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188-535: Logic is the study of correct reasoning . It includes both formal and informal logic . Formal logic is the study of deductively valid inferences or logical truths . It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies , critical thinking , and argumentation theory . Informal logic examines arguments expressed in natural language whereas formal logic uses formal language . When used as

376-499: A r y ) ∧ Q ( J o h n ) ) {\displaystyle \exists Q(Q(Mary)\land Q(John))} " . In this case, the existential quantifier is applied to the predicate variable " Q {\displaystyle Q} " . The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories. But it has drawbacks in regard to its meta-logical properties and ontological implications, which

564-444: A countable noun , the term "a logic" refers to a specific logical formal system that articulates a proof system . Logic plays a central role in many fields, such as philosophy , mathematics , computer science , and linguistics . Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to

752-400: A domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, each predicate is assigned a property of objects, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms, predicates, and formulas of the language. The study of the interpretations of formal languages

940-400: A formal language and usually belong to deductive reasoning. Their fault lies in the logical form of the argument, i.e. that it does not follow a valid rule of inference. A well-known formal fallacy is affirming the consequent . It has the following form: (1) q ; (2) if p then q ; (3) therefore p . This fallacy is committed, for example, when a person argues that "the burglars entered by

1128-493: A rigorous way. It happens in the form of inferences or arguments by starting from a set of premises and reasoning to a conclusion supported by these premises. The premises and the conclusion are propositions , i.e. true or false claims about what is the case. Together, they form an argument. Logical reasoning is norm-governed in the sense that it aims to formulate correct arguments that any rational person would find convincing. The main discipline studying logical reasoning

1316-410: A central role in logical reasoning. If one lacks important information, it is often better to delay a decision and look for new information before coming to a conclusion. If the decision is time-sensitive, on the other hand, logical reasoning may imply making a fast decision based on the currently available evidence even if it is very limited. For example, if a friend yells "Duck!" during a baseball game

1504-445: A central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments

1692-432: A central role in science when researchers discover unexplained phenomena. In this case, they often resort to a form of guessing to come up with general principles that could explain the observations. The hypotheses are then tested and compared to discover which one provides the best explanation. This pertains particularly to cases of causal reasoning that try to discover the relation between causes and effects. Abduction

1880-478: A certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth. This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to non-monotonicity and defeasibility : it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn. Ampliative reasoning plays

2068-434: A chain is formed in which the conclusions of earlier arguments act as premises for later arguments. Each link in this chain has to be successful for a complex argument to succeed. An argument is correct or incorrect depending on whether the premises offer support for the conclusion. This is often understood in terms of probability : if the premises of a correct argument are true, it raises the probability that its conclusion

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2256-571: A complex argument to be successful, each link of the chain has to be successful. Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning . The strongest form of support corresponds to deductive reasoning . But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases,

2444-422: A conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion. These general characterizations apply to logic in the widest sense, i.e., to both formal and informal logic since they are both concerned with assessing the correctness of arguments. Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic. Formal logic

2632-415: A conjunct , and the fallacy of the undistributed middle . Informal fallacies are expressed in natural language. Their main fault usually lies not in the form of the argument but has other sources, like its content or context. Some informal fallacies, like some instances of false dilemmas and strawman fallacies , even involve correct deductive reasoning on the formal level. The content of an argument

2820-453: A definite truth value. Quantifiers can be applied to variables in a formula. The variable x in the previous formula can be universally quantified, for instance, with the first-order sentence "For every x , if x is a philosopher, then x is a scholar". The universal quantifier "for every" in this sentence expresses the idea that the claim "if x is a philosopher, then x is a scholar" holds for all choices of x . The negation of

3008-418: A fixed, infinite set of non-logical symbols for all purposes: When the arity of a predicate symbol or function symbol is clear from context, the superscript n is often omitted. In this traditional approach, there is only one language of first-order logic. This approach is still common, especially in philosophically oriented books. A more recent practice is to use different non-logical symbols according to

3196-506: A formal language together with a set of axioms and a proof system used to draw inferences from these axioms. In logic, axioms are statements that are accepted without proof. They are used to justify other statements. Some theorists also include a semantics that specifies how the expressions of the formal language relate to real objects. Starting in the late 19th century, many new formal systems have been proposed. A formal language consists of an alphabet and syntactic rules. The alphabet

3384-685: A formal language while informal logic investigates them in their original form. On this view, the argument "Birds fly. Tweety is a bird. Therefore, Tweety flies." belongs to natural language and is examined by informal logic. But the formal translation "(1) ∀ x ( B i r d ( x ) → F l i e s ( x ) ) {\displaystyle \forall x(Bird(x)\to Flies(x))} ; (2) B i r d ( T w e e t y ) {\displaystyle Bird(Tweety)} ; (3) F l i e s ( T w e e t y ) {\displaystyle Flies(Tweety)} "

3572-406: A formula such as Phil( x ) is true must depend on what x represents. But the sentence ∃ x Phil( x ) will be either true or false in a given interpretation. In mathematics, the language of ordered abelian groups has one constant symbol 0, one unary function symbol −, one binary function symbol +, and one binary relation symbol ≤. Then: The axioms for ordered abelian groups can be expressed as

3760-400: A general law or principle from the observations of particular instances." For example, starting from the empirical observation that "all ravens I have seen so far are black", inductive reasoning can be used to infer that "all ravens are black". In a slightly weaker form, induction can also be used to infer an individual conclusion about a single case, for example, that "the next raven I will see

3948-461: A generalization about human beings, the sample should include members of different races, genders, and age groups. A lot of reasoning in everyday life is inductive. For example, when predicting how a person will react to a situation, inductive reasoning can be employed based on how the person reacted previously in similar circumstances. It plays an equally central role in the sciences , which often start with many particular observations and then apply

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4136-413: A given argument is valid. Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be translated into formal language before their validity can be assessed. The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning

4324-549: A given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In chess , for example, the definitory rules dictate that bishops may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's king . It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning. A formal system of logic consists of

4512-402: A great variety of topics. They include metaphysical theses about ontological categories and problems of scientific explanation. But in a more narrow sense, it is identical to term logic or syllogistics. A syllogism is a form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a subject , a predicate, and a copula connecting

