Philosophy of logic is the area of
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 ...
that studies the scope and nature of
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterization,
philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all.
Metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like
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 ...
and
completeness.
Various characterizations of the
nature of logic are found in the academic literature. Logic is often seen 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 ...
, 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 ...
,
valid 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 ...
, or
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 ...
. 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 ...
that investigates how conclusions follow from
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 manner, i.e. independent of the specific subject matter discussed. One form of inquiring into the nature of logic focuses on the commonalities between various 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 ...
s and on how they differ from non-logical formal systems. Important considerations in this respect are whether the formal system in question is compatible with fundamental logical intuitions and whether it is complete. Different conceptions of logic can be distinguished according to whether they define logic as the study of valid inference or logical truth. A further distinction among conceptions of logic is based on whether the criteria of valid inference and logical truth are specified in terms of syntax or semantics.
Different
types of logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
are often distinguished. Logic is usually understood as
formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
and is treated as such for most of this article. Formal logic is only interested in the form of
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, expressed in a
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of symb ...
, and focuses on deductive inferences.
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. ...
, on the other hand, addresses a much wider range of arguments found also in
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 ...
, which include non-deductive arguments. The correctness of arguments may depend on other factors than their form, like their content or their context. Various logical formal systems or logics have been developed in the 20th century and it is the task of the philosophy of logic to classify them, to show how they are related to each other, and to address the problem of how there can be a manifold of logics in contrast to one universally true logic. These logics can be divided into classical logic, usually identified with
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 ...
, extended logics, and deviant logics. Extended logics accept the basic formalism and the axioms of classical logic but extend them with new logical vocabulary. Deviant logics, on the other hand, reject certain core assumptions of classical logic and are therefore incompatible with it.
The philosophy of logic also investigates the nature and philosophical implications of the
fundamental concepts of logic. This includes the problem of
truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
, especially of logical truth, which may be defined as truth depending only on the meanings of the logical terms used. Another question concerns the nature of premises and conclusions, i.e. whether to understand them as thoughts, propositions, or sentences, and how they are composed of simpler constituents. Together, premises and a conclusion constitute an inference, which can be either deductive and ampliative depending on whether it is necessarily truth-preserving or introduces new and possibly false information. A central concern in logic is whether a deductive inference is valid or not. Validity is often defined in terms of necessity, i.e. an inference is valid if and only if it is impossible for the premises to be true and the conclusion to be false. Incorrect inferences and arguments, on the other hand, fail to support their conclusion. They can be categorized as formal or informal fallacies depending on whether they belong to formal or informal logic. Logic has mostly been concerned with definitory rules, i.e. with the question of which rules of inference determine whether an argument is valid or not. A separate topic of inquiry concerns the strategic rules of logic: the rules governing how to reach an intended conclusion given a certain set of premises, i.e. which inferences need to be drawn to arrive there.
The
metaphysics of logic
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 consci ...
is concerned with the metaphysical status of the laws and objects of logic. An important dispute in this field is between realists, who hold that logic is based on facts that have mind-independent existence, and anti-realists like conventionalists, who hold that the laws of logic are based on the conventions governing the use of language. Logic is closely related to
various disciplines. A central issue in regard to ontology concerns the ontological commitments associated with the use of logic, for example, with
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
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 ...
s. An important question in mathematics is whether all mathematical truths can be grounded in the axioms of logic together with set theory. Other related fields include computer science and psychology.
Definition and related disciplines
Philosophy of logic is the area of
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 ...
that studies the nature of
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
.
Like many other disciplines, logic involves various philosophical presuppositions which are addressed by the philosophy of logic.
The philosophy of logic can be understood in analogy to other discipline-specific branches of philosophy: just like the
philosophy of science
Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultim ...
investigates philosophical problems raised by science, so the philosophy of logic investigates philosophical problems raised by logic.
An important question studied by the philosophy of logic is how logic is to be defined, for example, in terms of valid inference or of logical truth.
This includes the issue of how to distinguish logical from non-logical formal systems.
It is especially relevant for clarifying the relation between the various proposed logical systems, both classical and non-classical, and for evaluating whether all of these systems actually qualify as ''logical systems''.
The philosophy of logic also investigates how to understand the most fundamental concepts of logic, like
truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
,
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, conclusions,
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 ...
,
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 ...
, and
validity
Validity or Valid may refer to:
Science/mathematics/statistics:
* Validity (logic), a property of a logical argument
* Scientific:
** Internal validity, the validity of causal inferences within scientific studies, usually based on experiments
** ...
.
It tries to clarify the relation between logic and other fields, such as
ontology
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 ...
,
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 ...
, and
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 ...
.
The philosophy of logic is closely related to
philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
but there is no general agreement about how these disciplines stand to each other.
Some theorists use these two terms for the same discipline while others see them as distinct disciplines.
According to the latter view, philosophical logic contrasts with the philosophy of logic in that it is usually seen as the application of logical methods to philosophical problems, often by developing deviant or extended logics.
In this sense, philosophical logic is one area of inquiry within the philosophy of logic, i.e. a part of the general study of philosophical problems raised by logic.
But this form of distinction is not universally accepted and some authors have proposed different characterizations.
The intimate connection between logic and philosophy is also reflected in the fact that many famous logicians were also philosophers.
The philosophy of logic is closely related to
metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
but not identical to it. Metalogic investigates the properties of formal logical systems, like whether a given logical system is consistent or complete.
It usually includes the study of the semantics and syntax of formal languages and formal systems.
Nature of logic
The term "logic" is based on the Greek word "", which is associated with various different senses, such as reason, discourse, or language. There are many disagreements about what logic is and how it should be defined.
Various characteristics are generally ascribed to logic, like that it studies the relation between premises and conclusions and that it does so in a topic-neutral manner. An important task of the philosophy of logic is to investigate the criteria according to which a formal system should count as logic.
