB, ∈(x,B), ⊆AB, ⊆(A,B), ⊂AB, ⊂(A,B) | Bx, B(x), Rxy, R(x,y) |
| Operators | ~p, p∧q, (∀x)(*x) |
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)