formal logic

TheInfoList

OR:

Logic is the study of correct
reason Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, ...
ing. It includes both formal and
informal logic Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. ...
. Formal logic is the science of deductively valid inferences or of
logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whic ...
s. It is a
formal science Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, ga ...
premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
s in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
that articulates a
proof system In mathematical logic, a proof calculus or a proof system is built to prove statements. Overview A proof system includes the components: * Language: The set ''L'' of formulas admitted by the system, for example, propositional logic or first-order ...
. Formal logic contrasts with informal logic, which is associated with
informal fallacies Informal fallacies are a type of incorrect argument in natural language. The source of the error is not just due to the ''form'' of the argument, as is the case for formal fallacies, but can also be due to their ''content'' and ''context''. Fall ...
,
critical thinking Critical thinking is the analysis of available facts, evidence, observations, and arguments to form a judgement. The subject is complex; several different definitions exist, which generally include the rational, skeptical, and unbiased analysis ...
, and
argumentation theory Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be supported or undermined by premises through logical reasoning. With historical origins in logic, dialectic, and rhetoric, argumentation theory, includ ...
. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
s are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
,
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, and
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually understood either as
sentences ''The Four Books of Sentences'' (''Libri Quattuor Sententiarum'') is a book of theology written by Peter Lombard in the 12th century. It is a systematic compilation of theology, written around 1150; it derives its name from the ''sententiae'' o ...
or as
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s and are characterized by their internal structure; complex propositions are made up of simpler propositions linked to each other by propositional connectives like $\land$ (and) or $\to$ (if...then). The truth of a proposition usually depends on the
denotation In linguistics and philosophy, the denotation of an expression is its literal meaning. For instance, the English word "warm" denotes the property of being warm. Denotation is contrasted with other aspects of meaning including connotation. For inst ...
s of its constituents. Logically true propositions constitute a special case since their truth depends only on the logical vocabulary used in them and not on the denotations of other terms. Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. The strongest form of support is found in deductive arguments: it is impossible for their premises to be true and their conclusion to be false. Deductive arguments contrast with
ampliative Ampliative (from Latin ''ampliare'', "to enlarge"), a term used mainly in logic, meaning "extending" or "adding to that which is already known". This terminology was often used by medieval logicians in the analyses of the temporal content of thei ...
arguments, which may arrive in their conclusion at new information that is not present in the premises. However, it is possible for all their premises to be true while their conclusion is still false. Many arguments found in everyday discourse and the sciences are ampliative arguments, sometimes divided into inductive and
abductive Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th centu ...
arguments. Inductive arguments usually take the form of
statistical Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...
generalizations while abductive arguments are ''
inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
s to the best explanation''. Arguments that fall short of the standards of correct reasoning are called
fallacies A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was intr ...
. Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. Logic has been studied since Antiquity; early approaches include Aristotelian logic,
Stoic logic Stoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece. It was one of the two great systems of logic in the classical world. It was largely built and shaped by Chrysippus, the third head of the Stoi ...
,
Anviksiki Ānvīkṣikī is a term in Sanskrit denoting roughly the "science of inquiry" and it should have been recognized in India as a distinct branch of learning as early as 650 BCE. However, over the centuries its meaning and import have undergone con ...
, and the
mohists Mohism or Moism (, ) was an ancient Chinese philosophy of ethics and logic, rational thought, and science developed by the academic scholars who studied under the ancient Chinese philosopher Mozi (c. 470 BC – c. 391 BC), embodied in an epon ...
. Modern formal logic has its roots in the work of late 19th-century mathematicians such as
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
. While Aristotelian logic focuses on reasoning in the form of
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
s, in the modern era its traditional dominance was replaced by
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
, a set of fundamental logical intuitions shared by most logicians. It consists of
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, which only considers the logical relations on the level of propositions, and
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
, which also articulates the internal structure of propositions using various linguistic devices, such as predicates and quantifiers. Extended logics accept the basic intuitions behind classical logic and extend it to other fields, such as
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
,
ethics Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concerns m ...
, and
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
. Deviant logics, on the other hand, reject certain classical intuitions and provide alternative accounts of the fundamental laws of logic.