Different conceptions of logic understand it as either based on valid inference or logical truth. The criteria of valid inference and logical truth can themselves be specified in different ways: based on syntactic or semantic considerations.
General characteristics
Traditionally, logic is often understood as the discipline investigating laws of thought.
One problem for this characterization is that logic is not an empirical discipline studying the regularities found in actual human thinking: this subject belongs to psychology.
This is better captured by another characterization sometimes found in the literature: that logic concerns the laws of correct thinking or, more specifically, correct reasoning.
This reflects the practical significance of logic as a tool to improve one's reasoning by drawing good inferences and becoming aware of possible mistakes.
Logic has also been defined as the science of valid argumentation.
This mirrors the definition in terms of reasoning since argumentation may be understood as an outward expression of inward reasoning.
Logic is often seen as a formal foundation of all knowledge.
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 ...
, it stands in contrast to the material or empirical sciences, like physics or biology, since it is mainly concerned with entailment relations between propositions but not with whether these propositions actually are true.
For example, deducing from the proposition "all moons are made of cheese" that "Earth's moon is made of cheese" is a valid inference. The error in this example is due to a false premise belonging to empirical astronomy.
A central feature of logic is that it is topic-neutral.
This means that it is concerned with the validity of arguments independent of the subject matter of these arguments.
In this sense, regular sciences are concerned with correct reasoning within a specific area of inquiry, for example, concerning material bodies for classical mechanics or living beings for biology, while logic is concerned with correct reasoning in general as applicable to all these disciplines.
One problem with this characterization is that it is not always clear how the terms "topic-neutral" and "subject matter" are to be understood in this context.
For example, it could be argued that first-order logic has individuals as its subject matter, due to its usage of singular terms and quantifiers, and is therefore not completely topic-neutral.
A closely related characterization holds that logic is concerned with the ''form'' of arguments rather than their ''contents''. On this view, the regular sciences could be seen as seeking true premises while logic studies how to draw conclusions from these or any premises. But this characterization also has its problems due to difficulties in distinguishing between form and content. For example, since temporal logic talks about time, this would lead to the implausible conclusion that time belongs to the form and not to the content of arguments.
These difficulties have led some theorists to doubt that logic has a clearly specifiable scope or an essential character.
Logical and non-logical formal systems
One approach to determining the nature of logic is to study the different formal systems, referred to as "logics", in order to determine what is essential to all of them, i.e. what makes them logics.
Formal systems of logic are systematizations of logical truths based on certain principles called axioms.
As for formal logic, a central question in the philosophy of logic is what makes a formal system into a system of logic rather than a collection of mere marks together with rules for how they are to be manipulated.
It has been argued that one central requirement is that the marks and how they are manipulated can be interpreted in such a way as to reflect the basic intuitions about valid arguments. This would mean, for example, that there are truth values and that the behavior of some marks corresponds to that of logical operators such as negation or conjunction.
Based on this characterization, some theorists hold that certain formal systems, such as three-valued logic or
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 ...
, stray too far from the common concept of logic to be considered logical systems.
Such a position may be defended based on the idea that by rejecting some basic logical assumptions, they include a too radical departure from fundamental logical intuitions to be considered logics. It has been suggested that rejecting the principle of the bivalence of truth, i.e. that propositions are either true or false, constitutes such a case.
Metalogicians sometimes hold that
logical completeness is a necessary requirement of logical systems.
A formal system is complete if it is possible to derive from its axioms every theorem belonging to it.
This would mean that only formal systems that are complete should be understood as constituting logical systems. One controversial argument for this approach is that incomplete theories cannot be fully formalized, which stands in contrast to the formal character of logic. On this view, first-order logic constitutes a logical system.
But this would also mean that higher-order "logics" are not logics strictly speaking, due to their incompleteness.
Conceptions based on valid inference or logical truth
Logic is often defined as the study of valid or correct inferences.
On this conception, it is the task of logic to provide a general account of the difference between correct and incorrect inferences. An inference is a set of premises together with a conclusion. An inference is valid if the conclusion follows from the premises, i.e. if the truth of the premises ensures the truth of the conclusion.
Another way to define logic is as the study of logical truth.
Logical truth is a special form of truth since it does not depend on how things are, i.e. on which possible world is actual. Instead, a logically true proposition is true in all possible worlds.
Their truth is based solely on the meanings of the terms they contain, independent of any empirical matters of fact.
There is an important link between these two conceptions: an inference from the premises to a conclusion 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 the premises to the conclusion is logically true.
For example, the inference from "roses are red and grass is green" to "roses are red" is valid since the material conditional "if roses are red and grass is green, then roses are red" is logically true.
Conceptions based on syntax or semantics
Whether logic is defined as the study of valid inference or of logical truth leaves open their exact criteria. There are two important ways of specifying these criteria: the syntactic and the semantic approach, sometimes also called the deductive-theoretic and the model-theoretic approach.
In this sense, a logic can be defined as a formal language together with either a deductive-theoretic or a model-theoretic account of logical consequence.
The syntactic approach tries to capture these features based only on syntactic or formal features of the premises and the conclusion.
This is usually achieved by expressing them through a formal symbolism to make these features explicit and independent of the ambiguities and irregularities of natural language.
In this formalism, the validity of arguments only depends on the structure of the argument, specifically on the logical constants used in the premises and the conclusion.
On this view, a proposition is a logical consequence of a group of premises if and only if the proposition is deducible from these premises.
This deduction happens by using
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 ...
.
This means that for a valid argument, it is not possible to produce true premises with a false conclusion by substituting their constituents with elements belonging to similar categories while keeping the logical constants in place.
In the case of logical truths, such a substitution cannot make them false. Different sets of rules of inference constitute different deductive systems, for example, the ones associated with classical logic or with intuitionistic logic. So whether the proposition is a logical consequence depends not just on the premises but also on the deductive system used.