4700-416: A history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001). While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification . A predicate evaluates to true or false for an entity or entities in the domain of discourse . Consider the two sentences " Socrates is a philosopher" and " Plato

4888-611: A logical connective like "and" to form a new complex proposition. In Aristotelian logic, the subject can be universal , particular , indefinite , or singular . For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates". Aristotelian logic only includes predicates for simple properties of entities. But it lacks predicates corresponding to relations between entities. The predicate can be linked to

5076-451: A proposition since it can be true or false. The sentences "Is the water boiling?" or "Boil the water!", on the other hand, express no propositions since they are neither true nor false. The propositions used as the starting point of logical reasoning are called the premises. The proposition inferred from them is called the conclusion. For example, in the argument "all puppies are dogs; all dogs are animals; therefore all puppies are animals",

5264-469: A red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers . This means that they have a truth value : they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences . Propositions are the denotations of sentences and are usually seen as abstract objects . For example, the English sentence "the tree

5452-663: A sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, p {\displaystyle p} ("yesterday was Sunday") and q {\displaystyle q} ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are ¬ {\displaystyle \lnot } ( not ), ∨ {\displaystyle \lor } ( or ), → {\displaystyle \to } ( if...then ), and ↑ {\displaystyle \uparrow } ( Sheffer stroke ). Given

5640-468: A set of premises to reach a conclusion. It can be defined as "selecting and interpreting information from a given context, making connections, and verifying and drawing conclusions based on provided and interpreted information and the associated rules and processes." Logical reasoning is rigorous in the sense that it does not generate any conclusion but ensures that the premises support the conclusion and act as reasons for believing it. One central aspect

5828-657: A set of sentences in first-order logic. The term "first-order" distinguishes first-order logic from higher-order logic , in which there are predicates having predicates or functions as arguments, or in which quantification over predicates, functions, or both, are permitted. In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. There are many deductive systems for first-order logic which are both sound , i.e. all provable statements are true in all models; and complete , i.e. all statements which are true in all models are provable. Although

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6016-482: A set of sentences in the language. For example, the axiom stating that the group is commutative is usually written ( ∀ x ) ( ∀ y ) [ x + y = y + x ] . {\displaystyle (\forall x)(\forall y)[x+y=y+x].} An interpretation of a first-order language assigns a denotation to each non-logical symbol (predicate symbol, function symbol, or constant symbol) in that language. It also determines

6204-477: A simple way. For example, the Bohr model explains the interactions of sub-atomic particles in analogy to how planets revolve around the sun. A fallacy is an incorrect argument or a faulty form of reasoning. This means that the premises provide no or not sufficient support for the conclusion. Fallacies often appear to be correct on the first impression and thereby seduce people into accepting and using them. In logic,

6392-410: A single symbol on the left side), except that the set of symbols may be allowed to be infinite and there may be many start symbols, for example the variables in the case of terms . The set of terms is inductively defined by the following rules: Only expressions which can be obtained by finitely many applications of rules 1 and 2 are terms. For example, no expression involving a predicate symbol

6580-399: A single variable. In general, predicates can take several variables. In the first-order sentence "Socrates is the teacher of Plato", the predicate "is the teacher of" takes two variables. An interpretation (or model) of a first-order formula specifies what each predicate means, and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which

6768-429: A ternary predicate symbol. However, ∀ x x → {\displaystyle \forall x\,x\rightarrow } is not a formula, although it is a string of symbols from the alphabet. The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way—by following the inductive definition (i.e., there is a unique parse tree for each formula). This property

6956-426: A topic, such as set theory, a theory for groups, or a formal theory of arithmetic , is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold about them. "Theory" is sometimes understood in a more formal sense as just

7144-410: Is Polish notation , in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } and so on in front of their arguments rather than between them. This convention is advantageous in that it allows all punctuation symbols to be discarded. As such, Polish notation is compact and elegant, but rarely used in practice because it

7332-426: Is logic . Distinct types of logical reasoning differ from each other concerning the norms they employ and the certainty of the conclusion they arrive at. Deductive reasoning offers the strongest support: the premises ensure the conclusion, meaning that it is impossible for the conclusion to be false if all the premises are true. Such an argument is called a valid argument, for example: all men are mortal; Socrates

7520-421: Is sound if it is valid and all its premises are true. For example, inferring the conclusion "no cats are frogs" from the premises "all frogs are amphibians" and "no cats are amphibians" is a sound argument. But even arguments with false premises can be deductively valid, like inferring that "no cats are frogs" from the premises "all frogs are mammals" and "no cats are mammals". In this regard, it only matters that

7708-420: Is sound when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by

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7896-400: Is a form of generalization that infers a universal law from a pattern found in many individual cases. It can be used to conclude that "all ravens are black" based on many individual observations of black ravens. Abductive reasoning, also known as "inference to the best explanation", starts from an observation and reasons to the fact explaining this observation. An example is a doctor who examines

8084-489: Is a man; therefore, Socrates is mortal. For valid arguments, it is not important whether the premises are actually true but only that, if they were true, the conclusion could not be false. Valid arguments follow a rule of inference , such as modus ponens or modus tollens . Deductive reasoning plays a central role in formal logic and mathematics . For non-deductive logical reasoning, the premises make their conclusion rationally convincing without ensuring its truth . This

8272-452: Is a philosopher" alone does not have a definite truth value of true or false, and is akin to a sentence fragment. Relationships between predicates can be stated using logical connectives . For example, the first-order formula "if x is a philosopher, then x is a scholar", is a conditional statement with " x is a philosopher" as its hypothesis, and " x is a scholar" as its conclusion, which again needs specification of x in order to have

8460-416: Is a philosopher". In propositional logic , these sentences themselves are viewed as the individuals of study, and might be denoted, for example, by variables such as p and q . They are not viewed as an application of a predicate, such as isPhil {\displaystyle {\text{isPhil}}} , to any particular objects in the domain of discourse, instead viewing them as purely an utterance which

8648-440: Is a restricted version of classical logic. It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that A {\displaystyle A} follows from ¬ ¬ A {\displaystyle \lnot \lnot A} . This is a valid rule of inference in classical logic but it

