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56-466: Logical consequence is necessary and formal , by way of examples that explain with formal proof and models of interpretation . A sentence is said to be a logical consequence of a set of sentences, for a given language , if and only if , using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must be true if every sentence in the set is true. Logicians make precise accounts of logical consequence regarding

112-466: A boundary between analytic and synthetic statements simply has not been drawn. That there is such a distinction to be drawn at all is an unempirical dogma of empiricists, a metaphysical article of faith. Although the soundness of Quine's proposition remains uncertain, it had a powerful effect on the project of explaining the a priori in terms of the analytic. The metaphysical distinction between necessary and contingent truths has also been related to

168-589: A cause to produce its effect]. Contrary to contemporary usages of the term, Kant believes that a priori knowledge is not entirely independent of the content of experience. Unlike the rationalists , Kant thinks that a priori cognition, in its pure form, that is without the admixture of any empirical content, is limited to the deduction of the conditions of possible experience . These a priori , or transcendental, conditions are seated in one's cognitive faculties, and are not provided by experience in general or any experience in particular (although an argument exists that

224-472: A given language L {\displaystyle {\mathcal {L}}} , either by constructing a deductive system for L {\displaystyle {\mathcal {L}}} or by formal intended semantics for language L {\displaystyle {\mathcal {L}}} . The Polish logician Alfred Tarski identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on

280-425: A kind. Logical constants, including logical connectives and quantifiers , can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible if, and only if 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 true if, and only if its opposite

336-548: A posteriori arguments for the existence of God appear in his Monadology (1714). George Berkeley outlined the distinction in his 1710 work A Treatise Concerning the Principles of Human Knowledge (para. XXI). The 18th-century German philosopher Immanuel Kant (1781) advocated a blend of rationalist and empiricist theories. Kant says, "Although all our cognition begins with experience, it does not follow that it arises from [is caused by] experience." According to Kant,

392-427: A priori and a posteriori knowledge. A proposition that is necessarily true is one in which its negation is self-contradictory; it is true in every possible world . For example, considering the proposition "all bachelors are unmarried:" its negation (i.e. the proposition that some bachelors are married) is incoherent due to the concept of being unmarried (or the meaning of the word "unmarried") being tied to part of

448-428: A priori cognition is transcendental , or based on the form of all possible experience, while a posteriori cognition is empirical, based on the content of experience: It is quite possible that our empirical knowledge is a compound of that which we receive through impressions, and that which the faculty of cognition supplies from itself sensuous impressions [sense data] giving merely the occasion [opportunity for

504-420: A priori intuitions can be "triggered" by experience). Kant nominated and explored the possibility of a transcendental logic with which to consider the deduction of the a priori in its pure form. Space , time and causality are considered pure a priori intuitions. Kant reasoned that the pure a priori intuitions are established via his transcendental aesthetic and transcendental logic. He claimed that

560-479: A priori is to "Demonstrate Proper Effects from Proper Efficient Causes" and likewise to demonstrate a posteriori is to demonstrate "Proper Efficient Causes from Proper Effects", according to his 1696 work The Method to Science Book III, Lesson IV, Section 7. G. W. Leibniz introduced a distinction between a priori and a posteriori criteria for the possibility of a notion in his (1684) short treatise "Meditations on Knowledge, Truth, and Ideas". A priori and

616-485: A priori would, according to Stephen Palmquist , best fit into Kant's epistemological framework by calling it "analytic a posteriori." Aaron Sloman presented a brief defence of Kant's three distinctions (analytic/synthetic, apriori/empirical and necessary/contingent), in that it did not assume "possible world semantics" for the third distinction, merely that some part of this world might have been different. The relationship between aprioricity, necessity and analyticity

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672-519: A set Γ {\displaystyle \Gamma } of formulas if there is a formal proof in F S {\displaystyle {\mathcal {FS}}} of A {\displaystyle A} from the set Γ {\displaystyle \Gamma } . This is denoted Γ ⊢ F S A {\displaystyle \Gamma \vdash _{\mathcal {FS}}A} . The turnstile symbol ⊢ {\displaystyle \vdash }