# Definition

The word "logic" originates from the Greek word "logos", which has a variety of translations, such as
reason Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, ...
,
discourse Discourse is a generalization of the notion of a conversation to any form of communication. Discourse is a major topic in social theory, with work spanning fields such as sociology, anthropology, continental philosophy, and discourse analysis. ...
, or
language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
. Logic is traditionally defined as the study of the
laws of thought The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally th ...
or correct
reasoning Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...
, and is usually understood in terms of
inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
s or
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
s. Reasoning may be seen as the activity of drawing inferences whose outward expression is given in arguments. An inference or an argument is a set of premises together with a conclusion. Logic is interested in whether arguments are good or inferences are valid, i.e. whether the premises support their conclusions. These general characterizations apply to logic in the widest sense since they are true both for formal and informal logic, but many definitions of logic focus on the more paradigmatic formal logic. In this narrower sense, logic is a formal science that studies how conclusions follow from premises in a topic-neutral way. In this regard, logic is sometimes contrasted with the theory of
rationality Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...
, which is wider since it covers all forms of good reasoning. As a
formal science Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, ga ...
, logic contrasts with both the
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
and
social science Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of soc ...
s in that it tries to characterize the inferential relations between premises and conclusions based on their structure alone. This means that the actual content of these propositions, i.e. their specific topic, is not important for whether the inference is valid or not. Valid inferences are characterized by the fact that the truth of their premises ensures the truth of their conclusion: it is impossible for the premises to be true and the conclusion to be false. The general logical structures characterizing valid inferences are called
rules of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
. In this sense, logic is often defined as the study of valid inference. This contrasts with another prominent characterization of logic as the science of
logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whic ...
s. 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 world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional logic, intensional and mod ...
s and under all interpretations of its non-logical terms. These two characterizations of logic are closely related to each other: an inference is valid if the
material conditional The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q is ...
from its premises to its conclusion is logically true. 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 A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
. Different logics differ from each other concerning the
formal languages In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (formal languages), alphabet and are well-formedness, well-formed ...
used to express them and, most importantly, concerning the rules of inference they accept as valid. Starting in the 20th century, many new formal systems have been proposed. There are various disagreements concerning what makes a formal system a logic. For example, it has been suggested that only logically complete systems qualify as logics. For such reasons, some theorists deny that
higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more express ...
s and
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
are logics in the strict sense.

