Existential Quantification - Properties - Rules of Inference

Transformation rules
Propositional calculus

Rules of inference
Modus ponens
Modus tollens
Biconditional introduction
Biconditional elimination
Conjunction introduction
Disjunction introduction
Disjunction elimination
Disjunctive syllogism
Hypothetical syllogism
Constructive dilemma
Destructive dilemma

Rules of replacement

Double negation
De Morgan's laws
Material implication
Predicate logic
Universal generalization
Universal instantiation
Existential generalization
Existential instantiation

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 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,

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 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:

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.

Read more about this topic:  Existential Quantification, Properties

Other articles related to "rules of inference":

Logical Biconditional - Rules of Inference - Biconditional Elimination
... For example, if it's true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, I'm alive likewise, it's true that if I'm alive, I'm breathing ... Formally ( A ↔ B ) ∴ ( A → B ) also ( A ↔ B ) ∴ ( B → A ) ...

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