144-483: Willard Van Orman Quine ( / k w aɪ n / ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition , recognized as "one of the most influential philosophers of the twentieth century". He served as the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978. Quine was a teacher of logic and set theory . He
288-501: A r y ) ∧ Q ( J o h n ) ) {\displaystyle \exists Q(Q(Mary)\land Q(John))} " . In this case, the existential quantifier is applied to the predicate variable " Q {\displaystyle Q} " . The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories. But it has drawbacks in regard to its meta-logical properties and ontological implications, which
432-527: A behaviorist theory of meaning . Quine grew up in Akron, Ohio , where he lived with his parents and older brother Robert Cloyd. His father, Cloyd Robert, was a manufacturing entrepreneur (founder of the Akron Equipment Company, which produced tire molds) and his mother, Harriett E., was a schoolteacher and later a housewife . Quine became an atheist around the age of 9 and remained one for
576-523: A Sheldon Fellowship, meeting Polish logicians (including Stanislaw Lesniewski and Alfred Tarski ) and members of the Vienna Circle (including Rudolf Carnap ), as well as the logical positivist A. J. Ayer . It was in Prague that Quine developed a passion for philosophy, thanks to Carnap, whom he defined as his "true and only maître à penser ". Quine arranged for Tarski to be invited to
720-445: A central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments
864-480: A certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth. This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to non-monotonicity and defeasibility : it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn. Ampliative reasoning plays
1008-412: A chapter of psychology and hence of natural science. It studies a natural phenomenon, viz., a physical human subject. This human subject is accorded a certain experimentally controlled input—certain patterns of irradiation in assorted frequencies, for instance—and in the fullness of time the subject delivers as output a description of the three-dimensional external world and its history. The relation between
1152-573: A complex argument to be successful, each link of the chain has to be successful. Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning . The strongest form of support corresponds to deductive reasoning . But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases,
1296-425: A conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion. These general characterizations apply to logic in the widest sense, i.e., to both formal and informal logic since they are both concerned with assessing the correctness of arguments. Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic. Formal logic
1440-509: A desire to minimize posits; each innovation is pushed as far as it can be pushed before further innovations are introduced. For Quine, there is but one connective, the Sheffer stroke , and one quantifier, the universal quantifier . All polyadic predicates can be reduced to one dyadic predicate, interpretable as set membership. His rules of proof were limited to modus ponens and substitution. He preferred conjunction to either disjunction or
1584-486: A difficult position. Just as he challenged the dominant analytic–synthetic distinction, Quine also took aim at traditional normative epistemology . According to Quine, traditional epistemology tried to justify the sciences, but this effort (as exemplified by Rudolf Carnap ) failed, and so we should replace traditional epistemology with an empirical study of what sensory inputs produce what theoretical outputs: Epistemology, or something like it, simply falls into place as
SECTION 10
#17328014784321728-510: A formal language together with a set of axioms and a proof system used to draw inferences from these axioms. In logic, axioms are statements that are accepted without proof. They are used to justify other statements. Some theorists also include a semantics that specifies how the expressions of the formal language relate to real objects. Starting in the late 19th century, many new formal systems have been proposed. A formal language consists of an alphabet and syntactic rules. The alphabet
1872-686: A formal language while informal logic investigates them in their original form. On this view, the argument "Birds fly. Tweety is a bird. Therefore, Tweety flies." belongs to natural language and is examined by informal logic. But the formal translation "(1) ∀ x ( B i r d ( x ) → F l i e s ( x ) ) {\displaystyle \forall x(Bird(x)\to Flies(x))} ; (2) B i r d ( T w e e t y ) {\displaystyle Bird(Tweety)} ; (3) F l i e s ( T w e e t y ) {\displaystyle Flies(Tweety)} "
2016-415: A given argument is valid. Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be translated into formal language before their validity can be assessed. The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning
2160-551: A given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In chess , for example, the definitory rules dictate that bishops may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's king . It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning. A formal system of logic consists of