## Formal and informal logic

Logic encompasses both formal and
informal logic Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. ...
. 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. The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments. On this view, formal logic studies arguments expressed in formal languages while informal logic studies arguments expressed in informal or
natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
s. This means that the inference from the formulas and to the conclusion is studied by formal logic. The inference from the
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
sentences "Al lit a cigarette" and "Bill stormed out of the room" to the sentence "Al lit a cigarette and Bill stormed out of the room", on the other hand, belongs to informal logic. Formal languages are characterized by their precision and simplicity. They normally contain a very limited vocabulary and exact rules on how their symbols can be used to construct sentences, usually referred to as
well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be ...
s. This simplicity and exactness of formal logic make it capable of formulating precise rules of inference that determine whether a given argument is valid. This approach brings with it the need to translate natural language arguments into the formal language before their validity can be assessed, a procedure that comes with various problems of its own. Informal logic avoids some of these problems by analyzing natural language arguments in their original form without the need of translation. But it faces problems associated with the
ambiguity Ambiguity is the type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty. It is thus an attribute of any idea or statement ...
,
vagueness In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is ...
, and context-dependence of natural language expressions. A closely related approach applies the terms "formal" and "informal" not just to the language used, but more generally to the standards, criteria, and procedures of argumentation. Another approach draws the distinction according to the different types of inferences analyzed. This perspective understands formal logic as the study of deductive inferences in contrast to informal logic as the study of non-deductive inferences, like inductive or abductive inferences. The characteristic of deductive inferences is that the truth of their premises ensures the truth of their conclusion. This means that if all the premises are true, it is impossible for the conclusion to be false. For this reason, deductive inferences are in a sense trivial or uninteresting since they do not provide the thinker with any new information not already found in the premises. Non-deductive inferences, on the other hand, are
ampliative Ampliative (from Latin ''ampliare'', "to enlarge"), a term used mainly in logic, meaning "extending" or "adding to that which is already known". This terminology was often used by medieval logicians in the analyses of the temporal content of thei ...
: they help the thinker learn something above and beyond what is already stated in the premises. They achieve this at the cost of certainty: even if all premises are true, the conclusion of an ampliative argument may still be false. One more approach tries to link the difference between formal and informal logic to the distinction between formal and
informal fallacies Informal fallacies are a type of incorrect argument in natural language. The source of the error is not just due to the ''form'' of the argument, as is the case for formal fallacies, but can also be due to their ''content'' and ''context''. Fall ...
. This distinction is often drawn in relation to the ''
form Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens. Form also refers to: *Form (document), a document (printed or electronic) with spaces in which to write or enter data ...
'', ''content'', and ''
context Context may refer to: * Context (language use), the relevant constraints of the communicative situation that influence language use, language variation, and discourse summary Computing * Context (computing), the virtual environment required to su ...
'' of arguments. In the case of formal fallacies, the error is found on the level of the argument's form, whereas for informal fallacies, the content and context of the argument are responsible. Formal logic abstracts away from the argument's content and is only interested in its form, specifically whether it follows a valid rule of inference. In this regard, it is not important for the validity of a formal argument whether its premises are true or false. Informal logic, on the other hand, also takes the content and context of an argument into consideration. A
false dilemma A false dilemma, also referred to as false dichotomy or false binary, is an informal fallacy based on a premise that erroneously limits what options are available. The source of the fallacy lies not in an invalid form of inference but in a false ...
, for example, involves an error of content by excluding viable options, as in "you are either with us or against us; you are not with us; therefore, you are against us". For the
strawman fallacy A straw man (sometimes written as strawman) is a form of argument and an informal fallacy of having the impression of refuting an argument, whereas the real subject of the argument was not addressed or refuted, but instead replaced with a false o ...
, on the other hand, the error is found on the level of context: a weak position is first described and then defeated, even though the opponent does not hold this position. But in another context, against an opponent that actually defends the strawman position, the argument is correct. Other accounts draw the distinction based on investigating general forms of arguments in contrast to particular instances or on the study of
logical constant In logic, a logical constant of a language \mathcal is a symbol that has the same semantic value under every interpretation of \mathcal. Two important types of logical constants are logical connectives and quantifiers. The equality predicate (us ...
concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by s ...
s. A further approach focuses on the discussion of logical topics with or without formal devices or on the role of
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
for the assessment of arguments.