A problem with the syntactic approach is that the use of
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of symb ...
is central to it. But the problem of logic, i.e. of valid inference and logical truth, is found not just in formal languages but also in natural languages.
However, even within the scope of formal languages, the problem of truth poses a variety of problems, which often call for a richer meta-language to be properly addressed. This threatens the syntactic approach even when restricted to formal languages.
Another difficulty is posed by the fact that it is often not clear how to distinguish formal from non-formal features, i.e. logical from non-logical symbols. This distinction lies at the very heart of the syntactic approach due to its role in the definition of valid inference or logical truth.
The semantic approach, on the other hand, focuses on the relation between language and reality. In logic, the study of this relationship is often termed
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 ...
.
For this reason, the semantic approach is also referred to as the model-theoretic conception of logic.
It was initially conceived by
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 ...
and characterizes logical truth not in relation to the logical constants used in sentences, but based on
set-theoretic
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 concern ...
structures that are used to interpret these sentences.
The idea behind this approach is that sentences are not true or false by themselves but only true or false in relation to an
interpretation.
Interpretations are usually understood in set-theoretic terms as
functions between symbols used in the sentence and a domain of objects. Such a function assigns individual constants to individual elements of the domain 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, ...
to
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s of elements of the domain.
An interpretation of a sentence (or of a theory comprising various sentences) is called a ''model'' of this sentence if the sentence is true according to this interpretation.
A sentence is ''logically true'' if it is true in every interpretation, i.e. if every interpretation is a model of this sentence. In this case, no matter how the interpretation-function and the domain of objects to which it points are defined, the sentence is always true.
If interpretations are understood in terms of possible worlds, logically true sentences can be seen as sentences that are true in every possible world.
Expressed in terms of valid arguments: an argument is valid if and only if its conclusion is true in all possible worlds in which its premises are true.
This conception avoids the problems of the syntactic approach associated with the difficulty of distinguishing between logical and non-logical symbols. But it faces other problems of its own.
On the one hand, it shares the problem with the syntactic approach of being in need of a meta-language to address the problem of truth.
It therefore presupposes a formal language that can be studied from a perspective outside itself. This poses problems for generalizing its insights to the logic of language in general as an all-encompassing medium.
On the other hand, it ignores the relationship between language and world, since it defines truth based on the interpretation that takes place only between symbols and set-theoretic objects.
Types of logics
The problem of having to choose between a manifold of rival logical systems is rather recent. For a long time in history, Aristotelian syllogistics was treated as the canon of logic and there were very few substantial improvements to it for over two thousand years until the works of
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ire ...
,
Bernard Bolzano
Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liber ...
,
Franz Brentano
Franz Clemens Honoratus Hermann Josef Brentano (; ; 16 January 1838 – 17 March 1917) was an influential German philosopher, psychologist, and former Catholic priest (withdrawn in 1873 due to the definition of papal infallibility in matters of F ...
,
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 ...
, and others.
These developments were often driven by a need to increase the expressive flexibility of logic and to adapt it to specific areas of usage.
A central problem in the philosophy of logic, raised by the contemporary proliferation of logical systems, is to explain how these systems are related to each other.
This brings with it the question of why all these formal systems deserve the title "logic". Another question is whether only one of these systems is the right one or how a multiplicity of logical systems is possible instead of just one universal logic.
Monism is the thesis that only one logic is correct while pluralism allows different alternative logical systems to be correct for different areas of discourse.
It has also been suggested that there may be one universal concept of logic that underlies and unifies all the different logical systems.
Formal and informal
Logic and the philosophy of logic have traditionally focused primarily on formal arguments, i.e. arguments expressed in a formal language. But they also include the study of informal arguments found in natural language.
Formal logic is usually seen as the paradigmatic form of logic but various modern developments have emphasized the importance of informal logic for many practical purposes where formal logic alone is unable to solve all issues by itself.
Both formal and informal logic aim at evaluating the correctness of arguments.
But formal logic restricts itself concerning the factors that are used in order to provide exact criteria for this evaluation.
Informal logic tries to take various additional factors into account and is therefore relevant for many arguments outside the scope of formal logic, but does so at the cost of precision and general rules.
Arguments that fail this evaluation are called fallacies. Formal fallacies are fallacies within the scope of formal logic whereas informal fallacies belong to informal logic.
Formal logic is concerned with the validity of inferences or arguments based only on their form, i.e. independent of their specific content and the context in which they are used.
This usually happens through abstraction by seeing particular arguments as instances of a certain form of argument. Forms of arguments are defined by how their logical constants and variables are related to each other. In this way, different arguments with very different contents may have the same logical form.
Whether an argument is valid only depends on its form. An important feature of formal logic is that for a valid argument, the truth of its premises ensures the truth of its conclusion, i.e. it is impossible for the premises to be true and the conclusion to be false.
A serious problem associated with the usage of formal logic for expressing theories from various fields is that these theories have to be translated into a formal language, usually the language of first-order logic.
This is necessary since formal logic is only defined for specific formal language: it is therefore not directly applicable to many arguments expressed differently. Such translations can be challenging since formal languages are often quite restrictive. For example, they frequently lack many of the informal devices found in natural language.
One recurrent problem concerns the word "is" in the English language, which has a variety of meanings depending on the context, such as identity, existence, predication, class-inclusion, or location.
Informal logic, on the other hand, has a more concrete orientation in that it tries to evaluate whether a specific instance of an argument is good or bad.
This brings with it the need to study not just the general form of the argument in question, but also the contents used as premises of this argument and the context in which this argument is used.
This means that the same argument may be both good, when used in one context, and bad, when used in another context. For example, a
strawman argument tries to overcome the opponent's position by attributing a weak position to them and then proving this position to be false.
In a context where the opponent does not hold this position, the argument is bad, while it may be a good argument against an opponent who actually defends the strawman position.