728-447: A system without any thing-in-itself. Consequently, he rejected the assumption of anything that was not through and through merely our representation , and therefore let the knowing subject be all in all or at any rate produce everything from its own resources. For this purpose, he at once did away with the essential and most meritorious part of the Kantian doctrine, the distinction between

784-409: Is Mike's brother's son. Therefore Fred is Mike's nephew." Since this argument depends on the meanings of the words "brother", "son", and "nephew", the statement "Fred is Mike's nephew" is a so-called material consequence of "Fred is Mike's brother's son", not a formal consequence. A formal consequence must be true in all cases , however this is an incomplete definition of formal consequence, since even

840-426: Is a consequence of Γ {\displaystyle \Gamma } , then A {\displaystyle A} is a consequence of any superset of Γ {\displaystyle \Gamma } . It is also possible to specify non-monotonic consequence relations to capture the idea that, e.g., 'Tweety can fly' is a logical consequence of but not of Logical truth Logical truth

896-494: Is a frog; and (c) Kermit is not green. Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea: The accounts considered above are all "truth-preservational", in that they all assume that the characteristic feature of a good inference is that it never allows one to move from true premises to an untrue conclusion. As an alternative, some have proposed " warrant -preservational" accounts, according to which

952-539: Is a logical consequence of P {\displaystyle P} cannot be influenced by empirical knowledge . Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a priori property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing

1008-602: Is both necessarily true , because water and H 2 O are the same thing, they are identical in every possible world, and truths of identity are logically necessary; and a posteriori , because it is known only through empirical investigation. Following such considerations of Kripke and others (see Hilary Putnam ), philosophers tend to distinguish the notion of aprioricity more clearly from that of necessity and analyticity. Kripke's definitions of these terms diverge in subtle ways from Kant's. Taking these differences into account, Kripke's controversial analysis of naming as contingent and

1064-421: 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. The concept of logical truth is closely connected to the concept of a rule of inference . Logical positivism was a movement in the early 20th century that tried to reduce the reasoning processes of science to pure logic. Among other things,

1120-446: 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

1176-451: 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

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1232-533: Is more formally known as Kant's transcendental deduction and it is the central argument of his major work, the Critique of Pure Reason . The transcendental deduction argues that time, space and causality are ideal as much as real. In consideration of a possible logic of the a priori , this most famous of Kant's deductions has made the successful attempt in the case for the fact of subjectivity , what constitutes subjectivity and what relation it holds with objectivity and

1288-404: Is no model I {\displaystyle {\mathcal {I}}} in which all members of Γ {\displaystyle \Gamma } are true and A {\displaystyle A} is false. This is denoted Γ ⊨ F S A {\displaystyle \Gamma \models _{\mathcal {FS}}A} . Or, in other words, the set of

1344-470: Is not easy to discern. Most philosophers at least seem to agree that while the various distinctions may overlap, the notions are clearly not identical: the a priori / a posteriori distinction is epistemological ; the analytic/synthetic distinction is linguistic ; and the necessary/contingent distinction is metaphysical . The term a priori is Latin for 'from what comes before' (or, less literally, 'from first principles, before experience'). In contrast,

1400-399: Is not self-contradictory. Thus, it is said not to be true in every possible world. As Jason Baehr suggests, it seems plausible that all necessary propositions are known a priori , because "[s]ense experience can tell us only about the actual world and hence about what is the case; it can say nothing about what must or must not be the case." Following Kant, some philosophers have considered

1456-485: Is one of the most fundamental concepts in logic . Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions . In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other than its logical constants ). Thus, logical truths such as "if p, then p" can be considered tautologies . Logical truths are thought to be

1512-585: Is something that one knows a priori because it expresses a statement that one can derive by reason alone. Consider the proposition: "George V reigned from 1910 to 1936." This is something that (if true) one must come to know a posteriori because it expresses an empirical fact unknowable by reason alone. Several philosophers, in reaction to Immanuel Kant , sought to explain a priori knowledge without appealing to, as Paul Boghossian describes as "a special faculty [intuition]   ... that has never been described in satisfactory terms." One theory, popular among