8836-416: Is a term. The set of formulas (also called well-formed formulas or WFFs ) is inductively defined by the following rules: Only expressions which can be obtained by finitely many applications of rules 1–5 are formulas. The formulas obtained from the first two rules are said to be atomic formulas . For example: is a formula, if f is a unary function symbol, P a unary predicate symbol, and Q

9024-411: Is about making judgments and drawing conclusions after careful evaluation and contrasts in this regard with uncritical snap judgments and gut feelings. Other core skills linked to logical reasoning are to assess reasons before accepting a claim and to search for new information if more is needed to reach a reliable conclusion. It also includes the ability to consider different courses of action and compare

9212-415: Is also known as symbolic logic and is widely used in mathematical logic . It uses a formal approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the logical form of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content. Formal logic

9400-444: Is also true. Forms of logical reasoning can be distinguished based on how the premises support the conclusion. Deductive arguments offer the strongest possible support. Non-deductive arguments are weaker but are nonetheless correct forms of reasoning. The term "proof" is often used for deductive arguments or very strong non-deductive arguments. Incorrect arguments offer no or not sufficient support and are called fallacies , although

9588-404: Is also very common in everyday life. It is used there in a similar but less systematic form. This relates, for example, to the trust people put in what other people say. The best explanation of why a person asserts a claim is usually that they believe it and have evidence for it. This form of abductive reasoning is relevant to why one normally trusts what other people say even though this inference

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9776-453: Is an example of the existential quantifier " ∃ {\displaystyle \exists } " applied to the individual variable " x {\displaystyle x} " . In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula " ∃ Q ( Q ( M

9964-401: Is based on a small set of axioms from which all essential properties of natural numbers can be inferred using deductive reasoning. Non-deductive reasoning is an important form of logical reasoning besides deductive reasoning. It happens in the form of inferences drawn from premises to reach and support a conclusion, just like its deductive counterpart. The hallmark of non-deductive reasoning

10152-430: Is black". Inductive reasoning is closely related to statistical reasoning and probabilistic reasoning . Like other forms of non-deductive reasoning, induction is not certain. This means that the premises support the conclusion by making it more probable but do not ensure its truth. In this regard, the conclusion of an inductive inference contains new information not already found in the premises. Various aspects of

10340-413: Is blurry in some cases, such as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between. The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term " induction " to cover all forms of non-deductive arguments. But in a more narrow sense, induction

10528-469: Is bound in φ if all occurrences of x in φ are bound. Intuitively, a variable symbol is free in a formula if at no point is it quantified: in ∀ y P ( x , y ) , the sole occurrence of variable x is free while that of y is bound. The free and bound variable occurrences in a formula are defined inductively as follows. For example, in ∀ x ∀ y ( P ( x ) → Q ( x , f ( x ), z )) , x and y occur only bound, z occurs only free, and w

10716-418: Is commonly defined in terms of arguments or inferences as the study of their correctness. An argument is a set of premises together with a conclusion. An inference is the process of reasoning from these premises to the conclusion. But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on

10904-405: Is controversial because it belongs to the field of psychology , not logic, and because appearances may be different for different people. Fallacies are usually divided into formal and informal fallacies. For formal fallacies, the source of the error is found in the form of the argument. For example, denying the antecedent is one type of formal fallacy, as in "if Othello is a bachelor, then he

11092-552: Is deductively valid no matter what p and q stand for. For example, the argument "today is Sunday; if today is Sunday then I don't have to go to work today; therefore I don't have to go to work today" is deductively valid because it has the form of modus ponens . Other popular rules of inference include modus tollens (not q ; if p then q ; therefore not p ) and the disjunctive syllogism ( p or q ; not p ; therefore q ). The rules governing deductive reasoning are often expressed formally as logical systems for assessing

11280-407: Is either true or false. However, in first-order logic, these two sentences may be framed as statements that a certain individual or non-logical object has a property. In this example, both sentences happen to have the common form isPhil ( x ) {\displaystyle {\text{isPhil}}(x)} for some individual x {\displaystyle x} , in the first sentence

11468-399: Is established by verification using a proof. Intuitionistic logic is especially prominent in the field of constructive mathematics , which emphasizes the need to find or construct a specific example to prove its existence. Logical reasoning Sound Unsound Unsound Cogent Uncogent Uncogent Logical reasoning is a mental activity that aims to arrive at a conclusion in

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11656-608: Is green" is different from the German sentence "der Baum ist grün" but both express the same proposition. Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions. These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like

11844-424: Is hard for humans to read. In Polish notation, the formula: becomes "∀x∀y→Pfx¬→ PxQfyxz". In a formula, a variable may occur free or bound (or both). One formalization of this notion is due to Quine, first the concept of a variable occurrence is defined, then whether a variable occurrence is free or bound, then whether a variable symbol overall is free or bound. In order to distinguish different occurrences of

12032-418: Is impossible to make people give up drinking alcohol. This is a strawman fallacy since the suggestion was merely to ban advertisements and not to stop all alcohol consumption. Ambiguous and vague expressions in natural language are often responsible for the faulty reasoning in informal fallacies. For example, this is the case for fallacies of ambiguity , like the argument "(1) feathers are light; (2) light

12220-430: Is interested in deductively valid arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false. For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a rule of inference . For example, modus ponens is a rule of inference according to which all arguments of

12408-414: Is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the law of excluded middle . It states that for every sentence, either it or its negation is true. This means that every proposition of the form A ∨ ¬ A {\displaystyle A\lor \lnot A} is true. These deviations from classical logic are based on the idea that truth

12596-412: Is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability. For convenience, conventions have been developed about the precedence of

12784-445: Is male; Othello is not a bachelor; therefore Othello is not male". But most fallacies fall into the category of informal fallacies, of which a great variety is discussed in the academic literature. The source of their error is usually found in the content or the context of the argument. Informal fallacies are sometimes categorized as fallacies of ambiguity, fallacies of presumption, or fallacies of relevance. For fallacies of ambiguity,

12972-403: Is more common in everyday life than deductive reasoning. Non-deductive reasoning is ampliative and defeasible . Sometimes, the terms non-deductive reasoning , ampliative reasoning , and defeasible reasoning are used synonymously even though there are slight differences in their meaning. Non-deductive reasoning is ampliative in the sense that it arrives at information not already present in

