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s of implication apply only to complete statements rules of replacement can be applied to any part of a compound statement 22 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 23 similarly the term is occasionally interpreted broadly to include general standards of research such as the principle that scientific experiments should be replicable 24 one of the first discussions of formal rules of inference dates to antiquity in aristotle s logic his explanations of valid 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 25 basic concepts edit rules of inference describe the structure of arguments which consist of premises that support a conclusion 26 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 27 an inference is a step of reasoning from premises to a conclusion while an argument is the outward expression of an inference 28 logic is the study of correct reasoning and examines how to distinguish good from bad arguments 29 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 30 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 31 they combine formal languages with rules of inference to construct formal systems frameworks for formulating propositions and drawing conclusions j different formal systems may employ different formal languages or different rules of inference 33 k 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 35 statements that can be deduced in a formal system are called theorems of this formal system 36 widely used systems of logic include propositional logic first order logic and modal logic 37 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 38 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 and of all premises and connecting it through implication if then 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 39 rules of inference are not the only way to demonstrate that an argument is valid alternative methods include the use of truth tables which apply to propositional logic and truth trees which can also be employed in first order logic 40 systems of logic edit classical edit propositional logic edit main article propositional logic propositional logic examines the inferential patterns of simple and compound propositions it uses letters such as p displaystyle p and q displaystyle q to represent simple propositions compound propositions are formed by modifying or combining simple propositions with logical operators such as displaystyle lnot not displaystyle land and displaystyle lor or and displaystyle to if then for example if p displaystyle p stands for the statement it is raining and q displaystyle q stands for the statement the streets are wet then p displaystyle lnot p expresses it is not raining and p q displaystyle p to q expresses if it is raining then the streets are wet these logical operators are truth functional meaning that the truth value of a compound proposition depends only on the truth values of the simple propositions composing it for instance the compound proposition p q displaystyle p land q is only true if both p displaystyle p and q displaystyle q are true in all other cases it is false propositional logic is not concerned with the concrete meaning of propositions other than their truth values 41 key rules of inference in propositional logic are modus ponens modus tollens hypothetical syllogism disjunctive syllogism and double negation elimination further rules include conjunction introduction conjunction elimination disjunction introduction disjunction elimination constructive dilemma destructive dilemma absorption and de morgan s laws 42 notable rules of inference 43 rule of inference form example modus ponens p q p q displaystyle begin array l p to q p hline q end array if kim is in seoul then kim is in south korea kim is in seoul therefore kim is in south korea displaystyle begin array l text if kim is in seoul then kim is in south korea text kim is in seoul hline text therefore kim is in south korea end array modus tollens p q q p displaystyle begin array l p to q lnot q hline lnot p end array if koko is a koala then koko is cuddly koko is not cuddly therefore koko is not a koala displaystyle begin array l text if koko is a koala then koko is cuddly text koko is not cuddly hline text therefore koko is not a koala end array hypothetical syllogism p q q r p r displaystyle begin array l p to q q to r hline p to r end array if leo is a lion then leo roars if leo roars then leo is fierce therefore if leo is a lion then leo is fierce displaystyle begin array l text if leo is a lion then leo roars text if leo roars then leo is fierce hline text therefore if leo is a lion then leo is fierce end array disjunctive syllogism p q p q displaystyle begin array l p lor q lnot p hline q end array the book is on the shelf or on the table the book is not on the shelf therefore the book is on the table displaystyle begin array l text the book is on the shelf or on the table text the book is not on the shelf hline text therefore the book is on the table end array double negation elimination p p displaystyle begin array l lnot lnot p hline p end array we were not unable to meet the deadline we were able to meet the deadline displaystyle begin array l text we were not unable to meet the deadline hline text we were able to meet the deadline end array first order logic edit main article first order logic as one of the founding fathers of modern logic gottlob frege 1848 1925 explored some of the foundational concepts of first order logic 44 first order logic also employs the logical operators from propositional logic but includes additional devices to articulate the internal structure of propositions basic propositions in first order logic consist of a predicate symbolized with uppercase letters like p displaystyle p and q displaystyle q which is applied to singular terms symbolized with lowercase letters like a displaystyle a and b displaystyle b for example if a displaystyle a stands for aristotle and p displaystyle p stands for is a philosopher then the formula p a displaystyle p a means that aristotle is a philosopher another innovation of first order