logical truth

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Logical truth is one of the most fundamental
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 in
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 ...
. Broadly speaking, a logical truth is a statement which is
true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * ...
regardless of the truth or falsity of its constituent
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other than its
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). Thus, logical truths such as "if p, then p" can be considered tautologies. Logical truths are thought to be the simplest case of statements which are analytically true (or in other words, true by definition). All of
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 ...
can be thought of as providing accounts of the nature of logical truth, as well as
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 ...
. Logical truths are generally considered to be ''necessarily true''. This is to say that they are such that no situation could arise in which they could fail to be true. The view that logical statements are necessarily true is sometimes treated as equivalent to saying that logical truths are true in all
possible world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional logic, intensional and mod ...
s. However, the question of whether any statements are ''necessarily'' true remains the subject of continued debate. Treating logical truths, analytic truths, and necessary truths as equivalent, logical truths can be contrasted with
fact A fact is a datum about one or more aspects of a circumstance, which, if accepted as true and proven true, allows a logical conclusion to be reached on a true–false evaluation. Standard reference works are often used to check facts. Scient ...
s (which can also be called ''contingent claims'' or ''synthetic claims''). Contingent truths are true in ''this'' world, but could have turned out otherwise (in other words, they are false in at least one possible world). Logically true
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s such as "If p and q, then p" and "All married people are married" are logical truths because they are true due to their internal structure and not because of any facts of the world (whereas "All married people are happy", even if it were true, could not be true solely in virtue of its logical structure). Rationalist philosophers have suggested that the existence of logical truths cannot be explained by
empiricism In philosophy, empiricism is an epistemological theory that holds that knowledge or justification comes only or primarily from sensory experience. It is one of several views within epistemology, along with rationalism and skepticism. Empir ...
, because they hold that it is impossible to account for our
knowledge Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distinc ...
of logical truths on empiricist grounds. Empiricists commonly respond to this objection by arguing that logical truths (which they usually deem to be mere tautologies), are analytic and thus do not purport to describe the world. The latter view was notably defended by the
logical positivists Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, is a movement in Western philosophy whose central thesis was the verification principle (also known as the verifiability criterion o ...
in the early 20th century.

Logical truths and analytic truths

Logical truths, being analytic statements, do not contain any information about any matters of
fact A fact is a datum about one or more aspects of a circumstance, which, if accepted as true and proven true, allows a logical conclusion to be reached on a true–false evaluation. Standard reference works are often used to check facts. Scient ...
. 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 The literal translation of the Latin "''salva veritate''" is "with (or by) unharmed truth", using ablative of manner: "''salva''" meaning "rescue," "salvation," or "welfare," and "''veritate''" meaning "reality" or "truth". Thus, ''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 "Two Dogmas of Empiricism" is a paper by analytic philosopher Willard Van Orman Quine published in 1951. According to University of Sydney professor of philosophy Peter Godfrey-Smith, this "paper ssometimes regarded as the most important in all ...
, the philosopher
W. V. O. Quine W. may refer to: * SoHo (Australian TV channel) (previously W.), an Australian pay television channel * ''W.'' (film), a 2008 American biographical drama film based on the life of George W. Bush * "W.", the fifth track from Codeine's 1992 EP ''Bar ...
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 A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are all ...
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.

Truth values and tautologies

Considering different interpretations of the same statement leads to the notion of
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progr ...
. The simplest approach to truth values means that the statement may be "true" in one case, but "false" in another. In one sense of the term ''tautology'', it is any type of
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
or
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
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). This is synonymous to logical truth. However, the term ''tautology'' is also commonly used to refer to what could more specifically be called
truth-functional In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
tautologies. Whereas a tautology or logical truth is true solely because of the logical terms it contains in general (e.g. " every", "
some Some may refer to: *''some'', an English word used as a determiner and pronoun; see use of ''some'' *The term associated with the existential quantifier *"Some", a song by Built to Spill from their 1994 album ''There's Nothing Wrong with Love'' *S ...
", and "is"), a truth-functional tautology is true because of the logical terms it contains which are
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
s (e.g. " or", "
and or AND may refer to: Logic, grammar, and computing * Conjunction (grammar), connecting two words, phrases, or clauses * Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition * Bitwise AND, a boolea ...
", and " nor"). Not all logical truths are tautologies of such a kind.

Logical truth and logical constants

Logical constants, including
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
s and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible ''
if, and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...
'' their
conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...
is logically false. One statement logically implies another when it is logically incompatible with the
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
of the other. A statement is logically true if, and only if its opposite is logically false. The opposite statements must contradict one another. In this way all logical connectives can be expressed in terms of preserving logical truth. The logical form of a sentence is determined by its semantic or syntactic structure and by the placement of logical constants. Logical constants determine whether a statement is a logical truth when they are combined with a language that limits its meaning. Therefore, until it is determined how to make a distinction between all logical constants regardless of their language, it is impossible to know the complete truth of a statement or argument.

Logical truth and rules of inference

The concept of logical truth is closely connected to the concept of a
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 ...
.

Logical truth and logical positivism

Logical positivism Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, is a movement in Western philosophy whose central thesis was the verification principle (also known as the verifiability criterion o ...
was a movement in the early 20th century that tried to reduce the reasoning processes of science to pure logic. Among other things, the logical positivists claimed that any proposition that is not empirically verifiable is neither true nor false, but nonsense. This movement faded out due to various problems with their approach among which was a growing understanding that science does not work in the way that the positivists described. Another problem was that one of the favorite slogans of the movement: "any proposition that is not empirically verifiable is nonsense" was itself not empirically verifiable, and therefore, by its own terms, nonsense.

Non-classical logics

Non-classical logic is the name given to
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 which differ in a significant way from standard logical systems such as propositional and
predicate 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 ...
. 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 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 ...
and logical truth.
Theodore Sider Theodore "Ted" Sider is an American philosopher specializing in metaphysics and philosophy of language. He is Distinguished Professor of Philosophy at Rutgers University. Family Sider is the son of theologian Ronald Sider. He is the partner of ...
, (2010). ''Logic for philosophy''

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Contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
*
False (logic) In logic, false or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual n ...
*
Logical truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
, a mathematical table used in logic *
Satisfiability In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over ...
*
Tautology (logic) In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always ...
(for symbolism of logical truth) *
Theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
*
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 ** ...