Absorption / modus tollens / modus ponendo tollens
Negation introduction
Rules of replacement
Associativity
Commutativity
Distributivity
Double negation
De Morgan's laws
Transposition
Material implication
Exportation
Tautology
Predicate logic
Rules of inference
Universal generalization / instantiation
Existential generalization / instantiation
In predicate logic, universal instantiation[1][2][3] (UI; also called universal specification or universal elimination,[citation needed] and sometimes confused with dictum de omni)[citation needed] is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory.
Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."
Formally, the rule as an axiom schema is given as
for every formula A and every term t, where is the result of substituting t for each free occurrence of x in A. is an instance of
And as a rule of inference it is
from infer
Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934."[4]
^Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 978-0205820375.[page needed]
^Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.
^Moore and Parker[full citation needed]
^Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ
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(Existential instantiation) y ≠ x {\displaystyle y\not =x} (Existential instantiation) ∀ x ( x ≠ x ) {\displaystyle \forall x\,(x\not =x)} (Faulty universal generalization)...
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that which provides us with what is known as UniversalInstantiation: Axiom scheme for UniversalInstantiation. Given a formula ϕ {\displaystyle \phi } in...
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and there exists. Common proof rules used are modus ponens and universalinstantiation. In contrast, an indirect proof may begin with certain hypothetical...
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variables for probability densities Substitution property of equality Universalinstantiation Kaplan, Wilfred (1973). "Change of Variables in Integrals". Advanced...
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standard, one replaces universalinstantiation, ∀ x ϕ x → ϕ y {\displaystyle \forall x\,\phi x\rightarrow \phi y} , with universal specification ( ∀ x ϕ...
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conceptualism) that universals are contained within the mind of God. To complicate things, what is the nature of the instantiation or exemplification relation...
{\displaystyle P(a)\to \ \exists {x}{\in }\mathbf {X} \,P(x)} Existential instantiation, when conducted in a Fitch style deduction, proceeds by entering a new...
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