13160-687: Is necessary, then it is also possible. This means that ◊ A {\displaystyle \Diamond A} follows from ◻ A {\displaystyle \Box A} . Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that ◻ A {\displaystyle \Box A} is equivalent to ¬ ◊ ¬ A {\displaystyle \lnot \Diamond \lnot A} . Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, deontic logic concerns

13348-516: Is necessary. For example, if the formula B ( s ) {\displaystyle B(s)} stands for the sentence "Socrates is a banker" then the formula ◊ B ( s ) {\displaystyle \Diamond B(s)} articulates the sentence "It is possible that Socrates is a banker". To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something

13536-488: Is neither because it does not occur in the formula. Free and bound variables of a formula need not be disjoint sets: in the formula P ( x ) → ∀ x Q ( x ) , the first occurrence of x , as argument of P , is free while the second one, as argument of Q , is bound. A formula in first-order logic with no free variable occurrences is called a first-order sentence . These are the formulas that will have well-defined truth values under an interpretation. For example, whether

13724-400: Is not as secure as deductive reasoning. A closely related aspect is that non-deductive reasoning is defeasible or non-monotonic . This means that one may have to withdraw a conclusion upon learning new information. For example, if all birds a person has seen so far can fly, this person is justified in reaching the inductive conclusion that all birds fly. This conclusion is defeasible because

13912-404: Is not the best or most likely explanation. Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as fallacies . Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion. So the argument "it is sunny today; therefore spiders have eight legs" is fallacious even though

14100-471: Is not the only viable solution. The strawman fallacy is another informal fallacy. Its error happens on the level of the context. It consists in misrepresenting the view of an opponent and then refuting this view. The refutation itself is often correct but the error lies in the false assumption that the opponent actually defends this view. For example, an alcohol lobbyist may respond to the suggestion to ban alcohol advertisements on television by claiming that it

14288-448: Is often understood in terms of probability : the premises make it more likely that the conclusion is true and strong inferences make it very likely. Some uncertainty remains because the conclusion introduces new information not already found in the premises. Non-deductive reasoning plays a central role in everyday life and in most sciences . Often-discussed types are inductive , abductive , and analogical reasoning . Inductive reasoning

14476-536: Is only one type of ampliative argument alongside abductive arguments . Some philosophers, like Leo Groarke, also allow conductive arguments as another type. In this narrow sense, induction is often defined as a form of statistical generalization. In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains. In this sense, one may infer that "all elephants are gray" based on one's past observations of

14664-430: Is opposed to darkness; (3) therefore feathers are opposed to darkness". The error is found in the ambiguous term "light", which has one meaning in the first premise ("not heavy") and a different meaning in the second premise ("visible electromagnetic radiation"). Some theorists discuss logical reasoning in a very wide sense that includes its role as a broad skill responsible for high-quality thinking. In this sense, it

14852-486: Is possible and what is necessary. Temporal logic can be used to draw inferences about what happened before, during, and after an event. Classical logic and its extensions rest on a set of basic logical intuitions accepted by most logicians. They include the law of excluded middle , the double negation elimination , the principle of explosion , and the bivalence of truth. So-called deviant logics reject some of these basic intuitions and propose alternative rules governing

15040-406: Is roughly equivalent to critical thinking and includes the capacity to select and apply the appropriate rules of logic to specific situations. It encompasses a great variety of abilities besides drawing conclusions from premises. Examples are to understand a position, to generate and evaluate reasons for and against it as well as to critically assess whether to accept or reject certain information. It

15228-428: Is studied by formal logic. The study of natural language arguments comes with various difficulties. For example, natural language expressions are often ambiguous, vague, and context-dependent. Another approach defines informal logic in a wide sense as the normative study of the standards, criteria, and procedures of argumentation. In this sense, it includes questions about the role of rationality , critical thinking , and

15416-632: Is studied in the foundations of mathematics . Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory , respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line . Axiom systems that do fully describe these two structures, i.e. categorical axiom systems, can be obtained in stronger logics such as second-order logic . The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce . For

15604-399: Is that this support is fallible. This means that if the premises are true, it makes it more likely but not certain that the conclusion is also true. So for a non-deductive argument, it is possible for all its premises to be true while its conclusion is still false. There are various types of non-deductive reasoning, like inductive, abductive, and analogical reasoning. Non-deductive reasoning

15792-425: Is that this support is not restricted to a specific reasoner but that any rational person would find the conclusion convincing based on the premises. This way, logical reasoning plays a role in expanding knowledge . The main discipline studying logical reasoning is called logic . It is divided into formal and informal logic , which study formal and informal logical reasoning. Traditionally, logical reasoning

15980-531: Is the idea that is expressed in it. For example, a false dilemma is an informal fallacy that is based on an error in one of the premises. The faulty premise oversimplifies reality: it states that things are either one way or another way but ignore many other viable alternatives. False dilemmas are often used by politicians when they claim that either their proposal is accepted or there will be dire consequences. Such claims usually ignore that various alternatives exist to avoid those consequences, i.e. that their proposal

16168-409: Is the set of basic symbols used in expressions . The syntactic rules determine how these symbols may be arranged to result in well-formed formulas. For instance, the syntactic rules of propositional logic determine that " P ∧ Q {\displaystyle P\land Q} " is a well-formed formula but " ∧ Q {\displaystyle \land Q} " is not since

16356-413: Is to convince a person that something is the case by providing reasons for this belief. Many arguments in natural language do not explicitly state all the premises. Instead, the premises are often implicitly assumed, especially if they seem obvious and belong to common sense . Some theorists distinguish between simple and complex arguments. A complex argument is made up of many sub-arguments. This way,

16544-431: Is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference . They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach

16732-536: Is unable to address. Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies. Many characterizations of informal logic have been suggested but there is no general agreement on its precise definition. The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments. On this view, informal logic studies arguments that are in informal or natural language. Formal logic can only examine them indirectly by translating them first into