Arguments studied by informal logic are usually expressed in natural language.
Informal logic does not face the need to translate natural language arguments into a formal language in order to be able to evaluate them. This way, it avoids various problems associated with this translation. But this does not solve many of the problems that the usage of natural language brings with it, like ambiguities, vague expressions, or implicitly assuming premises instead of explicitly stating them.
Many of the fallacies discussed in informal logic arise directly from these features. This concerns, for example, the
fallacies of ambiguity
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 ...
and of
presumption
In the law of evidence, a presumption of a particular fact can be made without the aid of proof in some situations. The invocation of a presumption shifts the Legal burden of proof, burden of proof from one party to the opposing party in a court t ...
.
Classical and non-classical
Within the domain of formal logic, an important distinction is between classical and non-classical logic. The term
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 ...
refers primarily to
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 ...
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 ...
.
It is the dominant logical system accepted and used by most theorists. But the philosophy of logic is also concerned with non-classical or alternative logics.
They are sometimes divided into 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. Extended logics are extensions of classical logic, i.e. they accept the basic formalism and axioms of classical logic but extend them with new logical vocabulary, like introducing symbols for "possibility" and "necessity" in
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 ...
or symbols for "sometimes" and "always" in temporal logic.
Deviant logics, on the other hand, reject certain core assumptions of classical logic.
They use axioms different from classical logic, which are often more limiting concerning which inferences are valid. They are "deviant" in the sense that they are incompatible with classical logic and may be seen as its rivals.
Classical
The term
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 ...
refers primarily to propositional logic and first-order logic.
It is usually treated by philosophers as the paradigmatic form of logic and is used in various fields. It is concerned with a small number of central logical concepts and specifies the role these concepts play in making valid inferences.
These core notions include quantifiers, expressing ideas like "all" and "some", and propositional connectives, like "and", "or", and "if-then".
Among the non-logical concepts, an important distinction is between singular terms and predicates. Singular terms stand for objects and predicates stand for properties of or relations between these objects. In this respect, first-order logic differs from traditional Aristotelian logic, which lacked predicates corresponding to relations.
First-order logic allows quantification only over individuals, in contrast to higher-order logic, which allows quantification also over predicates.
Extended
Extended logics accept the axioms and the core vocabulary of classical logic. This is reflected in the fact that the theorems of classical logic are valid in them. But they go beyond classical logic by including additional new symbols and theorems. The goal of these changes is usually either to apply logical treatment to new areas or to introduce a higher level of abstraction, for example, in the form of quantification applied not just to singular terms but also to predicates or propositions, or through truth predicates.
In this sense, deviant logics are usually seen as rivals to classical logic while extended logics are supplements to classical logic. Important examples of extended logics include modal logic and higher-order logic.
The term "modal logic", when understood in its widest sense, refers to a variety of extended logics, such as alethic,
deontic
In moral philosophy, deontological ethics or deontology (from Greek language, Greek: + ) is the normative ethics, normative ethical theory that the morality of an action should be based on whether that action itself is right or wrong under a s ...
, or
temporal modal logic. In its narrow sense, it is identical with alethic modal logic.
While classical logic is only concerned with what is true or false, alethic modal logic includes new symbols to express what is possibly or necessarily true or false.
These symbols take the form of sentential operators. Usually, the symbols and are used to express that the sentence following them is possibly or necessarily true. Modal logics also include various new rules of inferences specifying how these new symbols figure in valid arguments.
One example is the formula
, i.e. that if something is necessarily true then it is also possibly true. The other forms of modal logic besides alethic modal logic apply the same principles to different fields.
In deontic modal logic, the symbols and are used to express which actions are permissible or obligatory; in temporal logic, they express what is the case at some time or at every time; in epistemic logic, they express what is compatible with a person's beliefs or what this person knows.
Various rules of inference have been suggested as the basic axioms of the different modal logics but there is no general agreement on which are the right ones.
An influential interpretation of modal operators, due to Saul Kripke, understands them as quantifiers over possible worlds. A possible world is a complete and consistent way how things could have been. On this view, to say that something is necessarily true is to say that it is true in all accessible possible worlds.
One problem for this type of characterization is that they seem to be circular since possible worlds are themselves defined in modal terms, i.e. as ways how things ''could'' have been.
Even when restricted to alethic modal logic, there are again different types of possibility and necessity that can be meant by these terms.
For example, according to physical modality, it is necessary that an object falls if dropped since this is what the laws of nature dictate. But according to logical modality, this is not necessary since the laws of nature might have been different without leading to a logical contradiction.
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 extend classical first-order predicate logic by including new forms of quantification.
In first-order logic, quantification is restricted to individuals, like in the formula
(''there are some'' apples that are sweet). Higher-order logics allow quantification not just over individuals but also over predicates, as in
(''there are some'' qualities that Mary and John share).
The increased expressive power of higher-order logics is especially relevant for mathematics. For example, an infinite number of axioms is necessary for Peano arithmetic and Zermelo-Fraenkel set theory in first-order logic, while second-order logic only needs a handful of axioms to do the same job.
But this increased expressive power comes at certain costs. On the one hand, higher-order theories are incomplete:
it is not possible to prove every true sentence based on the axioms of this theory.
For theories in first-order logic, on the other hand, this is possible. Another drawback is that higher-order logics seem to be committed to a form of
Platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at le ...
since they quantify not just over individuals but also over properties and relations.
Deviant
Deviant logics are forms of logic in that they have the same goal as
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 ...
: to give an account of which inferences are valid. They differ from classical logic by giving a different account.
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 ...
, for example, rejects 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 ...
, which is a valid form of inference in classical logic.
This rejection is based on the idea that mathematical truth depends on verification through a
proof. The law fails for cases where no such proof is possible, which exist in every sufficiently strong formal system, according to
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of 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 i ...
.
