Predicates and Quantified Statements
Universal |
∀x ∈ D, P(x) |
∃x∈ D, ~P(x) |
Existential |
∃(x,y)∈ D,x≠y|P(x,y) |
∀(x,y)∈ D,x≠y|~P(x,y) |
Universal Conditional |
∀x, P(x)→Q(x) |
∃x∈ D|P(x)∧~Q(x) |
~existential = universal
~universal = existential
More Formal Statements
Formal Contrapositive |
∀x ∈ D, ~Q(x)→~P(x) |
Formal Converse |
Q(x)→P(x) ∀x ∈ D |
Formal Inverse |
∀x ∈ D, ~P(x)→~Q(x) |
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MQ Invalid Arguments
Quantified Converse |
∀x, P(x)→Q(x) |
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Q(j) for a particular j |
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∴P(j) |
Quantified Inverse Error |
∀x, P(x)→Q(x) |
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~P(j) for a particular j |
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∴~Q(j) |
Multiple Quantifiers
Existential MQ |
∃x ∈ D|∀y ∈ E, P(x, y) |
Neg. MQ |
∀x ∈ D, ∃y ∈ E | P(x, y) [original] |
∃x ∈ D, ∀y ∈ E | ~P(x, y) [negation] |
Universal Modes Pones |
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∀x ∈ Z, P(x)→Q(x) |
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P(k), for a particular k ∈ Z |
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∴ ~Q(k) |
Universal Modus Tones |
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∀x ∈ D, P(x)→Q(x) |
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~Q(j), j ∈ D |
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∴ ~P(j) |
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