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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 due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a 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 together with 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" to the conclusion "I don't have to work".^[1] 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 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 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.^[2] Abductive arguments are inferences to the best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer.^[3] 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 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 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 the internal parts of propositions into account, like predicates and quantifiers. Extended logics accept the basic intuitions behind classical logic and extend 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.

Definition

The word "logic" originates from the Greek word "logos", which has a variety of translations, such as reason, discourse, or language.^[4] Logic is traditionally defined as the study of the laws of thought or correct reasoning,^[5] and is usually understood in terms of inferences or arguments. Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences.^[6] An argument is a set of premises together with a conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion.^[7] 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.^[8] Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic.^[9]

Formal logic

Formal logic 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.^[10]

Formal logic 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.^[11] For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a rule of inference.^[12] For example, modus ponens is a rule of inference according to which all arguments of the form "(1) p, (2) if p then q, (3) therefore q" are valid, independent of what the terms p and q stand for.^[13] 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.^[14] 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 the claim "either it is raining, or it is not".^[15] 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.^[16]

Formal logic uses formal languages to express and analyze arguments.^[17] 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.^[18] This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether a given argument is valid.^[19] 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.^[20]

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 the rules of inference they accept as valid and the formal languages used to express them.^[21] Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic.^[22] 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 the strict sense.^[23]

Informal logic

When understood in a wide sense, logic encompasses both formal and informal logic.^[24] Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse.^[25] Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments.^[26] In this regard, it considers problems that formal logic on its own is unable to address.^[27] Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies.^[28]

Many characterizations of informal logic have been suggested but there is no general agreement on its precise definition.^[29] 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.^[30] Formal logic can only examine them indirectly by translating them first into a formal language while informal logic investigates them in their original form.^[31] 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) ${\displaystyle \forall x(Bird(x)\to Flies(x))}$; (2) ${\displaystyle Bird(Tweety)}$; (3) ${\displaystyle Flies(Tweety)}$" is studied by formal logic.^[32] The study of natural language arguments comes with various difficulties. For example, natural language expressions are often ambiguous, vague, and context-dependent.^[33] 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 the psychology of argumentation.^[34]

Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic.^[35] 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 the conclusion "all ravens are black".^[36]

A further approach is to define informal logic as the study of informal fallacies.^[37] Informal fallacies are incorrect arguments in which errors are present in the content and the context of the argument.^[38] 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".^[39] Some theorists state that formal logic studies 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 the assessment of arguments.^[40]

Basic concepts

Premises, conclusions, and truth

Premises and conclusions

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.^[41] For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers.^[41][a] 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.^[43] Propositions are the denotations of sentences and are usually seen as abstract objects.^[44] For example, the English sentence "the tree is green" is different from the German sentence "der Baum ist grün" but both express the same proposition.^[45]

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.^[43] These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like 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.^[46] Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism. It was discussed at length around the turn of the 20th century but it is not widely accepted today.^[47]

Internal structure

Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex.^[48] 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 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.^[49][48] 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 the propositional connective "and".^[49]

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.^[49][50] 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.^[51] Whether 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.^[52]

Logical truth

Some complex propositions are true independently of the substantive meanings of their parts.^[53] 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 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.^[54] 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.^[55] Some theorists define logic as the study of logical truths.^[16]

Truth tables

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 the truth values "true" and "false".^[56] 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 "${\displaystyle p\land q}$" uses the logical connective ${\displaystyle \land }$ (and). It could be used to express a sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, ${\displaystyle p}$ ("yesterday was Sunday") and ${\displaystyle q}$ ("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).^[57] Given the conditional proposition ${\displaystyle p\to q}$, one can form truth tables of its converse ${\displaystyle q\to p}$, its inverse (${\displaystyle \lnot p\to \lnot q}$), and its contrapositive (${\displaystyle \lnot q\to \lnot p}$). Truth tables can also be defined for more complex expressions that use several propositional connectives.^[58]

Truth table of various expressions

p	q	p ∧ q	p ∨ q	p → q	¬p → ¬q	p ${\displaystyle \uparrow }$ q
T	T	T	T	T	T	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	F	T	T	T

Arguments and inferences

Logic is commonly defined in terms of arguments or inferences as the study of their correctness.^[59] An argument is a set of premises together with a conclusion.^[60] An inference is the process of reasoning from these premises to the conclusion.^[43] 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 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.^[61] 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 a complex argument to be successful, each link of the chain has to be successful.^[43]

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.^[62] 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, the term ampliative or inductive reasoning is used.^[63] Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic.^[64]

Deductive

A deductively valid argument is one whose premises guarantee the truth of its conclusion.^[11] 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 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.^[65]

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 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.^[66]

Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference.^[67] Rules of inference specify the form of 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.^[68] The modus ponens is a prominent rule of inference. It has the form "p; if p, then q; therefore q".^[69] Knowing that it has just rained (${\displaystyle p}$) and that after rain the streets are wet (${\displaystyle p\to q}$), one can use modus ponens to deduce that the streets are wet (${\displaystyle q}$).^[70]

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.^[71] 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 the premises.^[72] 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 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.^[73]

Ampliative

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 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.^[74] 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 the light of new inferences drawn.^[75] Ampliative reasoning plays a central role for 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 is blurry in some cases, 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.^[76]

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.^[77] But in a more narrow sense, induction is only one type of ampliative argument alongside abductive arguments.^[78] Some philosophers, like Leo Groarke, also allow conductive arguments^[b] as one more type.^[79] In this narrow sense, induction is often defined as a form of statistical generalization.^[80] 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.^[81] In this sense, one may infer that "all elephants are gray" based on one's past observations of the color of elephants.^[78] 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.^[81] Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference.^[78]

Abductive inference may or may not take statistical observations into consideration. In either case, the premises offer support for the conclusion because the conclusion is the best explanation of why the premises are true.^[82] In this sense, abduction is also called the inference to the best explanation.^[83] 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 the table. This conclusion is justified because it is the best explanation of the current state of the kitchen.^[78] 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 is not the best or most likely explanation.^[82][83]

Fallacies

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