In Buddhism, refuge or taking refuge
refers to the prayer or recitation performed at the beginning of the
day or of a practice session. In Sutrayana, refuge is taken in the Three Jewels (also known as the Triple Gem or Three Refuges). In Vajrayana, refuge may be taken in the Three Jewels and Three Roots.
These are the three supports in which a Sutrayana Buddhist takes refuge:
Since early Buddhism, devotees expressed their faith through the act of taking refuge, which is threefold.
In this, it centres on the authority of a Buddha as a supremely
awakened being, by assenting to a role for a Buddha as a teacher of both
humans and devās
(heavenly beings). This often includes other Buddhas from the past, and
Buddhas who have not yet arisen. Secondly, the taking of refuge honours
the truth and efficacy of the Buddha's spiritual doctrine, which includes the characteristics of phenomenon (Pali: saṅkhāra) such as their impermanence (Pali: anicca), and the Noble Eightfold Path to liberation. The taking of refuge ends with the acceptance of worthiness of the community of spiritually developed followers (the saṅgha), which is mostly defined as the monastic community, but may also include lay people and even devās provided they are nearly or completely enlightened. Early Buddhism did not include bodhisattvas in the Three Refuges, because they were considered to still be on the path to enlightenment.
Early texts describe the saṅgha as a "field of merit", because early Buddhists regard offerings to them as particularly karmically fruitful.
Lay devotees support and revere the saṅgha, of which they believe it
will render them merit and bring them closer to enlightenment.
At the same time, the Buddhist monk is given a significant role in
promoting and upholding faith among laypeople. Although many examples in
the canon are mentioned of well-behaved monks, there are also cases of
monks misbehaving. In such cases, the texts describe that the Buddha
responds with great sensitivity to the perceptions of the lay community.
When the Buddha sets out new rules in the monastic code
to deal with the wrongdoings of his monastics, he usually states that
such behavior should be curbed, because it would not "persuade
non-believers" and "believers will turn away". He expects monks, nuns
and novices to not only to lead the spiritual life for their own
benefit, but also to uphold the faith of the people. On the other hand,
they are not to take the task of inspiring faith to the extent of
hypocrisy or inappropriateness, for example, by taking on other
professions apart from being a monastic, or by courting favours by
giving items to the laypeople.
Thus, taking refuge is a form of aspiration to lead a life with
the Triple Gem at its core. Taking refuge is done by a short formula in
which one names the Buddha, the dharma and the saṅgha as refuges.
In early Buddhist scriptures, taking refuge is an expression of
determination to follow the Buddha's path, but not a relinquishing of
responsibility. Refuge is common to all major schools of Buddhism.
The most used recitation in the Pali language goes:
Buddham saranam gacchami.
I take refuge in the Buddha.
Dhammam saranam gacchami.
I take refuge in the Dharma.
Sangham saranam gacchami.
I take refuge in the Sangha.
Dutiyampi Buddham saranam gacchami.
For the second time, I take refuge in the Buddha. Dutiyampi Dhammam saranam gacchami.
For the second time, I take refuge in the Dharma. Dutiyampi Sangham saranam gacchami.
For the second time, I take refuge in the Sangha.
Tatiyampi Buddham saranam gacchami.
For the third time, I take refuge in the Buddha. Tatiyampi Dhammam saranam gacchami.
For the third time, I take refuge in the Dharma. Tatiyampi Sangham saranam gacchami.
For the third time, I take refuge in the Sangha.
Except this there are various recitations mentioned in Pali
literature for taking refuge in the Three Jewels. Brett Shults proposes
that Pali texts may employ the Brahmanical motif of a group of three refuges, as found in Rig Veda 9.97.47, Rig Veda 6.46.9 and Chandogya Upanishad 2.22.3-4.
Faith is an important teaching element in both Theravada and Mahayana traditions. In contrast to perceived Western notions of faith, faith in Buddhism arises from accumulated experience and reasoning.
In the Kalama Sutra, the Buddha explicitly argues against simply following authority or tradition, particularly those of religions contemporary to the Buddha's time.
There remains value for a degree of trusting confidence and belief in
Buddhism, primarily in the spiritual attainment and salvation or enlightenment. Faith in Buddhism centres on belief in the Three Jewels.
Lay followers often undertake five precepts in the same ceremony as they take the refuges. Monks administer the precepts to the laypeople, which creates an additional psychological effect. The five precepts are:
to refrain from killing;
to refrain from stealing;
to refrain from lying;
to refrain from improper sexual conduct;
to refrain from consuming intoxicants.
In Early Buddhist Texts,
the role of the five precepts gradually developed. First of all, the
precepts were combined with a declaration of faith in the triple gem
(the Buddha, his teaching and the monastic community). Next, the precepts developed to become the foundation of lay practice. The precepts were seen as a preliminary condition for the higher development of the mind.
At a third stage in the texts, the precepts were actually mentioned
together with the triple gem, as though they were part of it. Lastly,
the precepts, together with the triple gem, became a required condition
for the practice of Buddhism, as lay people had to undergo a formal
initiation to become a member of the Buddhist religion.