Free logic
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter propert ...
differs from classical logic since it has fewer existential presuppositions: it allows non-denoting expressions, i.e. individual terms that do not refer to objects within the domain.
A central motivation for this type of modification is that free logic can be used to analyze discourse with empty singular terms, like in the expression "Santa Clause does not exist".
Many-valued logic
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" ...
is a logic that allows for additional truth values besides ''true'' and ''false'' in classical logic.
In this sense, it rejects the principle of the bivalence of truth.
In a simple form of
three-valued 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 ...
, for example, a third truth value is introduced: ''undefined''.
Fundamental concepts
Truth
In logic,
truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
is usually seen as a property of propositions or sentences. It plays a central role in logic since validity is often defined in terms of truth: an inference is valid if and only if it is impossible for its premises to be true and its conclusion to be false.
Theories of truth try to characterize the nature of truth.
According to
correspondence theories, a proposition is true if it corresponds to reality, i.e. if it represents things how they actually are.
Coherence theories, on the other hand, identify truth with coherence. On this view, a proposition is true if it is a coherent part of a specified set of propositions, i.e. if these propositions are consistent with each other and provide mutual inferential support for each other.
According to
pragmatic theories of truth, whether a proposition is true depends on its relation to practice. Some versions claim that a proposition is true if believing it is useful, if it is the ideal result of an endless inquiry, or if it meets the standards of warranted assertibility.
Deflationary theories of truth see truth as a rather empty notion that lacks an interesting nature of its own. On this view, to assert that a proposition is true is the same as asserting the proposition by itself.
Other important topics in the philosophy of logic concerning truth are the value of truth, the
liar paradox
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth ...
, and the
principle of bivalence of truth.
Logical truth
Central to logic is the notion 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 ...
. Logical truth is often understood in terms of the
analytic-synthetic distinction: a proposition is analytically true if its truth only depends on the meanings of the terms composing it. Synthetic propositions, on the other hand, are characterized by the fact that their truth depends on non-logical or empirical factors.
This is sometimes expressed by stating that analytical truths are tautologies, whose denial would imply a contradiction, while it is possible for synthetic propositions to be true or false.
In this sense, the proposition "all bachelors are unmarried" is analytically true since being unmarried is part of how the term "bachelor" is defined. The proposition "some bachelors are happy", on the other hand, is synthetically true since it depends on empirical factors not included in the meaning of its terms. But whether this distinction is tenable has been put into question. For example,
Willard Van Orman Quine
Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". ...
has argued that there are no purely analytic truths, i.e. that all propositions are to some extent empirical.
But others have explicitly defended the analytic-synthetic distinction against Quine's criticism.
But whether logical truths can be identified with analytical truths is not always accepted.
A different approach characterizes logical truths regarding a small subset of the meanings of all terms: the so-called
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 ...
s or
syncategoremata.
They include propositional connectives, like "and" or "if-then", quantifiers, like "for some" or "for all", and identity.
Propositional logic is only concerned with truth in virtue of propositional connectives, while predicate logic also investigates truths based on the usage of quantifiers and identity.
Extended logics introduce even more logical constants, like possibility and necessity in modal logic.
A sentence is true in virtue of the logical constants alone if all non-logical terms can be freely replaced by other terms of the appropriate type without affecting any change in the truth value of the sentence.
For example, the sentence "if it rains, then it rains" is true due to its logical form alone because all such replacements, like substituting the expression "Socrates is wise" for the expression "it rains", also result in true sentences. One problem with this characterization of logic is that it is not always clear how to draw the distinction between logical constants and other symbols. While there is little controversy in the paradigmatic cases, there are various borderline cases in which there seem to be no good criteria for deciding the issue.
Premises and conclusions
There are various discussions about the nature of premises and conclusions. It is widely agreed that they have to be
bearers of truth, i.e. that they are either true or false.
This is necessary so they can fulfill their logical role.
They are traditionally understood as thoughts or propositions, i.e. as mental or abstract objects. This approach has been rejected by various philosophers since it has proven difficult to specify clear identity criteria for these types of entities.
An alternative approach holds that only sentences can act as premises and conclusions.
Propositions are closely related to sentences since they are the meaning of sentences: sentences express propositions.
But this approach faces various problems of its own. One is due to the fact that the meaning of sentences usually is context-dependent. Because of this, it could be the case that the same inference is valid in one context and invalid in another.
Another problem consists in the fact that some sentences are ambiguous, i.e. that it sometimes depends on one's interpretation whether an inference is valid or not.
An important aspect both of propositions and of sentences is that they can be either simple or complex.
Complex propositions are made up of simple propositions that are linked to each other through propositional connectives. Simple propositions do not have other propositions as their parts, but they are usually understood as being constituted by other entities as well: by subpropositional parts like singular terms and predicates.
For example, the simple proposition "Mars is red" is made of the singular term "Mars", to which the predicate "red" is applied.
In contrast, the proposition "Mars is red and Venus is white" is made up of two propositions connected by the propositional connective "and".
In the simplest case, these connectives are truth-functional connectives: the truth value of the complex proposition is a function of the truth values of its constituents.
So the proposition "Mars is red and Venus is white" is true because the two propositions constituting it are true. The truth value of simple propositions, on the other hand, depends on their subpropositional parts.
This is usually understood in terms of reference: their truth is determined by how their subpropositional parts are related to the world, i.e. to the extra-linguistic objects they refer to. This relation is studied by theories of reference, which try to specify how singular terms refer to objects and how predicates apply to these objects.
In the case of singular terms, popular suggestions include that the singular term refers to its object either through a definite description or based on causal relations with it.
In the former sense, the name "
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
" may be understood as the definite description "the pupil of
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
who taught Alexander".
As for predicates, they are often seen as referring either to
universals
In metaphysics, a universal is what particular things have in common, namely characteristics or qualities. In other words, universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things. For exa ...
, to
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, or to classes of objects.
