Definition
The word "logic" originates from the Greek word "logos", which has a variety of translations, such as reason, discourse, orFormal and informal logic
Logic encompasses both formal and informal logic. Formal logic is the traditionally dominant field, but applying its insights to actual everyday arguments has prompted modern developments of informal logic, which considers problems that formal logic on its own is unable to address. Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies. Various suggestions have been made concerning how to draw the distinction between the two but there is no universally accepted answer.Fundamental 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. For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". It is generally accepted that premises and conclusions have to be truth-bearers.Though see imperative logic, dynamic semantics, and inquisitive semantics for logical systems which narrow or generalize the notion of valid inference to other kinds of objects. This means that they have a truth value: they are either true or false. Thus contemporary philosophy generally sees them either as '' propositions'' or as '' sentences''. Propositions are the denotations of sentences and are usually understood as abstract objects. Propositional theories of premises and conclusions are often criticized because of the difficulties involved in specifying the identity criteria of abstract objects or because of naturalist considerations. 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. In earlier work, premises and conclusions were understood in psychological terms as thoughts or judgments, in an approach known as " psychologism". This position was heavily criticized around the turn of the 20th century.Internal structure
Premises and conclusions have 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 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 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 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.Logical truth
In some cases, a simple or a complex proposition is true independently of the substantive meanings of its parts. For example, the complex proposition "if Mars is red, then Mars is red" is true independent of whether its parts, i.e. 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 notion can be understood equivalently as truth at all possible worlds. Logical truth plays an important role in logic and some theorists even define logic as the study of logical truths.Truth tables
Truth tables can be used to show how logical connectives work or how the truth 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. The final columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression uses the logical connective (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, ("yesterday was Sunday") and ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are (or), (if...then), and (not). Truth tables can also be defined for more complex expressions that use several propositional connectives. For example, given the conditional proposition , one can form truth tables of its inverse , and its contraposition .Arguments and inferences
Logic 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 the other hand, are true or false depending on whether they are in accord with reality. In formal logic, aDeductive
A ''deductively valid argument'' is one whose premises guarantee the truth of its conclusion. For instance, the argument "Victoria is tall; Victoria has brown hair; therefore Victoria is tall and has brown hair" is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "trees can speak the English language; therefore trees can speak a language" is valid because, if the premise were true, the conclusion would be true as well. Alfred Tarski holds that 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. Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference. Rules of inference specify how the premises and the conclusion 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 () and that after rain the streets are wet (), one can use modus ponens to deduce that the streets are wet (). 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 the premises. But this point is not always accepted since it would mean, for example, that most ofAmpliative
Ampliative inferences, on the other hand, are informative even on the depth level. They are more interesting in this sense since the thinker may acquire substantive information from them and thereby 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. 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. Ampliative reasoning is of central importance since many arguments found in everyday discourse and theFallacies
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 the conclusion is true. Some theorists 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 is controversial because it belongs to the field ofDefinitory and strategic rules
The main focus of most logicians is to investigate 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 constitute definitory rules, which determine whether a certain inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary in order to reach a given conclusion based on a certain set of premises. This distinction does not just apply to logic but also to various games as well. InFormal systems
A formal system of logic consists of aFormal language
A ''formal language'' consists of an ''alphabet'' and syntactic rules. The alphabet 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 is a well-formed formula but is not.Proof system
A ''proof system'' is a collection of formal rules which define when a conclusion follows from given premises. For instance, the classical rule of conjunction introduction states that follows from the premises and . Rules in a proof systems are always defined in terms of formulas' syntactic form, never in terms of their meanings. Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are a number of different types of proof systems including natural deduction and sequent calculi. Proof systems are closely linked to philosophical work which characterizes logic as the study of valid inference.Semantics
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, the semantics for classical propositional logic assigns the formula the denotation "true" whenever and are true. Entailment is a semantic relation which holds between formulas when the first cannot be true without the second being true as well. Semantics is closely tied to the philosophical characterization of logic as the study of logical truth.Soundness and completeness
A system of logic 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 the semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. The study of properties of formal systems is called ''metalogic''. Other important metalogical properties include '' consistency'', '' decidability'', and '' expressive power''.Systems of logic