When Buddhism spread to different places and people, the role of the
precepts began to vary. In countries in which Buddhism was adopted as
the main religion without much competition from other religious
disciplines, such as Thailand, the relation between the initiation of a
lay person and the five precepts has been virtually non-existent, and
the taking of the precepts has become a sort of ritual cleansing
ceremony. In such countries, people are presumed Buddhist from birth
without much of an initiation. The precepts are often committed to by
new followers as part of their installment, yet this is not very
pronounced. However, in some countries like China, where Buddhism was
not the only religion, the precepts became an ordination ceremony to
initiate lay people into the Buddhist religion.
A layperson who upholds the precepts is described in the texts as a "jewel among laymen".
In Tibetan Buddhism there are three refuge formulations, the Outer, Inner, and Secret forms of the Three Jewels. The 'Outer' form is the 'Triple Gem', (Sanskrit:triratna), the 'Inner' is the Three Roots and the 'Secret' form is the 'Three Bodies' or trikaya of a Buddha. These alternative refuge formulations are employed by those undertaking deity yoga and other tantric practices within the Tibetan Buddhist Vajrayana tradition as a means of recognizing Buddha Nature.
Recitations in other languages
In Sanskrit
The most used recitation in the Sanskrit language goes:
Ahamittham (name) nama yavajjivam Buddham sharanam gacchami dvipadanamagram. Ahamittham (name) nama yavajjivam Dharmam sharanam gacchami viraganamagram. Ahamittham (name) nama yavajjivam Sangham sharanam gacchami gananamagram.
Dvitiyamapi ahamittham (name) nama yavajjivam Buddham sharanam gacchami dvipadanamagram. Dvitiyamapi ahamittham (name) nama yavajjivam Dharmam sharanam gacchami viraganamagram. Dvitiyamapi ahamittham (name) nama yavajjivam Sangham sharanam gacchami gananamagram.
Tritiyamapi ahamittham (name) nama yavajjivam Buddham sharanam gacchami dvipadanamagram. Tritiyamapi ahamittham (name) nama yavajjivam Dharmam sharanam gacchami viraganamagram. Tritiyamapi ahamittham (name) nama yavajjivam Sangham sharanam gacchami gananamagram.
In Chinese
The most used recitation in the Chinese language goes:
Philosophy of logic is the area of philosophy that studies the scope and nature of logic.
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 relate 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
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 is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency 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, correct reasoning, validinference, or logical truth. It is a formal science that investigates how conclusions follow from premises
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 systems
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 are often distinguished. Logic is usually understood as formal logic and is treated as such for most of this article. Formal logic is only interested in the form of arguments, expressed in a formal language, and focuses on deductive inferences. Informal logic, on the other hand, addresses a much wider range of arguments found also in natural language,
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,
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,
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
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 terms and existential quantifiers.
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 that studies the nature of logic.
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
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, premises, conclusions, inference, argument, and validity. It tries to clarify the relation between logic and other fields, such as ontology, mathematics, and psychology.
The philosophy of logic is closely related to philosophical logic 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
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 "logos",
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,
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, 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 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.
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
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. For this reason, the semantic approach is also referred to as the model-theoretic conception of logic. It was initially conceived by Alfred Tarski and characterizes logical truth not in relation to the logical constants used in sentences, but based on set-theoretic 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 to tuples 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, Bernard Bolzano, Franz Brentano, Gottlob Frege, 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 and of presumption.
Classical and non-classical
Within the domain of formal logic, an important distinction is between classical and non-classical logic. The term classical logic refers primarily to propositional logic and first-order logic.
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 logics.
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 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 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, or temporal modal logic. In its narrow sense, it is identical with alethic modal logic.[2]
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 is the right one.
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 terms of modal
notion, 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 logics 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 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: to give an account of which inferences are valid. They differ from classical logic by giving a different account. Intuitionistic logic, for example, rejects the law of excluded middle, which is a valid form of inference in classical logic.[1][2] 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. Free logic
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 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, for example, a third truth value is introduced: undefined.
Fundamental concepts
Truth
In logic, truth
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, and the principle of bivalence of truth.
Logical truth
Central to logic is the notion of logical truth. 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 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 constants 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" may be understood as the definite description "the pupil of Plato who taught Alexander". As for predicates, they are often seen as referring either to universals, to concepts, or to classes of objects.
Inference and argument
An inference is the process of reasoning from premises to a conclusion. The relation between the premises and the conclusion is called "entailment" or "logical consequence". An argument
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,
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.
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, 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 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 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, 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. Formal fallacies pertain to formal logic and involve only errors of form by employing an invalid rule of inference. Denying the antecedent
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 dilemmas,
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 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
consist 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, 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 or psychologism, 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 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
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, respectively. The most famous defender of this approach is Willard Van Orman Quine,
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 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, the view that only individuals exist.
Mathematics
Mathematics and logic are related in various ways. Both are considered formal sciences and in many cases, developments in these two fields happened in parallel. Propositional logic, for example, is an instance of Boolean algebra. It is often claimed that mathematics can, in principle, be grounded in only first-order logic together with set theory. Metamath 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 and has already proved a significant amount of mathematical theorems based on these axioms. Closely related to this project is logicism: the thesis defended by Gottfried Wilhelm Leibniz and Gottlob Frege that arithmetic 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 like addition. 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 gates in computer science: they both follow the laws of Boolean algebra.
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 is an example of an automated theorem prover for first-order logic.
Psychology
A very close connection between psychology and logic can be drawn if logic is seen as the science of the laws of thought.[2]
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
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.
