Logical truth
Encyclopedia
Logical truth is one of the most fundamental concept
s in logic
, and there are different theories on its nature. A logical truth is a statement
which is true and remains true under all reinterpretations
of its components other than its logical constants. It is a type of analytic statement.
Logical truths (including tautologies
) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. However, it is not universally agreed that there are any statements which are necessarily true.
A logical truth was considered by Ludwig Wittgenstein
to be a statement which is true in all possible world
s. This is contrasted with fact
s (which may also be referred to as contingent claims or synthetic claims) which are true in this world, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded. The proposition
“If p and q, then p” and the proposition “All married people are married” are logical truths because they are true due to their inherent meanings and not because of any facts of the world.
Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations.
The existence of logical truths is sometimes put forward as an objection to empiricism
because it is impossible to account for our knowledge
of logical truths on empiricist grounds.
. Other than logical truths, there is also a second class of analytic statements, typified by "No bachelor is married." The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate
. "No bachelor is married." can be turned into "No unmarried man is married." by substituting 'unmarried man' for its synonym 'bachelor.'
In his essay. Two Dogmas of Empiricism
, the philosopher W.V.O Quine
called into question the distinction between analytic and synthetic statements. It was this second class of analytic statements that caused him to note that the concept of analyticity itself stands in need of clarification, because it seems to depend on the concept of synonym
y, which stands in need of clarification. In his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, given a re-evaluation of the truth-values of every other statement in one's complete theory.
or proposition
which turns out to be true under any possible interpretation
of its terms (may also be called a valuation
or assignment depending upon the context).
However, the term "tautology" is also commonly used to refer to what could more specifically called truth-functional tautologies. Whereas a tautology or logical truth is true solely because of the logical terms it contains in general (e.g. "every
", "some
", and "is"), a truth-functional tautology is true because of the logical terms it contains which are logical connective
s (e.g. "or
", "and
", and "nor").
s and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible just in case their conjunction
is logically false. One statement logically implies another when it is logically incompatible with the negation
of the other. A statement is logically false just in case its negation is logically true, etc. In this way all logical connectives can be expressed in terms of preserving logical truth.
, the concept of logical truth is closely connected to the concept of a rule of inference
.
s which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.
Concept
The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...
s in logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, and there are different theories on its nature. A logical truth is a statement
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...
which is true and remains true under all reinterpretations
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
of its components other than its logical constants. It is a type of analytic statement.
Logical truths (including tautologies
Tautology (logic)
In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. However, it is not universally agreed that there are any statements which are necessarily true.
A logical truth was considered by Ludwig Wittgenstein
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
to be a statement which is true in all possible world
Possible world
In philosophy and logic, the concept of a possible world is used to express modal claims. The concept of possible worlds is common in contemporary philosophical discourse and has also been disputed.- Possibility, necessity, and contingency :...
s. This is contrasted with fact
Fact
A fact is something that has really occurred or is actually the case. The usual test for a statement of fact is verifiability, that is whether it can be shown to correspond to experience. Standard reference works are often used to check facts...
s (which may also be referred to as contingent claims or synthetic claims) which are true in this world, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded. The proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
“If p and q, then p” and the proposition “All married people are married” are logical truths because they are true due to their inherent meanings and not because of any facts of the world.
Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations.
The existence of logical truths is sometimes put forward as an objection to empiricism
Empiricism
Empiricism is a theory of knowledge that asserts that knowledge comes only or primarily via sensory experience. One of several views of epistemology, the study of human knowledge, along with rationalism, idealism and historicism, empiricism emphasizes the role of experience and evidence,...
because it is impossible to account for our knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...
of logical truths on empiricist grounds.
Logical truths and analytic truths
Logical truths, being analytic statements, do not contain any information about any matters of factFact
A fact is something that has really occurred or is actually the case. The usual test for a statement of fact is verifiability, that is whether it can be shown to correspond to experience. Standard reference works are often used to check facts...
. Other than logical truths, there is also a second class of analytic statements, typified by "No bachelor is married." The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate
Salva veritate
Salva veritate is the logical condition in virtue of which interchanging two expressions may be done without changing the truth-value of statements in which the expressions occur. The phrase occurs in two fragments from Gottfried Leibniz's General Science...
. "No bachelor is married." can be turned into "No unmarried man is married." by substituting 'unmarried man' for its synonym 'bachelor.'
In his essay. Two Dogmas of Empiricism
Two Dogmas of Empiricism
W. V. Quine's paper Two Dogmas of Empiricism, published in 1951, is one of the most celebrated papers of twentieth century philosophy in the analytic tradition. According to Harvard professor of philosophy Peter Godfrey-Smith, this "paper [is] sometimes regarded as the most important in all of...
, the philosopher W.V.O Quine
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
called into question the distinction between analytic and synthetic statements. It was this second class of analytic statements that caused him to note that the concept of analyticity itself stands in need of clarification, because it seems to depend on the concept of synonym
Synonym
Synonyms are different words with almost identical or similar meanings. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy. The word comes from Ancient Greek syn and onoma . The words car and automobile are synonyms...
y, which stands in need of clarification. In his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, given a re-evaluation of the truth-values of every other statement in one's complete theory.
Logical truths and tautologies
All tautologies are logical truths, but not all logical truths are tautologies. There are several senses in which the term "tautology" is used. In one sense, they are synonymous. In this sense, a tautology is any type of formulaWell-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
or proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
which turns out to be true under any possible interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
of its terms (may also be called a valuation
Valuation (logic)
In logic and model theory, a valuation can be:*In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables....
or assignment depending upon the context).
However, the term "tautology" is also commonly used to refer to what could more specifically called truth-functional tautologies. Whereas a tautology or logical truth is true solely because of the logical terms it contains in general (e.g. "every
Universal quantification
In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
", "some
Existential quantification
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ , which is called the existential quantifier...
", and "is"), a truth-functional tautology is true because of the logical terms it contains which are logical connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s (e.g. "or
Logical disjunction
In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...
", "and
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
", and "nor").
Logical truth and logical constants
Logical constants, including logical connectiveLogical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible just in case their conjunction
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
is logically false. One statement logically implies another when it is logically incompatible with the negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
of the other. A statement is logically false just in case its negation is logically true, etc. In this way all logical connectives can be expressed in terms of preserving logical truth.
Logical truth and rules of inference
In classical logicClassical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
, the concept of logical truth is closely connected to the concept of a rule of inference
Rule of inference
In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...
.
Non-classical logics
Non-classical logic is the name given to formal systemFormal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
s which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.