2304-402: A great variety of topics. They include metaphysical theses about ontological categories and problems of scientific explanation. But in a more narrow sense, it is identical to term logic or syllogistics. A syllogism is a form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a subject , a predicate, and a copula connecting
2448-403: A kind of studied ambiguity peculiar to themselves." Putting it another way, to say 'I hate everything' is a very different statement than saying 'I hate Bertrand Russell', because the words 'Bertrand Russell' are a proper name that refer to a very specific person. Whereas the word 'everything' is a placeholder. It does not refer to a specific entity or entities. Quine is able, therefore, to make
2592-614: A logical connective like "and" to form a new complex proposition. In Aristotelian logic, the subject can be universal , particular , indefinite , or singular . For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates". Aristotelian logic only includes predicates for simple properties of entities. But it lacks predicates corresponding to relations between entities. The predicate can be linked to
2736-432: A major philosopher. By the 1960s, he had worked out his " naturalized epistemology " whose aim was to answer all substantive questions of knowledge and meaning using the methods and tools of the natural sciences. Quine roundly rejected the notion that there should be a "first philosophy", a theoretical standpoint somehow prior to natural science and capable of justifying it. These views are intrinsic to his naturalism . Like
2880-444: A meaningful claim about Pegasus' nonexistence for the simple reason that the placeholder (a thing) happens to be empty. It just so happens that the world does not contain a thing that is such that it is winged and it is a horse. In the 1930s and 40s, discussions with Rudolf Carnap , Nelson Goodman and Alfred Tarski , among others, led Quine to doubt the tenability of the distinction between "analytic" statements—those true simply by
3024-459: A rabbit, the natives could as well refer to something like undetached rabbit-parts , or rabbit- tropes and it would not make any observable difference. The behavioural data the linguist could collect from the native speaker would be the same in every case, or to reword it, several translation hypotheses could be built on the same sensoric stimuli. Quine concluded his " Two Dogmas of Empiricism " as follows: As an empiricist I continue to think of
SECTION 20
#17328014784323168-445: A scientific error to believe otherwise. But in point of epistemological footing, the physical objects and the gods differ only in degree and not in kind. Both sorts of entities enter our conceptions only as cultural posits. Quine's ontological relativism (evident in the passage above) led him to agree with Pierre Duhem that for any collection of empirical evidence , there would always be many theories able to account for it, known as
3312-664: A sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, p {\displaystyle p} ("yesterday was Sunday") and q {\displaystyle q} ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are ¬ {\displaystyle \lnot } ( not ), ∨ {\displaystyle \lor } ( or ), → {\displaystyle \to } ( if...then ), and ↑ {\displaystyle \uparrow } ( Sheffer stroke ). Given
3456-665: A set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work". Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ∧ {\displaystyle \land } ( and ) or → {\displaystyle \to } ( if...then ). Simple propositions also have parts, like "Sunday" or "work" in
3600-539: A thorough re-examination of the two inferences [existential generalization and universal instantiation] may prove worth our while. Lejewski then goes on to offer a description of free logic , which he claims accommodates an answer to the problem. Lejewski also points out that free logic additionally can handle the problem of the empty set for statements like ∀ x F x → ∃ x F x {\displaystyle \forall x\,Fx\rightarrow \exists x\,Fx} . Quine had considered
3744-417: A way of thinking. It translates literally as "master for thinking". To take a maître à penser is therefore close to becoming a disciple . The phrase itself can be used to refer to a type of person — an inspirational genius, for example — who naturally would attract followers interested enough to absorb a whole intellectual approach. A maître à penser is therefore possibly something like
3888-617: Is From A Logical Point of View . Quine confined logic to classical bivalent first-order logic , hence to truth and falsity under any (nonempty) universe of discourse . Hence the following were not logic for Quine: Quine wrote three undergraduate texts on formal logic: Mathematical Logic is based on Quine's graduate teaching during the 1930s and 1940s. It shows that much of what Principia Mathematica took more than 1000 pages to say can be said in 250 pages. The proofs are concise, even cryptic. The last chapter, on Gödel's incompleteness theorem and Tarski's indefinability theorem , along with
4032-420: Is sound when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by
4176-476: Is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers . This means that they have a truth value : they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences . Propositions are the denotations of sentences and are usually seen as abstract objects . For example, the English sentence "the tree