1568-747: Is sometimes treated as equivalent to saying that logical truths are true in all possible worlds . However, the question of which 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 facts (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 propositions 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

1624-672: Is to make it possible to construct different models of logical consequence and logical truth. A priori and a posteriori A priori ('from the earlier') and a posteriori ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge , justification , or argument by their reliance on experience. A priori knowledge is independent from any experience . Examples include mathematics , tautologies and deduction from pure reason . A posteriori knowledge depends on empirical evidence . Examples include most fields of science and aspects of personal knowledge . The terms originate from

1680-438: The history of philosophy . Both terms are primarily used as modifiers to the noun knowledge (e.g., a priori knowledge). A priori can be used to modify other nouns such as truth . Philosophers may use apriority , apriorist and aprioricity as nouns referring to the quality of being a priori . Consider the proposition : "If George V reigned at least four days, then he reigned more than three days." This

1736-420: The logical form of the sentences: (2) The relation is a priori , i.e., it can be determined with or without regard to empirical evidence (sense experience); and (3) The logical consequence relation has a modal component. The most widely prevailing view on how best to account for logical consequence is to appeal to formality. This is to say that whether statements follow from one another logically depends on

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1792-415: The logical positivists of the early 20th century, is what Boghossian calls the "analytic explanation of the a priori." The distinction between analytic and synthetic propositions was first introduced by Kant. While his original distinction was primarily drawn in terms of conceptual containment, the contemporary version of such distinction primarily involves, as American philosopher W. V. O. Quine put it,

1848-541: The analytic methods found in Organon , a collection of works by Aristotle . Prior analytics ( a priori ) is about deductive logic , which comes from definitions and first principles. Posterior analytics ( a posteriori ) is about inductive logic , which comes from observational evidence. Both terms appear in Euclid 's Elements and were popularized by Immanuel Kant 's Critique of Pure Reason , an influential work in

1904-515: The argument " P is Q ' s brother's son, therefore P is Q ' s nephew" is valid in all cases, but is not a formal argument. If it is known that Q {\displaystyle Q} follows logically from P {\displaystyle P} , then no information about the possible interpretations of P {\displaystyle P} or Q {\displaystyle Q} will affect that knowledge. Our knowledge that Q {\displaystyle Q}

1960-392: The characteristic feature of a good inference is that it never allows one to move from justifiably assertible premises to a conclusion that is not justifiably assertible. This is (roughly) the account favored by intuitionists such as Michael Dummett . The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if A {\displaystyle A}

2016-404: The concept in terms of proofs and via models . The study of the syntactic consequence (of a logic) is called (its) proof theory whereas the study of (its) semantic consequence is called (its) model theory . A formula A {\displaystyle A} is a syntactic consequence within some formal system F S {\displaystyle {\mathcal {FS}}} of

2072-405: The concept of synonymy , 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. Considering different interpretations of the same statement leads to

2128-411: The concept of being a bachelor (or part of the definition of the word "bachelor"). To the extent that contradictions are impossible, self-contradictory propositions are necessarily false as it is impossible for them to be true. The negation of a self-contradictory proposition is, therefore, supposed to be necessarily true. By contrast, a proposition that is contingently true is one in which its negation

2184-401: The dialogue Meno , according to which something like a priori knowledge is knowledge inherent, intrinsic in the human mind. Albert of Saxony , a 14th-century logician, wrote on both a priori and a posteriori . The early modern Thomistic philosopher John Sergeant differentiates the terms by the direction of inference regarding proper causes and effects. To demonstrate something

2240-449: The empirical. After Kant's death, a number of philosophers saw themselves as correcting and expanding his philosophy, leading to the various forms of German Idealism . One of these philosophers was Johann Fichte . His student (and critic), Arthur Schopenhauer , accused him of rejecting the distinction between a priori and a posteriori knowledge: ... Fichte who, because the thing-in-itself had just been discredited, at once prepared