16920-413: Is used in the study and analysis of logical reasoning. Logical reasoning happens by inferring a conclusion from a set of premises. Premises and conclusions are normally seen as propositions . A proposition is a statement that makes a claim about what is the case. In this regard, propositions act as truth-bearers : they are either true or false. For example, the sentence "The water is boiling." expresses

17108-597: Is used to represent the ideas of knowing something in contrast to merely believing it to be the case. Higher-order logics extend classical logic not by using modal operators but by introducing new forms of quantification. Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula " ∃ x ( A p p l e ( x ) ∧ S w e e t ( x ) ) {\displaystyle \exists x(Apple(x)\land Sweet(x))} " ( some apples are sweet)

17296-549: Is usually not drawn in an explicit way. Something similar happens when the speaker's statement is ambiguous and the audience tries to discover and explain what the speaker could have meant. Abductive reasoning is also common in medicine when a doctor examines the symptoms of their patient in order to arrive at a diagnosis of their underlying cause. Analogical reasoning involves the comparison of two systems in relation to their similarity . It starts from information about one system and infers information about another system based on

17484-610: Is usually required to be a nonempty set. For example, consider the sentence "There exists x such that x is a philosopher." This sentence is seen as being true in an interpretation such that the domain of discourse consists of all human beings, and that the predicate "is a philosopher" is understood as "was the author of the Republic ." It is true, as witnessed by Plato in that text. There are two key parts of first-order logic. The syntax determines which finite sequences of symbols are well-formed expressions in first-order logic, while

17672-429: Is why first-order logic is still more commonly used. Deviant logics are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue. Intuitionistic logic

17860-459: The Löwenheim–Skolem theorem . Though signatures might in some cases imply how non-logical symbols are to be interpreted, interpretation of the non-logical symbols in the signature is separate (and not necessarily fixed). Signatures concern syntax rather than semantics. In this approach, every non-logical symbol is of one of the following types: The traditional approach can be recovered in

18048-419: The logical consequence relation is only semidecidable , much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory , such as the Löwenheim–Skolem theorem and the compactness theorem . First-order logic is the standard for the formalization of mathematics into axioms , and

18236-513: The semantics determines the meanings behind these expressions. Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is well formed . There are two key types of well-formed expressions: terms , which intuitively represent objects, and formulas , which intuitively express statements that can be true or false. The terms and formulas of first-order logic are strings of symbols , where all

18424-466: The Greek word "logos", which has a variety of translations, such as reason , discourse , or language . Logic is traditionally defined as the study of the laws of thought or correct reasoning , and is usually understood in terms of inferences or arguments . Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences. An argument is a set of premises together with

18612-441: The advantages and disadvantages of their consequences, to use common sense, and to avoid inconsistencies . The skills responsible for logical reasoning can be learned, trained, and improved. Logical reasoning is relevant both on the theoretical and practical level. On the theoretical level, it helps decrease the number of false beliefs. A central aspect concerns the abilities used to distinguish facts from mere opinions, like

18800-415: The agent. For each possible action, there can be conflicting reasons, some in favor of it and others opposed to it. In such cases, logical reasoning includes weighing the potential benefits and drawbacks as well as considering their likelihood in order to arrive at a balanced all-things-considered decision. For example, when a person runs out of drinking water in the middle of a hiking trip, they could employ

18988-415: The ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark". Fallacies of presumption have a wrong or unjustified premise but may be valid otherwise. In the case of fallacies of relevance, the premises do not support the conclusion because they are not relevant to it. The main focus of most logicians

19176-400: The application of the norms, i.e. agreement about whether and to what degree the premises support their conclusion. The types of logical reasoning differ concerning the exact norms they use as well as the certainty of the conclusion they arrive at. Deductive reasoning offers the strongest support and implies its conclusion with certainty, like mathematical proofs . For non-deductive reasoning,

19364-500: The application one has in mind. Therefore, it has become necessary to name the set of all non-logical symbols used in a particular application. This choice is made via a signature . Typical signatures in mathematics are {1, ×} or just {×} for groups , or {0, 1, +, ×, <} for ordered fields . There are no restrictions on the number of non-logical symbols. The signature can be empty , finite, or infinite, even uncountable . Uncountable signatures occur for example in modern proofs of

19552-442: The argument. For informal fallacies , like false dilemmas , the source of the faulty reasoning is usually found in the content or the context of the argument. Some theorists understand logical reasoning in a wide sense that is roughly equivalent to critical thinking . In this regard, it encompasses cognitive skills besides the ability to draw conclusions from premises. Examples are skills to generate and evaluate reasons and to assess

19740-400: The assessment of arguments. Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion. For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is

19928-412: The assumption that nature remains uniform. Abductive reasoning is usually understood as an inference from an observation to a fact explaining this observation. Inferring that it has rained after seeing that the streets are wet is one example. Often, the expression "inference to the best explanation" is used as a synonym. This expression underlines that there are usually many possible explanations of

20116-494: The basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like metaphysics , ethics , and epistemology . Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: ◊ {\displaystyle \Diamond } expresses that something is possible while ◻ {\displaystyle \Box } expresses that something

20304-486: The best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer. Arguments that fall short of the standards of correct reasoning often embody fallacies . Systems of logic are theoretical frameworks for assessing the correctness of arguments. Logic has been studied since antiquity . Early approaches include Aristotelian logic , Stoic logic , Nyaya , and Mohism . Aristotelian logic focuses on reasoning in

20492-447: The brain development of rats; (3) therefore they may also affect the brain development of humans. Through analogical reasoning, knowledge can be transferred from one situation or domain to another. Arguments from analogy provide support for their conclusion but do not guarantee its truth. Their strength depends on various factors. The more similar the systems are, the more likely it is that a given feature of one object also characterizes

20680-641: The claim "either it is raining, or it is not". These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q " is a logical truth. Formal logic uses formal languages to express and analyze arguments. They normally have a very limited vocabulary and exact syntactic rules . These rules specify how their symbols can be combined to construct sentences, so-called well-formed formulas . This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether

20868-492: The color of elephants. A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray. Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference. Abductive inference may or may not take statistical observations into consideration. In either case,

21056-501: The conclusion "I don't have to work". Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ∧ {\displaystyle \land } ( and ) or → {\displaystyle \to } ( if...then ). Simple propositions also have parts, like "Sunday" or "work" in