4320-441: Is a restricted version of classical logic. It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that A {\displaystyle A} follows from ¬ ¬ A {\displaystyle \lnot \lnot A} . This is a valid rule of inference in classical logic but it
4464-416: Is also known as symbolic logic and is widely used in mathematical logic . It uses a formal approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the logical form of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content. Formal logic
Willard Van Orman Quine - Misplaced Pages Continue
4608-453: Is an example of the existential quantifier " ∃ {\displaystyle \exists } " applied to the individual variable " x {\displaystyle x} " . In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula " ∃ Q ( Q ( M
4752-495: Is associated with informal fallacies , critical thinking , and argumentation theory . Informal logic examines arguments expressed in natural language whereas formal logic uses formal language . When used as a countable noun , the term "a logic" refers to a specific logical formal system that articulates a proof system . Logic plays a central role in many fields, such as philosophy , mathematics , computer science , and linguistics . Logic studies arguments, which consist of
4896-415: Is blurry in some cases, such as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between. The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term " induction " to cover all forms of non-deductive arguments. But in a more narrow sense, induction
5040-421: Is commonly defined in terms of arguments or inferences as the study of their correctness. An argument is a set of premises together with a conclusion. An inference is the process of reasoning from these premises to the conclusion. But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on
5184-407: Is controversial because it belongs to the field of psychology , not logic, and because appearances may be different for different people. Fallacies are usually divided into formal and informal fallacies. For formal fallacies, the source of the error is found in the form of the argument. For example, denying the antecedent is one type of formal fallacy, as in "if Othello is a bachelor, then he
5328-453: Is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises. According to an influential view by Alfred Tarski , deductive arguments have three essential features: (1) they are formal, i.e. they depend only on
5472-420: Is established by verification using a proof. Intuitionistic logic is especially prominent in the field of constructive mathematics , which emphasizes the need to find or construct a specific example to prove its existence. Ma%C3%AEtre %C3%A0 penser Maître à penser is a French-language phrase, denoting a teacher whom one chooses, in order to learn not just a set of facts or point of view, but
5616-610: Is green" is different from the German sentence "der Baum ist grün" but both express the same proposition. Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions. These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like
5760-432: Is interested in deductively valid arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false. For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a rule of inference . For example, modus ponens is a rule of inference according to which all arguments of
5904-415: Is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the law of excluded middle . It states that for every sentence, either it or its negation is true. This means that every proposition of the form A ∨ ¬ A {\displaystyle A\lor \lnot A} is true. These deviations from classical logic are based on the idea that truth
Willard Van Orman Quine - Misplaced Pages Continue
6048-415: Is justified by confirmation holism . Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the nominalist who wishes to exclude the existence of sets and non-Euclidean geometry , but to include the existence of quarks and other undetectable entities of physics, for example, in
6192-447: Is male; Othello is not a bachelor; therefore Othello is not male". But most fallacies fall into the category of informal fallacies, of which a great variety is discussed in the academic literature. The source of their error is usually found in the content or the context of the argument. Informal fallacies are sometimes categorized as fallacies of ambiguity, fallacies of presumption, or fallacies of relevance. For fallacies of ambiguity,
6336-688: Is necessary, then it is also possible. This means that ◊ A {\displaystyle \Diamond A} follows from ◻ A {\displaystyle \Box A} . Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that ◻ A {\displaystyle \Box A} is equivalent to ¬ ◊ ¬ A {\displaystyle \lnot \Diamond \lnot A} . Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, deontic logic concerns
6480-518: Is necessary. For example, if the formula B ( s ) {\displaystyle B(s)} stands for the sentence "Socrates is a banker" then the formula ◊ B ( s ) {\displaystyle \Diamond B(s)} articulates the sentence "It is possible that Socrates is a banker". To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something
6624-418: Is not sufficient for ontology since it depends on a theory in order to result in ontological commitments. Quine proposed that we should base our ontology on our best scientific theory. Various followers of Quine's method chose to apply it to different fields, for example to "everyday conceptions expressed in natural language". In philosophy of mathematics , he and his Harvard colleague Hilary Putnam developed