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. Classical logic is to be distinguished from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in the sense that it is based on various fundamental 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, like the distinction between necessity and possibility, the problem of ethical obligation and permission, or the relations between past, present, and future. Such issues are addressed by extended logics. They build on the fundamental 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 the scope of mathematics. Deviant logics, on the other hand, reject some of the fundamental 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. Informal logic is usually carried out in a less systematic way. It often focuses on more specific issues, like investigating a particular type of fallacy or studying a certain aspect of argumentation. Nonetheless, some systems of informal logic have also been presented that try to provide a systematic characterization of the correctness of arguments.Aristotelian
Aristotelian logic encompasses a great variety of topics, including metaphysical theses about ontological categories and problems of scientific explanation. But in a more narrow sense, it refers to term logic or syllogistics. A syllogism is a certain 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 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. In this sense, Aristotelian logic does not contain complex propositions made up of various simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using a logical connective like "and" to form a new complex proposition.Classical
Propositional logic
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 and as the complex formula . Unlike predicate logic where terms and predicates are 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; it cannot represent inferences that results from the inner structure of a proposition.First-order logic
Extended
Modal logic
Many extended logics take the form of modal logic by introducing modal operators. Modal logics were originally developed to represent statements about necessity and possibility. For instance the modal formula can be read as "possibly " while can be read as "necessarily ". Modal logics can be used to represent different phenomena depending on what ''flavor'' of necessity and possibility is under consideration. When is used to represent epistemic necessity, states that is known. When is used to represent deontic necessity, states that is a moral or legal obligation. Within philosophy, modal logics are widely used in formal epistemology, formal ethics, and metaphysics. Within linguistic semantics, systems based on modal logic are used to analyze linguistic modality in natural languages. Other fields such asHigher order logic
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 (''some'' apples are sweet) is an example of the existential quantifier applied to the individual variable . 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 . In this case, the existential quantifier is applied to the predicate variable . The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories. But it has various drawbacks in regard to its meta-logical properties and ontological implications, which is why first-order logic is still much more widely used.Deviant
A great variety of deviant logics have been proposed. One major paradigm isInformal
The ''pragmatic'' or ''dialogical approach'' to informal logic sees arguments as speech acts and not merely as a set of premises together with a conclusion. As speech acts, they occur in a certain context, like a dialogue, which affects the standards of right and wrong arguments. A prominent version by Douglas N. Walton understands a dialogue as a game between two players. The initial position of each player is characterized by the propositions to which they are committed and the conclusion they intend to prove. Dialogues are games of persuasion: each player has the goal of convincing the opponent of their own conclusion. This is achieved by making arguments: arguments are the moves of the game. They affect to which propositions the players are committed. A winning move is a successful argument that takes the opponent's commitments as premises and shows how one's own conclusion follows from them. This is usually not possible straight away. For this reason, it is normally necessary to formulate a sequence of arguments as intermediary steps, each of which brings the opponent a little closer to one's intended conclusion. Besides these positive arguments leading one closer to victory, there are also negative arguments preventing the opponent's victory by denying their conclusion. Whether an argument is correct depends on whether it promotes the progress of the dialogue. Fallacies, on the other hand, are violations of the standards of proper argumentative rules. These standards also depend on the type of dialogue. For example, the standards governing the scientific discourse differ from the standards in business negiotiations. The ''epistemic approach'' to informal logic, on the other hand, focuses on the epistemic role of arguments. It is based on the idea that arguments aim to increase our knowledge. They achieve this by linking justified beliefs to beliefs that are not yet justified. Correct arguments succeed at expanding knowledge while fallacies are epistemic failures: they do not justify the belief in their conclusion. In this sense, logical normativity consists in epistemic success or rationality. For example, the fallacy of begging the question is a ''fallacy'' because it fails to provide independent justification for its conclusion, even though it is deductively valid. The Bayesian approach is one example of an epistemic approach. Central to Bayesianism is not just whether the agent believes something but the degree to which they believe it, the so-called ''credence''. Degrees of belief are understood as subjective probabilities in the believed proposition, i.e. as how certain the agent is that the proposition is true. On this view, reasoning can be interpreted as a process of changing one's credences, often in reaction to new incoming information. Correct reasoning, and the arguments it is based on, follows the laws of probability, for example, the principle of conditionalization. Bad or irrational reasoning, on the other hand, violates these laws.Areas of research
Logic is studied in various fields. In many cases, this is done by applying its formal method to specific topics outside its scope, like to ethics or computer science. In other cases, logic itself is made the subject of research in another discipline. This can happen in diverse ways, like by investigating the philosophical presuppositions of fundamental logical concepts, by interpreting and analyzing logic through mathematical structures, or by studying and comparing abstract properties of formal logical systems.Philosophy of logic and philosophical logic