21244-507: The conclusion "all ravens are black". A further approach is to define informal logic as the study of informal fallacies . Informal fallacies are incorrect arguments in which errors are present in the content and the context of the argument. A false dilemma , for example, involves an error of content by excluding viable options. This is the case in the fallacy "you are either with us or against us; you are not with us; therefore, you are against us". Some theorists state that formal logic studies

21432-463: The conclusion could not be false if the premises are true and not whether they actually are true. Deductively valid arguments follow a rule of inference . A rule of inference is a scheme of drawing conclusions that depends only on the logical form of the premises and the conclusion but not on their specific content. The most-discussed rule of inference is the modus ponens . It has the following form: p ; if p then q ; therefore q . This scheme

21620-456: The conclusion is true. Some theorists, like John Stuart Mill , give a more restrictive definition of fallacies by additionally requiring that they appear to be correct. This way, genuine fallacies can be distinguished from mere mistakes of reasoning due to carelessness. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them. However, this reference to appearances

21808-400: The conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by logical necessity for the given propositions, independent of any other circumstances. Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference. Rules of inference specify the form of

21996-590: The conditional proposition p → q {\displaystyle p\to q} , one can form truth tables of its converse q → p {\displaystyle q\to p} , its inverse ( ¬ p → ¬ q {\displaystyle \lnot p\to \lnot q} ) , and its contrapositive ( ¬ q → ¬ p {\displaystyle \lnot q\to \lnot p} ) . Truth tables can also be defined for more complex expressions that use several propositional connectives. Logic

22184-437: The contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future. Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like ethics or epistemology that lie beyond

22372-682: The correctness of deductive arguments. Aristotelian logic is one of the earliest systems and was treated as the canon of logic in the Western world for over two thousand years. It is based on syllogisms , like concluding that "Socrates is a mortal" from the premises "Socrates is a man" and "all men are mortal". The currently dominant system is known as classical logic and covers many additional forms of inferences besides syllogisms. So-called extended logics are based on classical logic and introduce additional rules of inference for specific domains. For example, modal logic can be used to reason about what

22560-450: The depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs. Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with

22748-409: The example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have

22936-466: The explanation is simple, i.e. does not include any unnecessary claims, and that it is consistent with established knowledge. Other central criteria for a good explanation are that it fits observed and commonly known facts and that it is relevant, precise, and not circular. Ideally, the explanation should be verifiable by empirical evidence . If the explanation involves extraordinary claims then it requires very strong evidence. Abductive reasoning plays

23124-432: The field of ethics and introduces symbols to express the ideas of obligation and permission , i.e. to describe whether an agent has to perform a certain action or is allowed to perform it. The modal operators in temporal modal logic articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time. In epistemology, epistemic modal logic

23312-483: The form "(1) p , (2) if p then q , (3) therefore q " are valid, independent of what the terms p and q stand for. In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of logical truths . A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all possible worlds and under all interpretations of its non-logical terms, like

23500-458: The form of syllogisms . It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as Gottlob Frege . Today, the most commonly used system is classical logic . It consists of propositional logic and first-order logic . Propositional logic only considers logical relations between full propositions. First-order logic also takes

23688-399: The front door" based on the premises "the burglars forced the lock" and "if the burglars entered by the front door, then they forced the lock". This fallacy is similar to the valid rule of inference known as modus ponens. It is faulty because the first premise and the conclusion are switched around. Other well-known formal fallacies are denying the antecedent , affirming a disjunct , denying

23876-421: The general form of arguments while informal logic studies particular instances of arguments. Another approach is to hold that formal logic only considers the role of logical constants for correct inferences while informal logic also takes the meaning of substantive concepts into account. Further approaches focus on the discussion of logical topics with or without formal devices and on the role of epistemology for

24064-476: The identical symbol x , each occurrence of a variable symbol x in a formula φ is identified with the initial substring of φ up to the point at which said instance of the symbol x appears. Then, an occurrence of x is said to be bound if that occurrence of x lies within the scope of at least one of either ∃ x {\displaystyle \exists x} or ∀ x {\displaystyle \forall x} . Finally, x

24252-406: The internal parts of propositions into account, like predicates and quantifiers . Extended logics accept the basic intuitions behind classical logic and apply it to other fields, such as metaphysics , ethics , and epistemology . Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic. The word "logic" originates from

24440-406: The internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, predicates , which refer to properties and relations, and quantifiers, which treat notions like "some" and "all". For example, to express the proposition "this raven is black", one may use the predicate B {\displaystyle B} for the property "black" and

24628-482: The interpretation at hand. Logical symbols are a set of characters that vary by author, but usually include the following: Not all of these symbols are required in first-order logic. Either one of the quantifiers along with negation, conjunction (or disjunction), variables, brackets, and equality suffices. Other logical symbols include the following: Non-logical symbols represent predicates (relations), functions and constants. It used to be standard practice to use

24816-521: The logical conjunction ∧ {\displaystyle \land } requires terms on both sides. A proof system is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content. For instance, the classical rule of conjunction introduction states that P ∧ Q {\displaystyle P\land Q} follows from

25004-419: The logical operators, to avoid the need to write parentheses in some cases. These rules are similar to the order of operations in arithmetic. A common convention is: Moreover, extra punctuation not required by the definition may be inserted—to make formulas easier to read. Thus the formula: might be written as: In some fields, it is common to use infix notation for binary relations and functions, instead of

25192-419: The logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or", which is represented by the logical symbol ∨ {\displaystyle \lor } . However, a non-logical predicate symbol such as Phil( x ) could be interpreted to mean " x is a philosopher", " x is a man named Philip", or any other unary predicate depending on

25380-406: The modern approach, by simply specifying the "custom" signature to consist of the traditional sequences of non-logical symbols. The formation rules define the terms and formulas of first-order logic. When terms and formulas are represented as strings of symbols, these rules can be used to write a formal grammar for terms and formulas. These rules are generally context-free (each production has

25568-440: The most logical response may be to blindly trust them and duck instead of demanding an explanation or investigating what might have prompted their exclamation. Generally speaking, the less time there is, the more significant it is to trust intuitions and gut feelings. If there is more time, on the other hand, it becomes important to examine ambiguities and assess contradictory information. First-order logic A theory about