6768-407: Is not the best or most likely explanation. Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as fallacies . Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion. So the argument "it is sunny today; therefore spiders have eight legs" is fallacious even though
6912-541: Is only one type of ampliative argument alongside abductive arguments . Some philosophers, like Leo Groarke, also allow conductive arguments as another type. In this narrow sense, induction is often defined as a form of statistical generalization. In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains. In this sense, one may infer that "all elephants are gray" based on one's past observations of
7056-430: Is studied by formal logic. The study of natural language arguments comes with various difficulties. For example, natural language expressions are often ambiguous, vague, and context-dependent. Another approach defines informal logic in a wide sense as the normative study of the standards, criteria, and procedures of argumentation. In this sense, it includes questions about the role of rationality , critical thinking , and
7200-406: Is synonymity between "unmarried man" and "bachelor", you have proved that both sentences are logically true and therefore self evident. Quine however gives several arguments for why this is not possible, for instance that "bachelor" in some contexts mean a Bachelor of Arts , not an unmarried man. Colleague Hilary Putnam called Quine's indeterminacy of translation thesis "the most fascinating and
7344-400: Is that we can now make free use of empirical psychology. Logician Logic is the study of correct reasoning . It includes both formal and informal logic . Formal logic is the study of deductively valid inferences or logical truths . It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic
SECTION 50
#17328014784327488-410: Is the set of basic symbols used in expressions . The syntactic rules determine how these symbols may be arranged to result in well-formed formulas. For instance, the syntactic rules of propositional logic determine that " P ∧ Q {\displaystyle P\land Q} " is a well-formed formula but " ∧ Q {\displaystyle \land Q} " is not since
7632-434: Is there?' It can be answered, moreover, in a word—'Everything'—and everyone will accept this answer as true. More directly, the controversy goes: How can we talk about Pegasus ? To what does the word 'Pegasus' refer? If our answer is, 'Something', then we seem to believe in mystical entities; if our answer is, 'nothing', then we seem to talk about nothing and what sense can be made of this? Certainly when we said that Pegasus
7776-432: Is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference . They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach
7920-467: Is to turn 'Pegasus' into a predicate, to use a term of First-order logic : i.e. a property. As such, when we say 'Pegasus', we are really saying 'the thing that is Pegasus' or 'the thing that Pegasizes' . This introduces, to use another term from logic, bound variables (ex: 'everything', 'something,' etc.) As Quine explains, bound variables, "far from purpoting to be names specifically...do not purport to be names at all: they refer to entities generally, with
8064-540: Is unable to address. Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies. Many characterizations of informal logic have been suggested but there is no general agreement on its precise definition. The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments. On this view, informal logic studies arguments that are in informal or natural language. Formal logic can only examine them indirectly by translating them first into
8208-599: Is used to represent the ideas of knowing something in contrast to merely believing it to be the case. Higher-order logics extend classical logic not by using modal operators but by introducing new forms of quantification. Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula " ∃ x ( A p p l e ( x ) ∧ S w e e t ( x ) ) {\displaystyle \exists x(Apple(x)\land Sweet(x))} " ( some apples are sweet)
8352-431: Is why first-order logic is still more commonly used. Deviant logics are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue. Intuitionistic logic
8496-519: The Duhem–Quine thesis . However, Duhem's holism is much more restricted and limited than Quine's. For Duhem, underdetermination applies only to physics or possibly to natural science , while for Quine it applies to all of human knowledge. Thus, while it is possible to verify or falsify whole theories, it is not possible to verify or falsify individual statements. Almost any particular statement can be saved, given sufficiently radical modifications of
8640-533: The Quine–Putnam indispensability thesis , an argument for the reality of mathematical entities . The form of the argument is as follows. The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real
8784-500: The conditional , because conjunction has the least semantic ambiguity. He was delighted to discover early in his career that all of first order logic and set theory could be grounded in a mere two primitive notions: abstraction and inclusion . For an elegant introduction to the parsimony of Quine's approach to logic, see his "New Foundations for Mathematical Logic", ch. 5 in his From a Logical Point of View . Quine has had numerous influences on contemporary metaphysics . He coined
