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A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
In propositional logic, disjunction elimination [1] [2] (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.
In propositional logic, modus ponens (/ ˈ m oʊ d ə s ˈ p oʊ n ɛ n z /; MP), also known as modus ponendo ponens (from Latin 'mode that by affirming affirms'), [1] implication elimination, or affirming the antecedent, [2] is a deductive argument form and rule of inference. [3] It can be summarized as "P implies Q. P is true. Therefore, Q ...
For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics ), in the sense that if the premises are true (under ...
The rule allows one to introduce or eliminate a negation from a formal proof. The rule is based on the equivalence of, for example, It is false that it is not raining. and It is raining. The double negation introduction rule is: P P. and the double negation elimination rule is:
The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself. An example in English: It's raining and it's pouring. Therefore it's raining. The rule consists of two separate sub-rules, which can be expressed in formal language as: and
There are ten primitive rules of proof, which are the rule assumption, plus four pairs of introduction and elimination rules for the binary connectives, and the rule reductio ad adbsurdum. [38] Disjunctive Syllogism can be used as an easier alternative to the proper ∨-elimination, [ 38 ] and MTT and DN are commonly given rules, [ 108 ...
Example requires a quantifier over predicates, which cannot be implemented in single-sorted first-order logic: Zj → ∃X(Xj∧Xp). Quantification over properties Santa Claus has all the attributes of a sadist. Example requires quantifiers over predicates, which cannot be implemented in single-sorted first-order logic: ∀X(∀x(Sx → Xx) → ...