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hen 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 affirmed and denied meaning that a formal system contains both p displaystyle p and p displaystyle lnot p as theorems classical logic prohibits contradictions because classical rules of inference bring with them the principle of explosion an admissible rule of inference that makes it possible to infer q displaystyle q from the premises p displaystyle p and p displaystyle lnot p since q displaystyle q is unrelated to p displaystyle p any arbitrary statement can be deduced from a contradiction making the affected systems useless for deciding what is true and false 58 n paraconsistent logics solve this problem by modifying the rules of inference in such a way that the principle of explosion is not an admissible rule of inference as a result it is possible to reason about inconsistent information without deriving absurd conclusions 60 many valued logics modify classical logic by introducing additional truth values in classical logic a proposition is either true or false in many valued logics some propositions take values other than true and false kleene logic for example is a three valued logic that introduces the additional truth value undefined to describe situations where information is incomplete or uncertain 61 many valued logics have adjusted rules of inference to accommodate the additional truth values for instance the classical rule of replacement stating that p q displaystyle p to q is equivalent to p q displaystyle lnot p lor q is invalid in many three valued systems 62 some many valued logics take the form of probability logics which make it possible to reason from uncertain information to probabilistic conclusions 63 formalisms edit various formalisms or proof systems have been suggested as distinct ways of codifying reasoning and demonstrating the validity of arguments unlike different systems of logic these formalisms do not impact what can be proven they only influence how proofs are formulated influential frameworks include natural deduction systems hilbert systems and sequent calculi 64 natural deduction systems aim to reflect how people naturally reason by introducing many intuitive rules of inference to make logical derivations more accessible they break complex arguments into simple steps often using subproofs based on temporary premises the rules of inference in natural deduction target specific logical operators governing how an operator can be added with introduction rules or removed with elimination rules for example the rule of conjunction introduction asserts that one can infer p q displaystyle p land q from the premises p displaystyle p and q displaystyle q thereby producing a conclusion with the conjunction operator from premises that do not contain it conversely the rule of conjunction elimination asserts that one can infer p displaystyle p from p q displaystyle p land q thereby producing a conclusion that no longer includes the conjunction operator similar rules of inference are disjunction introduction and elimination implication introduction and elimination negation introduction and elimination and biconditional introduction and elimination as a result systems of natural deduction usually include many rules of inference 65 o hilbert systems by contrast aim to provide a minimal and efficient framework of logical reasoning by including as few rules of inference as possible many hilbert systems only have modus ponens as the sole rule of inference to ensure that all theorems can be deduced from this minimal foundation they introduce axiom schemes 67 an axiom scheme is a template to create axioms or true statements it uses metavariables placeholders that can be replaced by specific terms or formulas to generate an infinite number of true statements 68 for example propositional logic can be defined with the following three axiom schemes 1 p q p displaystyle p to q to p 2 p q r p q p r displaystyle p to q to r to p to q to p to r and 3 p q q p displaystyle lnot p to lnot q to q to p 69 to formulate proofs logicians create new statements from axiom schemes and then apply modus ponens to these statements to derive conclusions compared to natural deduction this procedure tends to be less intuitive since its heavy reliance on symbolic manipulation can obscure the underlying logical reasoning 70 sequent calculi another approach introduce sequents as formal representations of arguments a sequent has the form a 1 a m b 1 b n displaystyle a_ 1 dots a_ m vdash b_ 1 dots b_ n where a i displaystyle a_ i and b i displaystyle b_ i stand for propositions sequents are conditional assertions stating that at least one b i displaystyle b_ i is true if all a i displaystyle a_ i are true rules of inference operate on sequents to produce additional sequents sequent calculi define two rules of inference for each logical operator one to introduce it on the left side of a sequent and another to introduce it on the right side for example through the rule for introducing the operator displaystyle lnot on the left side one can infer r p q displaystyle lnot r p vdash q from p q r displaystyle p vdash q r the cut rule an additional rule of inference makes it possible to simplify sequents by removing certain propositions 71 formal fallacies edit main article formal fallacy affirming the consequent is a formal fallacy that resembles the valid rule of inference modus ponens 72 while rules of inference describe valid patterns of deductive reasoning formal fallacies are invalid argument forms that involve logical errors the premises of a formal fallacy do not properly support its conclusion the conclusion can be false even if all premises are true formal fallacies often mimic the structure of valid rules of inference and can thereby mislead people into unknowingly committing them and accepting their conclusions 73 the formal fallacy of affirming the consequent concludes p displaystyle p from the premises p q displaystyle p to q and q displaystyle q as in the argument if leo is a cat then leo is an animal leo is an animal therefore leo is a cat this fallacy resembles valid inferences following modus ponens with the key difference that the fallacy swaps the second premise and the conclusion 72 the formal fallacy of denying the antecedent concludes q displaystyle lnot q from the premises p q displaystyle p to q and p displaystyle lnot p as in the argument if laya saw the movie then laya had fun laya did not see the movie therefore laya did not have fun this fallacy resembles valid inferences following modus tollens with the key difference that the fallacy swaps the second premise and the conclusion 74 other formal fallacies include affirming a disjunct the existential fallacy and the fallacy of the undistributed middle 75 in various fields edit rules of inference are relevant to many fields especially the formal sciences such as mathematics and computer science where they are used to prove theorems 76 mathematical proofs often start with a set of axioms to describe the logical relationships between mathematical constructs to establish theorems mathematicians apply rules of inference to these axioms aiming to demonstrate that the theorems are logical consequences 77 they distinguish various types of proof based on the inferential strategy to arrive at a conclusion such as direct proof proof by contradiction and mathematical induction 78 mathematical logi...