User:Phlsph7/Rule of inference - Definition

Definition

A rule of inference is a way of drawing a conclusion from a set of premises. Also called inference rule and transformation rule, it is a norm of correct inferences that can be used to guide reasoning, justify conclusions, and criticize arguments. As part of deductive logic, rules of inference are argument forms that preserve the truth of the premises, meaning that the conclusion is always true if the premises are true. An inference is deductively correct or valid if it follows a rule of inference. Whether this is the case depends only on the form or syntactical structure of the premises and the conclusion. As a result, the actual content or concrete meaning of the statements does not affect validity. For instance, modus ponens is a rule of inference that connects two premises of the form "if p then q" and "p" to the conclusion "q", where p and q stand for statements. Any argument with this form is valid, independent of the specific meanings of p and q, such as the argument "if it is raining, then the ground is wet; it is raining; therefore, the ground is wet". In addition to modus ponens, there are many other rules of inference, such as modus tollens, disjunctive syllogism, hypothetical syllogism, constructive dilemma, and destructive dilemma.

Rules of inference belong to logical systems and distinct logical systems may use different rules of inference. For example, universal instantiation is a rule of inference in the system of first-order logic but not in propositional logic. Rules of inference play a central role in proofs as explicit procedures for arriving at a new line of a proof based on the preceding lines. Proofs involve a series of inferential steps and often use various rules of inference to establish the theorem they intend to demonstrate. As standards or procedures governing the transformation of symbolic expressions, rules of inference are similar to mathematical functions taking premises as input and producing a conclusion as output. According to one interpretation, rules of inference are inherent in logical operators found in statements, making the meaning and function of these operators explicit without adding any additional information.

Logicians distinguish two types of rules of inference: rules of implication and rules of replacement. Rules of implication, like modus ponens, operate only in one direction, meaning that the conclusion can be deduced from the premises but the premises cannot be deduced from the conclusion. Rules of replacement, by contrast, operate in both directions, stating that two expressions are equivalent and can be freely replaced with each other. In classical logic, for example, a proposition (p) is equivalent to the negation of its negation (¬¬p). As a result, one can infer one from the other in either direction, making it a rule of replacement. Other rules of replacement include De Morgan's laws as well as the commutative and associative properties of conjunction and disjunction. While rules of implication apply only to complete statements, rules of replacement can be applied to any part of a compound statement.

References

Sources

• Tourlakis, George (2011). Mathematical Logic. John Wiley & Sons. ISBN 978-1-118-03069-1.
• Shanker, Stuart (2003). "Glossary". In Shanker, Stuart (ed.). Philosophy of Science, Logic and Mathematics in the Twentieth Century. Psychology Press. ISBN 978-0-415-30881-6.