Inference and argument
An
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 ...
is the process of reasoning from premises to a conclusion.
The relation between the
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 and the conclusion is called "entailment" or "
logical consequence
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 on ...
". An
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 ...
consists of the premises, the conclusion, and the relation between them. But the terms "inference", "argument", "entailment", and "logical consequence" are often used interchangeably. A complex argument is an argument involving several steps, in which the conclusions of earlier steps figure as the premises of the following steps.
Inferences and arguments can be correct or incorrect. This depends on whether the premises actually support the conclusion or not, i.e. on whether the conclusion follows from the premises.
For example, it follows from "Kelly is not both at home and at work" and "Kelly is at home" that "Kelly is not at work". But it does not follow that "Kelly is a football fan".
An important distinction among inferences is between deductive and ampliative inferences, also referred to as monotonic and non-monotonic inferences.
According to
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 ...
, deductive inference has three central features: (1) it is formal, i.e. it depends only on the form of the premises and the conclusion; (2) it is a priori, i.e. no sense experience is needed to determine whether it obtains; (3) it is modal, i.e. that it holds by necessity for the given propositions, independent of any other circumstances.
Deductive inferences are necessarily truth-preserving: the conclusion cannot be false if all the premises are true.
For this reason, they are unable to introduce new information not already found in the premises and are uninformative in this sense. One problem with characterizing deductive inferences as uninformative is that this seems to suggest that they are useless, i.e. it fails to explain why someone would use or study them.
This difficulty can be addressed by distinguishing between depth information and surface information. On this view, deductive logic is uninformative on the level of depth information but may still lead to surprising results on the level of surface information by presenting certain aspects in a new way.
Ampliative inferences, on the other hand, are informative by aiming to provide new information. This happens at the cost of losing the necessarily truth-preserving nature.
The most prominent form of ampliative inference is
induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
. An inductive inference involves particular propositions as premises, which are used to infer either one more particular proposition or a generalization as the conclusion.
Deductive inferences are the paradigmatic form of inference and are the main focus of logic. But many inferences drawn in the empirical sciences and in everyday discourse are ampliative inferences.
Validity and fallacies
A central problem in logic is how to distinguish correct or valid arguments from incorrect or invalid ones.
The philosophy of logic investigates issues like what it means that an argument is valid.
This includes the question of how this type of support is to be understood or of what the criteria are under which a premise supports a conclusion.
Some logicians define valid inference or entailment in terms of logical necessity: the premises entail the conclusion if it is impossible for the premises to be true and the conclusion to be false.
This can also be expressed by saying that the conjunction of the premises and the negation of the conclusion is logically impossible.
This conception brings with it 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 ...
, i.e. that anything follows from a contradiction.
But valid inferences can also be characterized in terms of rules of inference.
Rules of inference govern the transition from the premises to the conclusion. On this view, an inference is valid if it is in accordance with an appropriate rule of inference.
Closely related to the notion of valid inference is that of confirmation.
Valid inferences belong to formal logic and is associated with deductively valid arguments. But many arguments found in the sciences and in everyday discourse support their conclusion without ensuring its truth. They fall in the purview of informal logic and can also be divided into good and bad arguments. In this sense, for example, observations may act as
empirical evidence
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 ...
supporting a scientific hypothesis. This is often understood in terms of probability, i.e. that the evidence increases the likelihood that the hypothesis is true.
Of special interest are the so-called fallacies, i.e. incorrect arguments that appear to be correct.
They are incorrect because the premises do not support the conclusion in the assumed way. Due to their misleading appearance, they can seduce people into accepting and using them. Often three factors are identified as the sources of the error: ''form'', ''content'', and ''context''.
The form of an argument refers to its structure, i.e. which
rule 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 ...
it employs.
Errors on the level of form involve the use of invalid rules of inference.
An argument that is incorrect on the level of content uses false propositions as its premises.
The context of an argument refers to the situation in which it is used and the role it is supposed to play. An argument can be fallacious if it fails to play the role intended for it, as in 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 ...
, when the arguer attacks an overly weak position not held by the opponent.
An important distinction among fallacies can be drawn based on these sources of error: that 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''. Fallaci ...
. Formal fallacies pertain to formal logic and involve only errors of form by employing an invalid rule of inference.
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, for example, "If Othello is a bachelor, then he is male. Othello is not a bachelor. Therefore, Othello is not male".
Informal fallacies belong to informal logic and their main source of error is found on the level of content and context.
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 ...
s, for example, are based on a false disjunctive premise that oversimplifies reality by excluding viable alternatives, as in "Stacey spoke out against capitalism; therefore, she must be a communist".
Since logic evaluates arguments as good or bad, logic faces the problem of the nature and justification of the norms guiding these evaluations.
This is similar to issues found in
metaethics
In metaphilosophy and ethics, meta-ethics is the study of the nature, scope, and meaning of moral judgment. It is one of the three branches of ethics generally studied by philosophers, the others being normative ethics (questions of how one ought ...
about how to justify moral norms.
One approach to this issue is to characterize the norms of logic as generalizations of the inferential practices found in natural language or the sciences. This way, justification is inherited from the evaluations of good and bad inferences used in the corresponding field.
Definitory and strategic rules
An important distinction among the rules of logic is that between definitory and strategic rules.
Rules of inferences are definitory rules: they govern which inferences are valid. And while it has been the main objective of logic to distinguish valid from invalid inferences, there is also a secondary objective often associated with logic: to determine which inferential steps are needed to prove or disprove a given proposition based on a set of premises.
This is the domain of strategic rules. The rules of inference specify which steps are allowed but they remain silent on which steps need to be taken to reach a certain conclusion. The difference between definitory and strategic rules is found not only in logic but in various games as well.
In chess, for example, the definitory rules specify that bishops may only move diagonally while strategic rules describe how the allowed moves may be used to win a game, e.g. by controlling the center or by protecting one's king. Following definitory rules determines whether one plays chess or something else while following strategic rules determines whether one is a good or a bad chess player.