2296-466: The human subject would not have the kind of experience that it has were these a priori forms not in some way constitutive of him as a human subject. For instance, a person would not experience the world as an orderly, rule-governed place unless time, space and causality were determinant functions in the form of perceptual faculties, i. e., there can be no experience in general without space, time or causality as particular determinants thereon. The claim

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2352-414: The interpretations that make all members of Γ {\displaystyle \Gamma } true is a subset of the set of the interpretations that make A {\displaystyle A} true. Modal accounts of logical consequence are variations on the following basic idea: Alternatively (and, most would say, equivalently): Such accounts are called "modal" because they appeal to

2408-436: The logical positivists claimed that any proposition that is not empirically verifiable is neither true nor false, but nonsense . Non-classical logic is the name given to formal systems 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

2464-422: The meaning of the proposition in question. More simply, proponents of this explanation claimed to have reduced a dubious metaphysical faculty of pure reason to a legitimate linguistic notion of analyticity. The analytic explanation of a priori knowledge has undergone several criticisms. Most notably, Quine argues that the analytic–synthetic distinction is illegitimate: But for all its a priori reasonableness,

2520-427: The modal notions of logical necessity and logical possibility . 'It is necessary that' is often expressed as a universal quantifier over possible worlds , so that the accounts above translate as: Consider the modal account in terms of the argument given as an example above: The conclusion is a logical consequence of the premises because we can not imagine a possible world where (a) all frogs are green; (b) Kermit

2576-427: The notion of truth value . 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 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). This is synonymous to logical truth. However,

2632-493: The notions of "true by virtue of meanings and independently of fact." Analytic propositions are considered true by virtue of their meaning alone, while a posteriori propositions by virtue of their meaning and of certain facts about the world. According to the analytic explanation of the a priori , all a priori knowledge is analytic; so a priori knowledge need not require a special faculty of pure intuition , since it can be accounted for simply by one's ability to understand

2688-518: The relationship between aprioricity , analyticity and necessity to be extremely close. According to Jerry Fodor , " positivism , in particular, took it for granted that a priori truths must be necessary." Since Kant, the distinction between analytic and synthetic propositions has slightly changed. Analytic propositions were largely taken to be "true by virtue of meanings and independently of fact", while synthetic propositions were not—one must conduct some sort of empirical investigation, looking to

2744-461: The simplest case of statements which are analytically true (or in other words, true by definition). All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence . 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

2800-412: The structure or logical form of the statements without regard to the contents of that form. Syntactic accounts of logical consequence rely on schemes using inference rules . For instance, we can express the logical form of a valid argument as: This argument is formally valid, because every instance of arguments constructed using this scheme is valid. This is in contrast to an argument like "Fred

2856-476: The term a posteriori is Latin for 'from what comes later' (or 'after experience'). They appear in Latin translations of Euclid 's Elements , a work widely considered during the early European modern period as the model for precise thinking. An early philosophical use of what might be considered a notion of a priori knowledge (though not called by that name) is Plato 's theory of recollection , related in

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2912-461: The term tautology is also commonly used to refer to what could more specifically be 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 connectives (e.g. " or ", " and ", and " nor "). Not all logical truths are tautologies of such

2968-541: 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 , because they hold that it is impossible to account for our knowledge 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

3024-442: The world, to determine the truth-value of synthetic propositions. Aprioricity, analyticity and necessity have since been more clearly separated from each other. American philosopher Saul Kripke (1972), for example, provides strong arguments against this position, whereby he contends that there are necessary a posteriori truths. For example, the proposition that water is H 2 O (if it is true): According to Kripke, this statement

3080-482: The world. The latter view was notably defended by the logical positivists in the early 20th century. Logical truths, being analytic statements, do not contain any information about any matters of fact . 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

3136-489: Was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935). Syntactic consequence does not depend on any interpretation of the formal system. A formula A {\displaystyle A} is a semantic consequence within some formal system F S {\displaystyle {\mathcal {FS}}} of a set of statements Γ {\displaystyle \Gamma } if and only if there

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