# 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-bearer A truth-bearer is an entity that is said to be either true or false and nothing else. The thesis that some things are true while others are false has led to different theories about the nature of these entities. Since there is divergence of o ...
s.Though see
imperative logic Imperative logic is the field of logic concerned with imperatives. In contrast to declaratives, it is not clear whether imperatives denote propositions or more generally what role truth and falsity play in their semantics. Thus, there is almost n ...
,
dynamic semantics Dynamic semantics is a framework in logic and natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynami ...
, and
inquisitive semantics Inquisitive semantics is a framework in logic and natural language semantics. In inquisitive semantics, the semantic content of a sentence captures both the information that the sentence conveys and the issue that it raises. The framework provides ...
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 In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progr ...
: they are either true or false. Thus contemporary philosophy generally sees them either as ''
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s'' or as ''
sentences ''The Four Books of Sentences'' (''Libri Quattuor Sententiarum'') is a book of theology written by Peter Lombard in the 12th century. It is a systematic compilation of theology, written around 1150; it derives its name from the ''sententiae'' o ...
''. Propositions are the
denotation In linguistics and philosophy, the denotation of an expression is its literal meaning. For instance, the English word "warm" denotes the property of being warm. Denotation is contrasted with other aspects of meaning including connotation. For inst ...
s of sentences and are usually understood as
abstract object In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, hum ...
s. 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 Ambiguity is the type of meaning (linguistics), meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations wikt:plausible#Adjective, plausible. A common aspect of ambiguity is uncertainty. It ...
, 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 Psychologism is a family of philosophical positions, according to which certain psychological facts, laws, or entities play a central role in grounding or explaining certain non-psychological facts, laws, or entities. The word was coined by Johann ...
". 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 term A singular term is a paradigmatic referring device in a language. Singular terms are of philosophical importance for philosophers of language, because they ''refer'' to things in the world, and the ability of words to refer calls for scrutiny. Over ...
s and
predicates Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, ...
. 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 In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
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 logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
s, 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 table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
s 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 $\land$ (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, $p$ ("yesterday was Sunday") and $q$ ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are $\lor$ (or), $\to$ (if...then), and $\lnot$ (not). Truth tables can also be defined for more complex expressions that use several propositional connectives. For example, given the conditional proposition $p \to q$, one can form truth tables of its inverse , and its
contraposition In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as Proof by contrapositive, proof by contraposition. The cont ...
.

## 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 Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary. The term is also used to refer to the ontological status of things, indicating their existence. In physical terms, r ...
. In formal logic, a
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
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. These simple arguments constitute a ''chain'' because the conclusions of the earlier arguments are used as premises in the later arguments. For a complex argument to be successful, each link of the chain has to be successful. Arguments and inferences are either are 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 Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
. But even arguments that are not deductively valid may still constitute good arguments because their premises offer non-deductive support to their conclusions. For such cases, 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.

### Deductive

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 Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
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 Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a Statement (logic), statement which is truth, true regardless of the truth or falsity of its constituent propositions. In other words, a logical ...
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 In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
. 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 In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
is a prominent rule of inference. It has the form "''p''; if ''p'', then ''q''; therefore ''q''". Knowing that it has just rained ($p$) and that after rain the streets are wet ($p \to q$), one can use modus ponens to deduce that the streets are wet ($q$). 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 Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
not already present in the premises. But this point is not always accepted since it would mean, for example, that most of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
is uninformative. A different characterization distinguishes between surface and depth information. On this view, deductive inferences are uninformative on the depth level but can be highly informative on the surface level, as may be the case for various mathematical proofs.

### Ampliative

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 the
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
s are ampliative. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. An important aspect of most ampliative arguments is that the support they provide for their conclusion comes in degrees. In this sense, 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. The terminology used to categorize ampliative arguments is inconsistent. Some authors use the term "induction" to cover all forms of non-deductive arguments. But in a more narrow sense, ''induction'' is only one type of ampliative argument besides ''abductive arguments''. Some authors also allow ''conductive arguments'' as one more type. In this narrow sense, induction is often defined as a form of
statistical Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...
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 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 stipulate that inductive inferences rest only on statistical considerations in order to distinguish them from abductive inference. 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 An explanation is a set of Statement (logic), statements usually constructed to description, describe a set of facts which clarifies the causality, causes, wiktionary:context, context, and Logical consequence, consequences of those facts. It may ...
of why the premises are true.On
abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th century ...
, see: * Magnani, L. 2001. ''Abduction, Reason, and Science: Processes of Discovery and Explanation''. New York: Kluwer Academic Plenum Publishers. xvii. . * Josephson, John R., and Susan G. Josephson. 1994. ''Abductive Inference: Computation, Philosophy, Technology''. New York: Cambridge University Press. viii. . * Bunt, H. and W. Black. 2000. ''Abduction, Belief and Context in Dialogue: Studies in Computational Pragmatics'', (''Natural Language Processing'' 1). Amsterdam:
John Benjamins John Benjamins Publishing Company is an independent academic publisher in social sciences and humanities with its head office in Amsterdam, Netherlands. The company was founded in the 1960s by John and Claire Benjamins and is currently managed ...
. vi. .
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 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 is not the best or most likely explanation.