logic is the use of the quantifiers displaystyle exists and displaystyle forall which express that a predicate applies to some or all individuals for instance the formula x p x displaystyle exists xp x expresses that philosophers exist while x p x displaystyle forall xp x expresses that everyone is a philosopher the rules of inference from propositional logic are also valid in first order logic 45 additionally first order logic introduces new rules of inference that govern the role of singular terms predicates and quantifiers in arguments key rules of inference are universal instantiation and existential generalization other rules of inference include universal generalization and existential instantiation 10 notable rules of inference 10 rule of inference form example universal instantiation x p x p a displaystyle begin array l forall xp x hline p a end array l everyone must pay taxes therefore wesley must pay taxes displaystyle begin array l text everyone must pay taxes hline text therefore wesley must pay taxes end array existential generalization p a x p x displaystyle begin array l p a hline exists xp x end array socrates is mortal therefore someone is mortal displaystyle begin array l text socrates is mortal hline text therefore someone is mortal end array modal logics edit main article modal logic modal logics are formal systems that extend propositional logic and first order logic with additional operators alethic modal logic introduces the operator displaystyle diamond to express that something is possible and the operator displaystyle box to express that something is necessary for example if p displaystyle p means that parvati works then p displaystyle diamond p means that it is possible that parvati works while p displaystyle box p means that it is necessary that parvati works these two operators are related by a rule of replacement stating that p displaystyle box p is equivalent to p displaystyle lnot diamond lnot p in other words if something is necessarily true then it is not possible that it is not true further rules of inference include the necessitation rule which asserts that a statement is necessarily true if it is provable in a formal system without any additional premises and the distribution axiom which allows one to derive p q displaystyle box p to box q from p q displaystyle box p to q these rules of inference belong to system k a weak form of modal logic with only the most basic rules of inference many formal systems of alethic modal logic include additional rules of inference such as system t which allows one to deduce p displaystyle p from p displaystyle box p 46 non alethic systems of modal logic introduce operators that behave like displaystyle diamond and displaystyle box in alethic modal logic following similar rules of inference but with different meanings deontic logic is one type of non alethic logic it uses the operator p displaystyle mathbf p to express that an action is permitted and the operator o displaystyle mathbf o to express that an action is required where p displaystyle mathbf p behaves similarly to displaystyle diamond and o displaystyle mathbf o behaves similarly to displaystyle box for instance the rule of replacement in alethic modal logic which asserts that q displaystyle box q is equivalent to q displaystyle lnot diamond lnot q also applies to deontic logic as a result one can deduce from o q displaystyle mathbf o q e g quinn has an obligation to help that p q displaystyle lnot mathbf p lnot q e g quinn is not permitted not to help 47 other systems of modal logic include temporal modal logic which has operators for what is always or sometimes the case as well as doxastic and epistemic modal logics which have operators for what people believe and know 48 other systems edit the rules of inference in aristotle s 384 322 bce logic have the form of syllogisms 49 many other systems of logic have been proposed 50 in ancient greece one of the earliest systems was aristotelian logic according to which each statement is made up of two terms a subject and a predicate connected by a copula for example the statement all humans are mortal has the subject all humans the predicate mortal and the copula is all rules of inference in aristotelian logic have the form of syllogisms which consist of two premises and a conclusion for instance the barbara rule of inference describes the validity of arguments of the form all men are mortal all greeks are men therefore all greeks are mortal 51 the nyaya school in ancient india also explored rules of inference in the form of syllogisms such as the argument all things which have smoke have fire this hill has smoke therefore this hill has fire 52 second order logic extends first order logic by allowing quantifiers to apply to predicates in addition to singular terms for example to express that the individuals adam a displaystyle a and bianca b displaystyle b share a property one can use the formula x x a x b displaystyle exists x x a land x b 53 second order logic also comes with new rules of inference m for instance one can infer p a displaystyle p a adam is a philosopher from x x a displaystyle forall xx a every property applies to adam 55 intuitionistic logic is a non classical variant of propositional and first order logic it shares with them many rules of inference such as modus ponens but excludes certain rules for example in classical logic one can infer p displaystyle p from p displaystyle lnot lnot p using the rule of double negation elimination however in intuitionistic logic this inference is invalid as a result every theorem that can be deduced in intuitionistic logic can also be deduced in classical logic but some theorems provable in classical logic cannot be proven in intuitionistic logic 56 one motivation for this modification is the idea that proofs should demonstrate that an object exists or can be constructed not merely that its nonexistence would lead to a contradiction 57 paraconsistent logics revise classical logic to allow the existence of contradictions in logic a contradiction happens if the same proposition is both...
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