Both definitory and strategic rules are to be distinguished from empirical descriptive rules, which generalize how people actually draw inferences, whether correct or incorrect. In this sense, definitory rules are permissive and strategic rules are prescriptive while empirical generalizations are descriptive.
Violating the definitory rules of logic results in committing fallacies.
It has been argued that the almost exclusive focus of logicians on the definitory rules of logic is not justified. On this view, more emphasis should be given to strategic rules instead, since many applications of logic, like the problem of rational belief change, depend more on strategic rules than on definitory rules.
Metaphysics of logic
The philosophy of logic is in many ways closely related to the philosophy of mathematics, especially in relation to their metaphysical aspects.
The metaphysics of logic is concerned with the metaphysical status of its objects and the laws governing them. The theories within the metaphysics of logic can roughly be divided into realist and non-realist positions.
Logical realists hold that the laws of logic are objective, i.e. independent of humans and their ways of thinking.
On this view, the structures found in logic are structures of the world itself.
According to a definition proposed by Sandra LaPointe, logical realism consists of two theses: that there are logical facts and that they are independent of our cognitive and linguistic make-up and practices.
Logical realism is often interpreted from the perspective of
Platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at le ...
, i.e. that there is an intelligible realm of abstract objects that includes the objects of logic.
On this view, logic is not invented but discovered.
An important consequence of this position is that there is a clear gap between the facts of logic themselves and our beliefs about these facts.
One difficulty of this position consists in clarifying which sense of independence is meant when saying that logic is independent of humans. If it is understood in the strictest sense possible, no knowledge of it would be possible since a fully independent reality could play no part in human consciousness.
Another problem is to explain the relation between the one world and the many different logical systems proposed. This would suggest that there is only one true logic and all other logical systems are either false or incomplete.
Logical realism is rejected by anti-realists, who hold that logic does not describe an objective feature of reality. Anti-realism about logic often takes the form of
conceptualism
In metaphysics, conceptualism is a theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. Intermediate between nominalism and realism, the conceptualist view approaches the metaphysical co ...
or
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 ...
, in which the objects of logic consist in mental conceptions or the logical laws are identified with psychological laws.
This can include the thesis that the laws of logic are not
knowable a priori, as is often held, but that they are discovered through the methods of experimental inquiry.
An argument for psychologism is based on the idea that logic is a sub-discipline of psychology: it studies not all laws of thought, but only the subset of laws corresponding to valid reasoning.
Another argument focuses on the thesis that we learn about logical truths through the feeling of self-evidence, which is in turn studied by psychology.
Various objections to psychologism have been raised, especially in
German philosophy
German philosophy, here taken to mean either (1) philosophy in the German language or (2) philosophy by Germans, has been extremely diverse, and central to both the analytic and continental traditions in philosophy for centuries, from Gottfried W ...
around the turn of the 20th century in the so-called "Psychologismus-Streit".
One objection focuses on the thesis that the laws of logic are known a priori, which is not true for the
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 ...
laws studied by psychology. Another points out that psychological laws are usually vague, whereas logic is an exact science with clear laws.
Conventionalism is another form of anti-realism, in which the logical truths depend on the meanings of the terms used, which in turn depend on linguistic conventions adopted by a group of agents.
One problem for this position consists in providing a clear definition of the term "convention". Conventions are widely observed regularities. But not every widely observed regularity is a convention: conventions include a certain normative factor that distinguishes right from wrong behavior, whereas irregular behavior is not automatically wrong.
Another problem concerns the fact that conventions are contingent, while logical truths are necessary. This casts doubt on the possibility of defining logical truth in terms of convention unless a plausible explanation could be given how contingent conventions can ground necessary truths.
Relation to other disciplines
Ontology
A central issue in ontology is the problem of existence, i.e. whether an entity or a certain kind of entity exists.
According to some theorists, the main goal of ontology is just to determine what exists and what does not exist.
The issue of existence is closely related to singular terms, like names, and existential quantifiers (
): it is often held that these devices carry existential presuppositions or ontological commitments with them.
On this view, sentences like and involve ontological commitments to the existence of apples and of
Pegasus
Pegasus ( grc-gre, Πήγασος, Pḗgasos; la, Pegasus, Pegasos) is one of the best known creatures in Greek mythology. He is a winged divine stallion usually depicted as pure white in color. He was sired by Poseidon, in his role as hor ...
, respectively. The most famous defender of this approach is
Willard Van Orman Quine
Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". ...
, who argues that the ontological commitments of any theory can be determined by translating it into first-order logic and reading them off from the existential quantifiers used in this translation.
One problem with this approach is that it can lead to various controversial ontological commitments.
Mathematics, for example, quantifies over numbers in sentences such as "there are prime numbers between 1000 and 1010".
This would mean that the ontological commitment to the existence of numbers, i.e.
realism
Realism, Realistic, or Realists may refer to:
In the arts
*Realism (arts), the general attempt to depict subjects truthfully in different forms of the arts
Arts movements related to realism include:
*Classical Realism
*Literary realism, a move ...
about numbers, is already built into mathematics.
Another problem is due to the fact that natural language contains many names for imaginary entities, such as Pegasus or Santa Clause.
But if names come with existential commitments, then sentences like "Santa Clause does not exist" would be contradictory. Within ontology, these problems are sometimes approached through Platonism or psychologism by holding that the problematic entities do exist, but only in the form of abstract or mental objects while lacking concrete or material existence.
Within logic, these problems can be avoided by using certain forms of non-classical logic. Free logic, for example, allows empty singular terms, which do not denote any object in the domain and therefore carry no ontological commitments.
This is often combined with an existence-predicate, which can be used to specify whether a singular term denotes an object in the domain.