''Philosophy of logic'' is the philosophical discipline studying the scope and nature of logic. It investigates many presuppositions implicit in logic, like how to define its fundamental concepts or the metaphysical assumptions associated with them. It is also concerned with how to classify the different logical systems and considers the ontological commitments they incur. ''Philosophical logic'' is one important area within the philosophy of logic. It studies the application of logical methods to philosophical problems in fields like metaphysics, ethics, and epistemology. This application usually happens in the form of extended or deviant logical systems.Mathematical logic
Mathematical logic is the study of logic within mathematics. Major subareas include model theory,Computational logic
Formal semantics of natural language
Formal semantics, a subfield of bothEpistemology of logic
The epistemology of logic investigates how one knows that an argument is valid or that a proposition is logically true. This includes questions like how to justify that modus ponens is a valid rule of inference or that contradictions are false. The traditionally dominant view is that this form of logical understanding belongs to knowledge a priori. In this regard, it is often argued that the mind has a special faculty to examine relations between pure ideas and that this faculty is also responsible for apprehending logical truths. A similar approach understands the rules of logic in terms of linguistic conventions. On this view, the laws of logic are trivial since they are true by definition: they just express the meanings of the logical vocabulary. Important objections to the view that logic is knowable a priori were presented in the 20th century by W. V. Quine and Hilary Putnam. In his paper " Is Logic Empirical?", Putnam builds on a suggestion by Quine and argues that, in general, the facts of propositional logic have a similar epistemological status as facts about the physical universe. This pertains, for example, to the laws of mechanics or of general relativity, and in particular to what physicists have learned about quantum mechanics. According to Putnam, these insights provide a compelling case for abandoning certain familiar principles of classical logic: if one wants to be a realist about the physical phenomena described by quantum theory, then one should abandon the principle of distributivity. He suggests that classical logic be replaced with the quantum logic proposed by Garrett Birkhoff and John von Neumann.History
Logic was developed independently in several cultures during antiquity. One major early contributor was Aristotle, who developed term logic in his '' Organon'' and '' Prior Analytics''. In this approach, ''judgements'' are broken down into ''propositions'' consisting of two terms that are related by one of a fixed number of relations. Inferences are expressed by means of syllogisms that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises. Aristotle's monumental insight was the notion that arguments can be characterized in terms of their form. The later logician Łukasiewicz described this insight as "one of Aristotle's greatest inventions". Aristotle's system of logic was also responsible for the introduction of hypothetical syllogism, temporal modal logic, and inductive logic, as well as influential vocabulary such as terms, predicables, syllogisms and propositions. Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century. It has now been superseded by later work, though many of its key insights are still present in modern systems of logic. Ibn Sina (Avicenna) (980–1037 CE) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world. It also had an important influence on Western medieval writers such as Albertus Magnus and William of Ockham. Ibn Sina wrote on the hypothetical syllogism and on the propositional calculus. He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic. He also made use of inductive logic, such as his methods of agreement, difference, and concomitant variation, which are critical to the scientific method. Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873). In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the High Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of scholasticism. Initially, medieval Christian scholars drew on the classics that had been preserved in Latin through commentaries by such figures such as Boethius. Later, the work of Islamic philosophers such as Ibn Sina and Ibn Rushd (Averroes 1126–1198 CE) were drawn on. This expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries. In 1323, William of Ockham's influential '' Summa Logicae'' was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in Holberg's satirical play '' Erasmus Montanus''. Friedrich Nietzsche criticized logic based on the claim that the logical structure of thought is a useful tool for human survival while " gic itself rests upon assumptions to which nothing in the world of reality corresponds". The Chinese logical philosopher Gongsun Long () proposed the paradox "One and one cannot become two, since neither becomes two". In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi. In India, the Anviksiki school of logic was founded by Medhātithi (c. 6th century BCE). Innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century with the Navya-Nyāya school. By the 16th century, it developed theories resembling modern logic, such as Gottlob Frege's "distinction between sense and reference of proper names" and his definition of number. Its development of the theory of ''restrictive conditions for universals'' anticipated some of the developments in modern set theory.Chakrabarti, Kisor Kumar. 1976. "Some Comparisons Between Frege's Logic and Navya-Nyaya Logic." '' Philosophy and Phenomenological Research'' 36(4):554–63. . "This paper consists of three parts. The first part deals with Frege's distinction between sense and reference of proper names and a similar distinction in Navya-Nyaya logic. In the second part we have compared Frege's definition of number to the Navya-Nyaya definition of number. In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory." Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and George Boole. In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively. The syllogistic logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the foundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1854, George Boole published '' The Laws of Thought'', introducing symbolic logic and the principles of what is now known as Boolean logic. In 1879, Gottlob Frege published '' Begriffsschrift'', which inaugurated modern logic with the invention of quantifier notation. This invention reconciled the Aristotelian and Stoic logics in a broader system, and solved problems for which Aristotelian logic was impotent, such as the problem of multiple generality. From 1910 to 1913, Alfred North Whitehead and Bertrand Russell published '' Principia Mathematica'' on the foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931, Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues. The development of logic since Frege, Russell, and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see analytic philosophy) and philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy, sociology, advertising and literature departments, often as a compulsory discipline.See also
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