## Fallacies

Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as
fallacies A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was intr ...
. 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 of
psychology Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries betwe ...
, 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 Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form: :If ''P'', then ''Q''. :Therefore, if not ...
is one type of formal fallacy, as in "if Othello is a bachelor, then he 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, 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.

## Definitory 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. In
chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...
, for example, the definitory rules dictate that
bishops A bishop is an ordained clergy member who is entrusted with a position of authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance of dioceses. The role or office of bishop is ca ...
may only move diagonally while the strategic rules describe how the allowed moves may be used to win a game, for example, by controlling the center and by defending one's
king King is the title given to a male monarch in a variety of contexts. The female equivalent is queen, which title is also given to the consort of a king. *In the context of prehistory, antiquity and contemporary indigenous peoples, the tit ...
. A third type of rules concerns
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
descriptive rules. They belong to the field of psychology and generalize how people actually draw inferences. It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning.

## Formal systems

A formal system of logic consists of a
language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
, a
proof system In mathematical logic, a proof calculus or a proof system is built to prove statements. Overview A proof system includes the components: * Language: The set ''L'' of formulas admitted by the system, for example, propositional logic or first-order ...
, and a
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy Philosophy (f ...
. The term "a logic" is often used a
countable noun In linguistics, a count noun (also countable noun) is a noun that can be modified by a quantity and that occurs in both singular and plural forms, and that can co-occur with quantificational determiners like ''every'', ''each'', ''several'', e ...
to refer to a particular formal system of logic. Starting in the 20th century, many new formal systems have been proposed.The term "a logic" is sometimes reserved for just the system's syntax, i.e. its language and proof theory. In the philosophical literature, the term is sometimes further restricted to refer only to particular logic-based formal systems such as those which are complete or motivated by intuitions close to those which motivated classical logic.

### Formal 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 Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
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 Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction) is a valid rule of inference of propositional logic. The rule makes it possible to introduce a conjunction into a logical proof. I ...
states that $P \land Q$ follows from the premises $P$ and $Q$. 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 In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axiom ...
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 $P \land Q$ the denotation "true" whenever $P$ and $Q$ are true.
Entailment Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...
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 In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
'', '' 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 logic Deviant logic is a type of logic incompatible with classical logic. Philosopher Susan Haack uses the term ''deviant logic'' to describe certain non-classical systems of logic. In these logics: * the set of well-formed formulas generated equals t ...
s. 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 In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...
, the
double negation elimination In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not (not ...
, the
principle of explosion In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (, 'from falsehood, anything ollows; or ), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a co ...
, 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 Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concerns m ...
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 Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
ontological In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
categories and problems of scientific explanation. But in a more narrow sense, it refers to
term logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, th ...
or syllogistics. A
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
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. Aristotelian logic differs from predicate logic in that the subject is either ''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". In predicate logic, on the other hand, universal and particular propositions would be expressed by using a quantifier and two predicates. Another important difference is that Aristotelian logic only includes predicates for simple
properties Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and ...
of entities, but lacks predicates corresponding to relations between entities. The predicate can be linked to 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 different 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 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

### Propositional logic

Propositional logic comprises formal systems in which formulae are built from
atomic propositions In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. ...
using
logical connectives In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary c ...
. For instance, propositional logic represents the
conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...
of two atomic propositions $P$ and $Q$ as the complex formula $P \land Q$. 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

First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates the internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects,
predicates Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, ...
, 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 $B$ for the property "black" and the singular term $r$ referring to the raven to form the expression $B\left(r\right)$. To express that some objects are black, the existential quantifier $\exists$ is combined with the variable $x$ to form the proposition $\exists x B\left(x\right)$. 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 $\exists x B\left(x\right)$ from $B\left(r\right)$. The development of first-order logic is usually attributed to
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
, who is also credited as one of the founders of
analytic philosophy Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United Sta ...
. However, the formulation of first-order logic most often used today is found in Principles of Mathematical Logic by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
and
Wilhelm Ackermann Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography ...
in 1928. The analytical generality of first-order logic allowed the formalization of mathematics, drove the investigation of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, and allowed the development of Alfred Tarski's approach to
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
. It provides the foundation of modern
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
.