SECTION 60
#17328014784328928-469: The Greek word "logos", which has a variety of translations, such as reason , discourse , or language . Logic is traditionally defined as the study of the laws of thought or correct reasoning , and is usually understood in terms of inferences or arguments . Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences. An argument is a set of premises together with
9072-711: The Harvard graduate theses of, among others, David Lewis , Gilbert Harman , Dagfinn Føllesdal , Hao Wang , Hugues LeBlanc , Henry Hiz and George Myro . For the academic year 1964–1965, Quine was a fellow on the faculty in the Center for Advanced Studies at Wesleyan University . In 1980 Quine received an honorary doctorate from the Faculty of Humanities at Uppsala University , Sweden. Quine's student Dagfinn Føllesdal noted that Quine suffered from memory loss towards his final years. The deterioration of his short-term memory
9216-864: The September 1939 Unity of Science Congress in Cambridge, for which the Jewish Tarski sailed on the last ship to leave Danzig before Nazi Germany invaded Poland and triggered World War II . Tarski survived the war and worked another 44 years in the US. During the war, Quine lectured on logic in Brazil , in Portuguese, and served in the United States Navy in a military intelligence role, deciphering messages from German submarines, and reaching
9360-418: The ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark". Fallacies of presumption have a wrong or unjustified premise but may be valid otherwise. In the case of fallacies of relevance, the premises do not support the conclusion because they are not relevant to it. The main focus of most logicians
9504-482: The article Quine (1946), became a launching point for Raymond Smullyan 's later lucid exposition of these and related results. Quine's work in logic gradually became dated in some respects. Techniques he did not teach and discuss include analytic tableaux , recursive functions , and model theory . His treatment of metalogic left something to be desired. For example, Mathematical Logic does not include any proofs of soundness and completeness . Early in his career,
9648-399: The assessment of arguments. Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion. For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars
9792-495: The basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like metaphysics , ethics , and epistemology . Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: ◊ {\displaystyle \Diamond } expresses that something is possible while ◻ {\displaystyle \Box } expresses that something
9936-401: The basis of meager sensory input". He also advocated holism in science, known as the Duhem–Quine thesis . His major writings include the papers "On What There Is" (1948), which elucidated Bertrand Russell 's theory of descriptions and contains Quine's famous dictum of ontological commitment, "To be is to be the value of a variable ", and " Two Dogmas of Empiricism " (1951), which attacked
10080-487: The best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer. Arguments that fall short of the standards of correct reasoning often embody fallacies . Systems of logic are theoretical frameworks for assessing the correctness of arguments. Logic has been studied since antiquity . Early approaches include Aristotelian logic , Stoic logic , Nyaya , and Mohism . Aristotelian logic focuses on reasoning in
10224-471: The bound variable x ranges over electrons, resulting in an ontological commitment to electrons. This approach is summed up by Quine's famous dictum that "[t]o be is to be the value of a variable". Quine applied this method to various traditional disputes in ontology. For example, he reasoned from the sentence "There are prime numbers between 1000 and 1010" to an ontological commitment to the existence of numbers, i.e. realism about numbers. This method by itself
10368-645: The claim "either it is raining, or it is not". These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q " is a logical truth. Formal logic uses formal languages to express and analyze arguments. They normally have a very limited vocabulary and exact syntactic rules . These rules specify how their symbols can be combined to construct sentences, so-called well-formed formulas . This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether
10512-495: The color of elephants. A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray. Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference. Abductive inference may or may not take statistical observations into consideration. In either case,
10656-502: The conceptual scheme of science as a tool, ultimately, for predicting future experience in the light of past experience. Physical objects are conceptually imported into the situation as convenient intermediaries not by definition in terms of experience, but simply as irreducible posits comparable, epistemologically, to the gods of Homer …. For my part I do, qua lay physicist, believe in physical objects and not in Homer's gods; and I consider it