But talk of existence as a predicate is controversial. Opponents of this approach often point out that existence is required for an object to have any predicates at all and can therefore not be one of them.
The issue of existence brings with it its own problems in the case of higher-order logics.
Second-order logic, for example, includes existential quantification not just for singular terms but also for predicates. This is often understood as entailing ontological commitments not just to regular objects but also to the properties and relations instantiated by these objects.
This position is known as realism and is often rejected in contemporary philosophy due to
naturalist considerations. It contrasts with
nominalism
In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universalsthings t ...
, the view that only individuals exist.
Mathematics
Mathematics and logic are related in various ways. Both are considered
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 ...
s and in many cases, developments in these two fields happened in parallel.
Propositional logic, for example, is an instance of
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
. It is often claimed that mathematics can, in principle, be grounded in only first-order logic together with set theory.
Metamath
Metamath is a formal language and an associated computer program (a proof checker) for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in ...
is one example of such a project. It is based on 20 axioms of propositional logic, first-order predicate logic, and
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
and has already proved a significant amount of mathematical theorems based on these axioms. Closely related to this project is
logicism
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...
: the thesis defended by
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
and
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 ...
that
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
is reducible to logic alone.
This would mean that any statement in arithmetic, like "2 + 2 = 4", can be expressed in purely logical terms, i.e. without using numbers or arithmetic
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
like
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
. In this case, all the theorems of arithmetic would be derivable from the axioms of logic.
Whether this thesis is correct depends on how the term "logic" is understood. If "logic" only refers to the axioms of first-order predicate logic, it is false. But if one includes set-theory in it or higher-order logic, then arithmetic is reducible to logic.
Computer science
An important relation between logic and computer science arises from the parallels between propositional connectives of propositional logic and
logic gate
A logic gate is an idealized or physical device implementing a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate, ...
s in computer science: they both follow the laws of
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
.
Propositions are either false or true while the inputs and outputs of logic gates are termed 0 and 1. Both use truth tables to illustrate the functioning of propositional connectives and logic gates. Another important relation to logic consists in the development of logic software that can assist logicians in formulating proofs or even automate the process.
Prover9 Prover9 is an automated theorem prover for first-order and equational logic developed by William McCune.
Description
Prover9 is the successor of the Otter theorem prover also developed by William McCune. Prover9 is noted for producing relatively ...
is an example of an
automated theorem prover
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a maj ...
for
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 ...
.
Psychology
A very close connection between psychology and logic can be drawn if logic is seen as the science 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 ...
.
One important difference between psychology and logic in the light of this characterization is that psychology is an empirical science that aims to study how humans actually think. Logic, on the other hand, has the objective of discovering the laws of correct reasoning, independently of whether actual human thinking often falls short of this ideal.
The psychologist
Jean Piaget
Jean William Fritz Piaget (, , ; 9 August 1896 – 16 September 1980) was a Swiss psychologist known for his work on child development. Piaget's theory of cognitive development and epistemological view are together called " genetic epistemolog ...
applied logic to psychology by using it to identify different stages of human psychological development. On his view, the ability to reason logically only arises at a certain stage in the child's development and can be used as a criterion to distinguish it from earlier stages.
See also
Important theorists
Philosophical theories of logic
Others
References
Further reading
*
Haack, Susan. 1978. ''Philosophy of Logics''.
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
. ()
*
Quine, W. V. O. 2004. ''Philosophy of Logic''. 2nd ed.
Harvard University Press
Harvard University Press (HUP) is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. It is a member of the Association of American University Presses. After the retirem ...
. ()
*
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 ...
. 1983. The concept of truth in formalized languages, pp. 152–278, Logic,semantics, metamathematics, papers from 1923 to 1938, ed.
John Corcoran (logician)
John Corcoran ( ; 20 March 1937 - 8 January 2021) was an American logician, philosopher, mathematician, and historian of logic. He is best known for his philosophical work on concepts such as the nature of inference, relations between condition ...
, Hackett,Indianapolis 1983.
* Fisher Jennifer, On the Philosophy of Logic, Thomson Wadworth, 2008,
* Goble, Lou, ed., 2001
''(The Blackwell Guide to) Philosophical Logic.''Oxford:
Blackwell Blackwell may refer to:
Places
;Canada
* Blackwell, Ontario
;United Kingdom
* Blackwell, County Durham, England
* Blackwell, Carlisle, Cumbria, England
* Blackwell (historic house), South Lakeland, Cumbria, England
* Blackwell, Bolsover, Alfre ...
. .
*
Grayling, A. C., 1997
''An Introduction to Philosophical Logic.''3rd ed. Oxford: Blackwell. .
* Jacquette, Dale, ed., 2002
'' A Companion to Philosophical Logic.''Oxford Blackwell. .
*
* McGinn, Colin, 2000. ''Logical Properties: Identity, Existence, Predication, Necessity, Truth''. Oxford:
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
. .
*
* Sainsbury, Mark, 2001
''Logical Forms: An Introduction to Philosophical Logic.''2nd ed. Oxford: Blackwell. .
*
*
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 ...
,1983. The concept of truth in formalized languages, pp. 152–278, Logic,semantics, metamathematics, papers from 1923 to 1938, ed.
John Corcoran (logician)
John Corcoran ( ; 20 March 1937 - 8 January 2021) was an American logician, philosopher, mathematician, and historian of logic. He is best known for his philosophical work on concepts such as the nature of inference, relations between condition ...
, Hackett,Indianapolis 1983.
* Wolfram, Sybil, 1989. ''Philosophical Logic: An Introduction.'' London:
Routledge
Routledge () is a British multinational publisher. It was founded in 1836 by George Routledge, and specialises in providing academic books, journals and online resources in the fields of the humanities, behavioural science, education, law, and ...
. 290 pages. ,
Journal of Philosophical Logic Springer SBM
{{logic
Concepts in logic
Logic
Philosophical logic
Philosophy of mathematics