## 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 $\Diamond P$ can be read as "possibly $P$" while $\Box P$ can be read as "necessarily $P$". Modal logics can be used to represent different phenomena depending on what ''flavor'' of necessity and possibility is under consideration. When $\Box$ is used to represent epistemic necessity, $\Box P$ states that $P$ is known. When $\Box$ is used to represent deontic necessity, $\Box P$ states that $P$ is a moral or legal obligation. Within philosophy, modal logics are widely used in
formal epistemology Formal epistemology uses formal methods from decision theory, logic, probability theory and computability theory to model and reason about issues of epistemological interest. Work in this area spans several academic fields, including philosophy, ...
,
formal ethics Formal ethics is a formal logical system for describing and evaluating the "form" as opposed to the "content" of ethical principles. Formal ethics was introduced by Harry J. Gensler, in part in his 1990 logic textbook ''Symbolic Logic: Classical an ...
, and metaphysics. Within
linguistic semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
, systems based on modal logic are used to analyze
linguistic modality In linguistics and philosophy, modality refers to the ways language can express various relationships to reality or truth. For instance, a modal expression may convey that something is likely, desirable, or permissible. Quintessential modal ex ...
in natural languages. Other fields such as
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
and set theory have applied the relational semantics for modal logic beyond its original conceptual motivation, using it to provide insight into patterns including the set-theoretic multiverse and
transition system In theoretical computer science, a transition system is a concept used in the study of computation. It is used to describe the potential behavior of discrete systems. It consists of states and transitions between states, which may be labeled wi ...
s in computation.

### Higher 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 In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
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 is
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
, which rejects the
law of the excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, Exclusive or, either this proposition or its negation is Truth value, true. It is one of the so-called Law of thought#The three tradit ...
. Intuitionism was developed by the Dutch mathematicians
L.E.J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
and
Arend Heyting __NOTOC__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Biography Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a foot ...
to underpin their constructive approach to mathematics, in which the existence of a mathematical object can only be proven by constructing it. Intuitionism was further pursued by
Gerhard Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...
,
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
,
Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He wa ...
, among others. Intuitionistic logic is of great interest to computer scientists, as it is a
constructive logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems o ...
and sees many applications, such as extracting verified programs from proofs and influencing the design of
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s through the formulae-as-types correspondence. It is closely related to nonclassical systems such as
Gödel–Dummett logic In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate ...
and inquisitive logic.
Multi-valued logics Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...
depart from classicality by rejecting the
principle of bivalence In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called ...
which requires all propositions to be either true or false. For instance,
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. He ...
and
Stephen Cole Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
both proposed
ternary logic In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating ''true'', ''false'' and some indeterminate ...
s which have a third truth value representing that a statement's truth value is indeterminate. These logics have seen applications including to presupposition in linguistics. Fuzzy logics are multivalued logics that have an infinite number of "degrees of truth", represented by a real number between 0 and 1. Paraconsistent logics are logical systems that can deal with contradictions. They are formulated to avoid the principle of explosion: for them, it is not the case that anything follows from a contradiction. They are often motivated by dialetheism, the view that contradictions are real or that reality itself is contradictory. Graham Priest is an important contemporary proponent of this position and similar views have been ascribed to Georg Wilhelm Friedrich Hegel.

## Informal

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 epistemology, 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 probability, 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 Bayesian epistemology#Principle of conditionalization, 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, extended or #Deviant, deviant logical systems.

