Rule of inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. Modus ponens, an influential rule of inference, connects two premises of the form "if then " and "" to the conclusion "", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as modus tollens, disjunctive syllogism, constructive dilemma, and existential generalization.
Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. They contrast with formal fallaciesinvalid argument forms involving logical errors.
Rules of inference belong to logical systems, and distinct systems use different rules. Propositional logic examines the inferential patterns of simple and compound statements. First-order logic extends propositional logic by analyzing the components or internal structure of propositions. It introduces new rules of inference governing how this internal structure affects valid arguments. Modal logics explore concepts like possibility and necessity, examining the inferential structure of these concepts. Intuitionistic, paraconsistent, and many-valued logics propose alternative inferential patterns that differ from the traditionally dominant approach associated with classical logic. Various formalisms are used to express logical systems. Natural deduction systems employ many intuitive rules of inference to reflect how people naturally reason, while Hilbert systems provide minimalistic frameworks to represent foundational principles without redundancy.
Rules of inference are relevant to many areas, such as proofs in mathematics and automated reasoning in computer science. Their conceptual and psychological underpinnings are studied by philosophers of logic and cognitive psychologists.
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 valid if it follows a correct rule of inference. Whether this is the case depends only on the form or syntactic structure of the premises and the conclusion, that is, 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 then " and "" to the conclusion "". The letters and in this example and in later formulas are so-called metavariables: they stand for any simple or compound proposition. Any argument following modus ponens is valid, independent of the specific meanings of and, such as the argument "If it is day, then it is light. It is day. Therefore, it is light." In addition to 'modus ponens, there are many other rules of inference, such as modus tollens, disjunctive syllogism, and constructive dilemma.There are different formats to represent rules of inference. A common approach is to use a new line for each premise and to separate the premises from the conclusion using a horizontal line. With this format, modus ponens is written as:
Some logicians employ the therefore sign either together with or instead of the horizontal line to indicate where the conclusion begins. The sequent notation, a different approach, uses a single line in which the premises are separated by commas and connected to the conclusion with the turnstile symbol, as in.
Rules of inference are part of logical systems and different systems employ distinct sets of rules. 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 deriving new lines of a proof from preceding lines. Proofs involve a series of inferential steps and often use various rules of inference to establish the theorem they intend to demonstrate. Rules of inference are definitory rulesrules about which inferences are allowed. They contrast with strategic rules, which govern the inferential steps needed to prove a certain theorem from a specific set of premises. Mastering definitory rules by itself is not sufficient for effective reasoning since they provide little guidance on how to reach the intended conclusion. 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 is equivalent to the negation of its negation. 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.
Deductive rules of inference differ from defeasible argumentation schemes, which describe patterns of reasoning that provide some support to a conclusion without guaranteeing its truth, such as the argument from authority and the argument from analogy. However, the term "rule of inference" is sometimes used in a looser sense to include non-deductive argumentation schemes. Similarly, the term is occasionally interpreted broadly to include general standards of research, such as the principle that scientific experiments should be replicable.
One of the first discussions of formal rules of inference dates to antiquity, in Aristotle's logic. His explanations of valid and invalid syllogisms were further refined in medieval and early modern philosophy. The development of symbolic logic in the 19th century, such as George Boole's articulation of Boolean algebra, led to the formulation of many additional rules of inference belonging to classical propositional and first-order logic. In the 20th and 21st centuries, logicians developed various non-classical systems of logic with alternative rules of inference.
Basic concepts
Rules of inference describe the structure of arguments, which consist of premises that support a conclusion. Premises and conclusions are statements or propositions about what is true. For instance, the assertion "The door is open." is a statement that is either true or false, while the question "Is the door open?" and the command "Open the door!" are not statements and have no truth value. An inference is a step of reasoning from premises to a conclusion, while an argument is the outward expression of an inference.Logic is the study of correct reasoning and examines how to distinguish good from bad arguments. Deductive logic is the branch that investigates the strongest arguments, called deductively valid arguments, for which the conclusion cannot be false if all the premises are true. This is expressed by saying that the conclusion is a logical consequence of the premises. Rules of inference belong to deductive logic and describe argument forms that fulfill this requirement. In order to precisely assess whether an argument follows a rule of inference, logicians use formal languages to express statements in a rigorous manner, similar to mathematical formulas. They combine formal languages with rules of inference to construct formal systems—frameworks for formulating propositions and drawing conclusions. Different formal systems may employ different formal languages or different rules of inference. The basic rules of inference within a formal system can often be expanded by introducing new rules, known as admissible rules. Admissible rules do not change which arguments in a formal system are valid but can simplify proofs. If an admissible rule can be expressed through a combination of the system's basic rules, it is called a derived or derivable rule. Statements that can be deduced in a formal system are called theorems of this formal system. Widely used systems of logic include propositional logic, first-order logic, and modal logic.
Rules of inference only ensure that the conclusion is true if the premises are true. An argument with false premises can still be valid, but its conclusion may be false. For example, the argument "If pigs can fly, then the sky is purple. Pigs can fly. Therefore, the sky is purple." is valid because it follows modus ponens, even though it contains false premises. A valid argument is called a sound argument if all of its premises are true.
Rules of inference are closely related to tautologies or logical truths. In logic, a tautology is a statement that is true only because of the logical vocabulary it uses, independent of the meanings of its non-logical vocabulary. For example, the statement "if the tree is green and the sky is blue then the tree is green" is true independently of the meanings of terms like tree and green, making it a tautology. Every argument following a rule of inference can be transformed into a tautology. This is achieved by forming a conjunction of all premises and connecting it through implication to the conclusion, thereby combining all the individual statements of the argument into a single statement. For example, the valid argument "The tree is green and the sky is blue. Therefore, the tree is green." can be transformed into the tautology "if the tree is green and the sky is blue then the tree is green".
Rules of inference are not the only way to demonstrate that an argument is valid. Alternative methods include the use of truth tables, which applies to propositional logic, and truth trees, which can also be employed in first-order logic.