10800-511: The conclusion "all ravens are black". A further approach is to define informal logic as the study of informal fallacies . Informal fallacies are incorrect arguments in which errors are present in the content and the context of the argument. A false dilemma , for example, involves an error of content by excluding viable options. This is the case in the fallacy "you are either with us or against us; you are not with us; therefore, you are against us". Some theorists state that formal logic studies
10944-458: The conclusion is true. Some theorists, like John Stuart Mill , give a more restrictive definition of fallacies by additionally requiring that they appear to be correct. This way, genuine fallacies can be distinguished from mere mistakes of reasoning due to carelessness. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them. However, this reference to appearances
11088-591: The conditional proposition p → q {\displaystyle p\to q} , one can form truth tables of its converse q → p {\displaystyle q\to p} , its inverse ( ¬ p → ¬ q {\displaystyle \lnot p\to \lnot q} ) , and its contrapositive ( ¬ q → ¬ p {\displaystyle \lnot q\to \lnot p} ) . Truth tables can also be defined for more complex expressions that use several propositional connectives. Logic
11232-455: The containing theory. For Quine, scientific thought forms a coherent web in which any part could be altered in the light of empirical evidence, and in which no empirical evidence could force the revision of a given part. The problem of non-referring names is an old puzzle in philosophy, which Quine captured when he wrote, A curious thing about the ontological problem is its simplicity. It can be put into three Anglo-Saxon monosyllables: 'What
11376-438: The contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future. Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like ethics or epistemology that lie beyond
11520-460: The data or being unworkably complex, there are many equally justifiable alternatives. While the Greeks ' assumption that (unobservable) Homeric gods exist is false, and our supposition of (unobservable) electromagnetic waves is true, both are to be justified solely by their ability to explain our observations. The gavagai thought experiment tells about a linguist, who tries to find out, what
11664-451: The depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs. Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with
11808-451: The edge of Occam’s razor. Quine was unsympathetic, however, to the claim that saying 'X does not exist' is a tacit acceptance of X's existence and, thus, a contradiction. Appealing to Bertrand Russell and his theory of "singular descriptions", Quine explains how Russell was able to make sense of "complex descriptive names" ('The Present King of France', 'The author of Waverly was a poet', etc.) by thinking about them as merely "fragments of
11952-409: The example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have
12096-458: The expression gavagai means, when uttered by a speaker of a yet unknown, native language upon seeing a rabbit. At first glance, it seems that gavagai simply translates with rabbit . Now, Quine points out that the background language and its referring devices might fool the linguist here, because he is misled in a sense that he always makes direct comparisons between the foreign language and his own. However, when shouting gavagai , and pointing at
12240-434: The field of ethics and introduces symbols to express the ideas of obligation and permission , i.e. to describe whether an agent has to perform a certain action or is allowed to perform it. The modal operators in temporal modal logic articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time. In epistemology, epistemic modal logic
12384-485: The form "(1) p , (2) if p then q , (3) therefore q " are valid, independent of what the terms p and q stand for. In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of logical truths . A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all possible worlds and under all interpretations of its non-logical terms, like
12528-513: The form of syllogisms . It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as Gottlob Frege . Today, the most commonly used system is classical logic . It consists of propositional logic and first-order logic . Propositional logic only considers logical relations between full propositions. First-order logic also takes
12672-523: The form of the premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid. The modus ponens is a prominent rule of inference. It has the form " p ; if p , then q ; therefore q ". Knowing that it has just rained ( p {\displaystyle p} ) and that after rain the streets are wet ( p → q {\displaystyle p\to q} ), one can use modus ponens to deduce that
12816-419: The form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by logical necessity for the given propositions, independent of any other circumstances. Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference. Rules of inference specify
12960-421: The general form of arguments while informal logic studies particular instances of arguments. Another approach is to hold that formal logic only considers the role of logical constants for correct inferences while informal logic also takes the meaning of substantive concepts into account. Further approaches focus on the discussion of logical topics with or without formal devices and on the role of epistemology for
13104-406: The internal parts of propositions into account, like predicates and quantifiers . Extended logics accept the basic intuitions behind classical logic and apply it to other fields, such as metaphysics , ethics , and epistemology . Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic. The word "logic" originates from
