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A mixed hypothetical syllogism has two premises: one conditional statement and one statement that either affirms or denies the antecedent or consequent of that conditional statement. For example, If P, then Q. P. ∴ Q. In this example, the first premise is a conditional statement in which "P" is the antecedent and "Q" is the consequent.
The form of a modus ponens argument is a mixed hypothetical syllogism, with two premises and a conclusion: If P, then Q. P. Therefore, Q. The first premise is a conditional ("if–then") claim, namely that P implies Q. The second premise is an assertion that P, the antecedent of the conditional claim, is the case.
Disjunctive syllogism (sometimes abbreviated DS) has one of the same characteristics as modus tollens in that it contains a premise, then in a second premise it denies a statement, leading to the conclusion. In Disjunctive Syllogism, the first premise establishes two options.
Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. 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.
Prototypical conditional sentences in English are those of the form "If X, then Y". The clause X is referred to as the antecedent (or protasis), while the clause Y is called the consequent (or apodosis). A conditional is understood as expressing its consequent under the temporary hypothetical assumption of its antecedent.
Phrased another way, denying the antecedent occurs in the context of an indicative conditional statement and assumes that the negation of the antecedent implies the negation of the consequent. It is a type of mixed hypothetical syllogism that takes on the following form: [1] If P, then Q. Not P. Therefore, not Q. which may also be phrased as
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science.Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language.
In the philosophy of logic and logic, specifically in deductive reasoning, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions).
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