Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Existential quantification
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== Rules of inference === {{Transformation rules}} A [[rule of inference]] is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the existential quantifier. ''[[Existential generalization|Existential introduction]]'' (∃I) concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically, :<math> P(a) \to\ \exists{x}{\in}\mathbf{X}\, P(x)</math> [[Existential elimination|Existential instantiation]], when conducted in a Fitch style deduction, proceeds by entering a new sub-derivation while substituting an existentially quantified variable for a subject—which does not appear within any active sub-derivation. If a conclusion can be reached within this sub-derivation in which the substituted subject does not appear, then one can exit that sub-derivation with that conclusion. The reasoning behind existential elimination (∃E) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is [[logical truth|necessarily true]], as long as it does not contain the name. Symbolically, for an arbitrary ''c'' and for a proposition ''Q'' in which ''c'' does not appear: :<math> \exists{x}{\in}\mathbf{X}\, P(x) \to\ ((P(c) \to\ Q) \to\ Q)</math> <math>P(c) \to\ Q</math> must be true for all values of ''c'' over the same domain ''X''; else, the logic does not follow: If ''c'' is not arbitrary, and is instead a specific element of the domain of discourse, then stating ''P''(''c'') might unjustifiably give more information about that object.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)