13248-407: The internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, predicates , which refer to properties and relations, and quantifiers, which treat notions like "some" and "all". For example, to express the proposition "this raven is black", one may use the predicate B {\displaystyle B} for the property "black" and
13392-522: The logical conjunction ∧ {\displaystyle \land } requires terms on both sides. A proof system is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content. For instance, the classical rule of conjunction introduction states that P ∧ Q {\displaystyle P\land Q} follows from
13536-444: The logical constants known as existential quantifiers (' ∃ '), whose meaning corresponds to expressions like "there exists..." or "for some...". They are used to bind the variables in the expression following the quantifier. The ontological commitments of the theory then correspond to the variables bound by existential quantifiers. For example, the sentence "There are electrons" could be translated as " ∃ x Electron ( x ) ", in which
13680-529: The majority of analytic philosophers, who were mostly interested in systematic thinking, Quine evinced little interest in the philosophical canon : only once did he teach a course in the history of philosophy, on David Hume , in 1946. Over the course of his career, Quine published numerous technical and expository papers on formal logic, some of which are reprinted in his Selected Logic Papers and in The Ways of Paradox . His most well-known collection of papers
13824-410: The matter to empirical discovery when it seems we should have a formal distinction between referring and non-referring terms or elements of our domain. Lejewski writes further: This state of affairs does not seem to be very satisfactory. The idea that some of our rules of inference should depend on empirical information, which may not be forthcoming, is so foreign to the character of logical inquiry that
13968-407: The meager input and the torrential output is a relation that we are prompted to study for somewhat the same reasons that always prompted epistemology: namely, in order to see how evidence relates to theory, and in what ways one's theory of nature transcends any available evidence... But a conspicuous difference between old epistemology and the epistemological enterprise in this new psychological setting
14112-638: The meanings of their words, such as "No bachelor is married"— and "synthetic" statements, those true or false by virtue of facts about the world, such as "There is a cat on the mat." This distinction was central to logical positivism . Although Quine is not normally associated with verificationism , some philosophers believe the tenet is not incompatible with his general philosophy of language, citing his Harvard colleague B. F. Skinner and his analysis of language in Verbal Behavior . But Quine believes, with all due respect to his "great friend" Skinner, that
14256-517: The most discussed philosophical argument since Kant 's Transcendental Deduction of the Categories ". The central theses underlying it are ontological relativity and the related doctrine of confirmation holism . The premise of confirmation holism is that all theories (and the propositions derived from them) are under-determined by empirical data (data, sensory-data , evidence); although some theories are not justifiable, failing to fit with
14400-491: The notation of his writings on logic was often idiosyncratic. His later writings nearly always employed the now-dated notation of Principia Mathematica . Set against all this are the simplicity of his preferred method (as exposited in his Methods of Logic ) for determining the satisfiability of quantified formulas, the richness of his philosophical and linguistic insights, and the fine prose in which he expressed them. Most of Quine's original work in formal logic from 1960 onwards
14544-440: The notion of truth by definition was unsatisfactory. Quine's chief objection to analyticity is with the notion of cognitive synonymy (sameness of meaning). He argues that analytical sentences are typically divided into two kinds; sentences that are clearly logically true (e.g. "no unmarried man is married") and the more dubious ones; sentences like "no bachelor is married". Previously it was thought that if you can prove that there
14688-450: The other hand, I seem to be in a predicament. I cannot admit that there are some things which McX countenances and I do not, for in admitting that there are such things I should be contradicting my own rejection of them...This is the old Platonic riddle of nonbeing. Nonbeing must in some sense be, otherwise what is it that there is not? This tangled doctrine might be nicknamed Plato's beard : historically it has proved tough, frequently dulling
14832-409: The other hand, are true or false depending on whether they are in accord with reality. In formal logic, a sound argument is an argument that is both correct and has only true premises. Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For
14976-444: The other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like singular terms and predicates . For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by