25756-408: The other hand, are true or false depending on whether they are in accord with reality. In formal logic, a sound argument is an argument that is both correct and has only true premises. Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For

25944-442: The other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like singular terms and predicates . For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by

26132-634: The other object. Another factor concerns not just the degree of similarity but also its relevance. For example, an artificial strawberry made of plastic may be similar to a real strawberry in many respects, including its shape, color, and surface structure. But these similarities are irrelevant to whether the artificial strawberry tastes as sweet as the real one. Analogical reasoning plays a central role in problem-solving , decision-making , and learning. It can be used both for simple physical characteristics and complex abstract ideas. In science, analogies are often used in models to understand complex phenomena in

26320-461: The person avoid the effects of propaganda or being manipulated by others. When important information is missing, it is often better to suspend judgment than to jump to conclusions. In this regard, logical reasoning should be skeptical and open-minded at the same time. On the practical level, logical reasoning concerns the issue of making rational and effective decisions. For many real-life decisions, various courses of action are available to

26508-442: The prefix notation defined above. For example, in arithmetic, one typically writes "2 + 2 = 4" instead of "=(+(2,2),4)". It is common to regard formulas in infix notation as abbreviations for the corresponding formulas in prefix notation, cf. also term structure vs. representation . The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention

26696-484: The premises P {\displaystyle P} and Q {\displaystyle Q} . Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are different types of proof systems including natural deduction and sequent calculi . A semantics is a system for mapping expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance,

26884-509: The premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid. The modus ponens is a prominent rule of inference. It has the form " p ; if p , then q ; therefore q ". Knowing that it has just rained ( p {\displaystyle p} ) and that after rain the streets are wet ( p → q {\displaystyle p\to q} ), one can use modus ponens to deduce that

27072-444: The premises are important to ensure that they offer significant support to the conclusion. In this regard, the sample size should be large to guarantee that many individual cases were considered before drawing the conclusion. An intimately connected factor is that the sample is random and representative. This means that it includes a fair and balanced selection of individuals with different key characteristics. For example, when making

27260-469: The premises make the conclusion more likely but do not ensure it. This support comes in degrees: strong arguments make the conclusion very likely, as is the case for well-researched issues in the empirical sciences. Some theorists give a very wide definition of logical reasoning that includes its role as a cognitive skill responsible for high-quality thinking. In this regard, it has roughly the same meaning as critical thinking . A variety of basic concepts

27448-411: The premises offer support for the conclusion because the conclusion is the best explanation of why the premises are true. In this sense, abduction is also called the inference to the best explanation . For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean

27636-408: The premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises. According to an influential view by Alfred Tarski , deductive arguments have three essential features: (1) they are formal, i.e. they depend only on the form of the premises and

27824-469: The premises. But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on

28012-401: The premises. Deductive reasoning, by contrast, is non-ampliative since it only extracts information already present in the premises without adding any additional information. So with non-deductive reasoning, one can learn something new that one did not know before. But the fact that new information is added means that this additional information may be false. This is why non-deductive reasoning

28200-408: The process of finding and evaluating reasons for and against a position to come to one's own conclusion. This includes being able to differentiate between reliable and unreliable sources of information. This matters for effective reasoning since it is often necessary to rely on information provided by other people instead of checking every single fact for oneself. This way, logical reasoning can help

28388-467: The process of generalization to arrive at a universal law. A well-known issue in the field of inductive reasoning is the so-called problem of induction . It concerns the question of whether or why anyone is justified in believing the conclusions of inductive inferences. This problem was initially raised by David Hume , who holds that future events need not resemble past observations. In this regard, inductive reasoning about future events seems to rest on

28576-481: The propositional connective "and". Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using truth-functional propositional connectives, their truth only depends on the truth values of their parts. But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects. Whether

28764-443: The propositions "all puppies are dogs" and "all dogs are animals" act as premises while the proposition "all puppies are animals" is the conclusion. A set of premises together with a conclusion is called an argument . An inference is the mental process of reasoning that starts from the premises and arrives at the conclusion. But the terms "argument" and "inference" are often used interchangeably in logic. The purpose of arguments

28952-403: The propositions are formed. For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid. Classical logic is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in

29140-415: The psychology of argumentation. Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic. Non-deductive arguments make their conclusion probable but do not ensure that it is true. An example is the inductive argument from the empirical observation that "all ravens I have seen so far are black" to

29328-445: The reasoner may have to revise it upon learning that penguins are birds that do not fly. Inductive reasoning starts from a set of individual instances and uses generalization to arrive at a universal law governing all cases. Some theorists use the term in a very wide sense to include any form of non-deductive reasoning, even if no generalization is involved. In the more narrow sense, it can be defined as "the process of inferring

29516-406: The reliability of information. Further factors are to seek new information, to avoid inconsistencies , and to consider the advantages and disadvantages of different courses of action before making a decision. Logical reasoning is a form of thinking that is concerned with arriving at a conclusion in a rigorous way. This happens in the form of inferences by transforming the information present in

29704-401: The resemblance between the two systems. Expressed schematically, arguments from analogy have the following form: (1) a is similar to b ; (2) a has feature F ; (3) therefore b probably also has feature F . Analogical reasoning can be used, for example, to infer information about humans from medical experiments on animals: (1) rats are similar to humans; (2) birth control pills affect

29892-434: The rules of inference they accept as valid and the formal languages used to express them. Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic. For example, it has been suggested that only logically complete systems, like first-order logic , qualify as logics. For such reasons, some theorists deny that higher-order logics are logics in

30080-438: The same fact and that the reasoner should only infer the best explanation . For example, a tsunami could also explain why the streets are wet but this is usually not the best explanation. As a form of non-deductive reasoning, abduction does not guarantee the truth of the conclusion even if the premises are true. The more plausible the explanation is, the stronger it is supported by the premises. In this regard, it matters that

30268-491: The scope of mathematics. Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives . For instance, propositional logic represents the conjunction of two atomic propositions P {\displaystyle P} and Q {\displaystyle Q} as the complex formula P ∧ Q {\displaystyle P\land Q} . Unlike predicate logic where terms and predicates are