## Mathematical logic

Mathematical logic is the study of logic within mathematics. Major subareas include model theory, proof theory, set theory, and computability theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics. The latter was a major concern in early 20th century mathematical logic, which pursued the program of logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell. Mathematical theories were supposed to be logical tautology (logic), tautologies, and the programme was to show this by means of a reduction of mathematics to logic. The various attempts to carry this out met with failure, from the crippling of Frege's project in his ''Grundgesetze'' by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems. Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic. They include Cantor's theorem, the status of the Axiom of Choice, the question of the independence of the continuum hypothesis, and the modern debate on large cardinal axioms. Recursion theory captures the idea of computation in logical and arithmetic terms; its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing, and his presentation of the Church–Turing thesis. Today recursion theory is mostly concerned with the more refined problem of complexity classes and the classification of Turing degree, degrees of unsolvability.

## Computational logic

In computer science, logic is studied as part of the theory of computation. Key areas of logic that are relevant to computing include computability theory, modal logic, and category theory. Early computer machinery was based on ideas from logic such as the lambda calculus. Computer scientists also apply concepts from logic to problems in computing and vice versa. The works of Claude Shannon were influential in this regard. He showed how boolean logic can be used to understand and implement computer circuits. The interaction between the two disciplines can be seen, for example, in how modern artificial intelligence builds on logicians' work in
argumentation theory Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be supported or undermined by premises through logical reasoning. With historical origins in logic, dialectic, and rhetoric, argumentation theory, includ ...
, while automated theorem proving can assist logicians in finding and checking proofs. In logic programming languages such as Prolog, a program computes the consequences of logical axioms and rules to answer a query.

## Formal semantics of natural language

Formal semantics, a subfield of both
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
and philosophy, uses logic to analyze meaning in natural language. It is an empirical field which seeks to characterize the denotations of linguistic expressions and explain how those denotations are compositionality, composed from the meanings of their parts. The field was developed by Richard Montague and Barbara Partee in the 1970s, and remains an active area of research. Central questions include scope (formal semantics), scope, binding (linguistics), binding, and linguistic modality.

# Epistemology 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 and a posteriori, 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 Conventionalism, 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 philosophical realism, 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 logic, temporal modal logic, and inductive logic, as well as influential vocabulary such as terminology, 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. Avicenna, Ibn Sina (Avicenna) (980–1037 CE) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of Logic in Islamic philosophy, 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 Mill's Methods, 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 Averroes, 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 ''Sum of Logic, Summa Logicae'' was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in Ludvig Holberg, 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 "[l]ogic itself rests upon assumptions to which nothing in the world of reality corresponds". The Logic in China, 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 Ānvīkṣikī is a term in Sanskrit denoting roughly the "science of inquiry" and it should have been recognized in India as a distinct branch of learning as early as 650 BCE. However, over the centuries its meaning and import have undergone con ...
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 Quantification (logic), 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 Digital electronics, 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.

# References

## Bibliography

* * * * * * * * * * * * * * * * * * * *

* * Introductions and tutorials ** aimed at beginners. *
forall x: an introduction to formal logic
by P.D. Magnus, covers sentential and quantified logic. *
Logic Self-Taught: A Workbook
(originally prepared for on-line logic instruction). * Essays *
"Symbolic Logic"
an
"The Game of Logic"
Lewis Carroll, 1896. * Online Tools *
Interactive Syllogistic Machine
A web-based syllogistic machine for exploring fallacies, figures, terms, and modes of syllogisms. *
A Logic Calculator
A web-based application for evaluating simple statements in symbolic logic. * Reference material *

by Peter Suber, for translating from English into logical notation. *
Ontology and History of Logic. An Introduction
with an annotated bibliography. {{Authority control Logic, Abstraction Critical thinking Formal sciences History of logic History of philosophy Intellectual history Philosophical logic Philosophy of logic Reasoning Thought