30456-417: The semantics for classical propositional logic assigns the formula P ∧ Q {\displaystyle P\land Q} the denotation "true" whenever P {\displaystyle P} and Q {\displaystyle Q} are true. From the semantic point of view, a premise entails a conclusion if the conclusion is true whenever the premise is true. A system of logic

30644-600: The semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. For over two thousand years, Aristotelian logic was treated as the canon of logic in the Western world, but modern developments in this field have led to a vast proliferation of logical systems. One prominent categorization divides modern formal logical systems into classical logic , extended logics, and deviant logics . Aristotelian logic encompasses

30832-516: The sense that it is based on basic logical intuitions shared by most logicians. These intuitions include the law of excluded middle , the double negation elimination , the principle of explosion , and the bivalence of truth. It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are

31020-415: The sentence "For every x , if x is a philosopher, then x is a scholar" is logically equivalent to the sentence "There exists x such that x is a philosopher and x is not a scholar". The existential quantifier "there exists" expresses the idea that the claim " x is a philosopher and x is not a scholar" holds for some choice of x . The predicates "is a philosopher" and "is a scholar" each take

31208-402: The simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true under all interpretations of its non-logical terms. In some modal logics , this means that the proposition is true in all possible worlds. Some theorists define logic as

31396-413: The simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by theories of reference . Some complex propositions are true independently of the substantive meanings of their parts. In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like

31584-869: The singular term r {\displaystyle r} referring to the raven to form the expression B ( r ) {\displaystyle B(r)} . To express that some objects are black, the existential quantifier ∃ {\displaystyle \exists } is combined with the variable x {\displaystyle x} to form the proposition ∃ x B ( x ) {\displaystyle \exists xB(x)} . First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer ∃ x B ( x ) {\displaystyle \exists xB(x)} from B ( r ) {\displaystyle B(r)} . Extended logics are logical systems that accept

31772-494: The skills associated with logical reasoning to decide whether to boil and drink water from a stream that might contain dangerous microorganisms rather than break off the trip and hike back to the parking lot. This could include considering factors like assessing how dangerous the microorganisms are and the likelihood that they survive the boiling procedure. It may also involve gathering relevant information to make these assessments, for example, by asking other hikers. Time also plays

31960-472: The smallest units, propositional logic takes full propositions with truth values as its most basic component. Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition. First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates

32148-416: The streets are wet ( q {\displaystyle q} ). The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false. Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in

32336-433: The strict sense. When understood in a wide sense, logic encompasses both formal and informal logic. Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse. Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments. In this regard, it considers problems that formal logic on its own

32524-549: The strongest form of support: if their premises are true then their conclusion must also be true. This is not the case for ampliative arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into inductive and abductive arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens. Abductive arguments are inferences to

32712-437: The study of logical truths. Truth tables can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for

32900-697: The subject in two ways: either by affirming it or by denying it. For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one predicate in each case. Thus, these three propositions contain three predicates, referred to as major term , minor term , and middle term . The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how

33088-415: The subject to the predicate. For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is". The subject and the predicate are the terms of the proposition. Aristotelian logic does not contain complex propositions made up of simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using

33276-422: The symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted. Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism . It

33464-430: The symbols together form the alphabet of the language. As with all formal languages , the nature of the symbols themselves is outside the scope of formal logic; they are often regarded simply as letters and punctuation symbols. It is common to divide the symbols of the alphabet into logical symbols , which always have the same meaning, and non-logical symbols , whose meaning varies by interpretation. For example,

33652-413: The symptoms of their patient to make a diagnosis of the underlying cause. Analogical reasoning compares two similar systems. It observes that one of them has a feature and concludes that the other one also has this feature. Arguments that fall short of the standards of logical reasoning are called fallacies . For formal fallacies , like affirming the consequent , the error lies in the logical form of

33840-408: The table. This conclusion is justified because it is the best explanation of the current state of the kitchen. For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it

34028-469: The term ampliative or inductive reasoning is used. Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic. A deductively valid argument is one whose premises guarantee the truth of its conclusion. For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs" is deductively valid. For deductive validity, it does not matter whether

34216-508: The term "fallacy" does not mean that the conclusion is false. Instead, it only means that some kind of error was committed on the way to reaching the conclusion. An argument can be a fallacy even if, by a fortuitous accident, the conclusion is true. Outside the field of logic, the term "fallacy" is sometimes used in a slightly different sense for a false belief or theory and not for an argument. Fallacies are usually divided into formal and informal fallacies . Formal fallacies are expressed in

34404-478: The truth values "true" and "false". The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression " p ∧ q {\displaystyle p\land q} " uses the logical connective ∧ {\displaystyle \land } ( and ). It could be used to express

34592-411: The use of incorrect arguments does not mean their conclusions are incorrect . Deductive reasoning is the mental process of drawing deductive inferences. Deductively valid inferences are the most reliable form of inference: it is impossible for their conclusion to be false if all the premises are true. This means that the truth of the premises ensures the truth of the conclusion. A deductive argument

34780-423: The validity of arguments. For example, intuitionistic logics reject the law of excluded middle and the double negation elimination while paraconsistent logics reject the principle of explosion. Deductive reasoning plays a central role in formal logic and mathematics . In mathematics, it is used to prove mathematical theorems based on a set of premises, usually called axioms. For example, Peano arithmetic

34968-417: The value of the variable x is "Socrates", and in the second sentence it is "Plato". Due to the ability to speak about non-logical individuals along with the original logical connectives, first-order logic includes propositional logic. The truth of a formula such as " x is a philosopher" depends on which object is denoted by x and on the interpretation of the predicate "is a philosopher". Consequently, " x

35156-403: Was discussed at length around the turn of the 20th century but it is not widely accepted today. Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex. A complex proposition has other propositions as its constituents, which are linked to each other through propositional connectives like "and" or "if...then". Simple propositions, on

35344-404: Was primarily associated with deductive reasoning studied by formal logic. But in a wider sense, it also includes forms of non-deductive reasoning, such as inductive , abductive , and analogical reasoning . The forms of logical reasoning have in common that they use premises to make inferences in a norm-governed way. As norm-governed practices, they aim at inter-subjective agreement about

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