15120-485: The premises P {\displaystyle P} and Q {\displaystyle Q} . Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are different types of proof systems including natural deduction and sequent calculi . A semantics is a system for mapping expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance,
15264-413: The premises offer support for the conclusion because the conclusion is the best explanation of why the premises are true. In this sense, abduction is also called the inference to the best explanation . For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean
15408-470: The premises. But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on
15552-509: The problem of the empty set unrealistic, which left Lejewski unsatisfied. The notion of ontological commitment plays a central role in Quine's contributions to ontology. A theory is ontologically committed to an entity if that entity must exist in order for the theory to be true. Quine proposed that the best way to determine this is by translating the theory in question into first-order predicate logic . Of special interest in this translation are
15696-485: The propositional connective "and". Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using truth-functional propositional connectives, their truth only depends on the truth values of their parts. But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects. Whether
15840-406: The propositions are formed. For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid. Classical logic is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in
15984-417: The psychology of argumentation. Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic. Non-deductive arguments make their conclusion probable but do not ensure that it is true. An example is the inductive argument from the empirical observation that "all ravens I have seen so far are black" to
16128-658: The rank of lieutenant commander. Quine could lecture in French, German, Italian, Portuguese, and Spanish as well as his native English. He had four children by two marriages. Guitarist Robert Quine was his nephew. Quine was politically conservative, but the bulk of his writing was in technical areas of philosophy removed from direct political issues. He did, however, write in defense of several conservative positions: for example, he wrote in defense of moral censorship ; while, in his autobiography, he made some criticisms of American postwar academics. At Harvard, Quine helped supervise
16272-510: The rest of his life. Quine received his B.A. summa cum laude in mathematics from Oberlin College in 1930, and his Ph.D. in philosophy from Harvard University in 1932. His thesis supervisor was Alfred North Whitehead . He was then appointed a Harvard Junior Fellow , which excused him from having to teach for four years. During the academic year 1932–33, he travelled in Europe thanks to
16416-436: The rules of inference they accept as valid and the formal languages used to express them. Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic. For example, it has been suggested that only logically complete systems, like first-order logic , qualify as logics. For such reasons, some theorists deny that higher-order logics are logics in
16560-492: The scope of mathematics. Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives . For instance, propositional logic represents the conjunction of two atomic propositions P {\displaystyle P} and Q {\displaystyle Q} as the complex formula P ∧ Q {\displaystyle P\land Q} . Unlike predicate logic where terms and predicates are
16704-418: The semantics for classical propositional logic assigns the formula P ∧ Q {\displaystyle P\land Q} the denotation "true" whenever P {\displaystyle P} and Q {\displaystyle Q} are true. From the semantic point of view, a premise entails a conclusion if the conclusion is true whenever the premise is true. A system of logic
16848-604: The semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. For over two thousand years, Aristotelian logic was treated as the canon of logic in the Western world, but modern developments in this field have led to a vast proliferation of logical systems. One prominent categorization divides modern formal logical systems into classical logic , extended logics, and deviant logics . Aristotelian logic encompasses
16992-518: The sense that it is based on basic logical intuitions shared by most logicians. These intuitions include the law of excluded middle , the double negation elimination , the principle of explosion , and the bivalence of truth. It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are
17136-404: The simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true under all interpretations of its non-logical terms. In some modal logics , this means that the proposition is true in all possible worlds. Some theorists define logic as
17280-415: The simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by theories of reference . Some complex propositions are true independently of the substantive meanings of their parts. In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like
17424-870: The singular term r {\displaystyle r} referring to the raven to form the expression B ( r ) {\displaystyle B(r)} . To express that some objects are black, the existential quantifier ∃ {\displaystyle \exists } is combined with the variable x {\displaystyle x} to form the proposition ∃ x B ( x ) {\displaystyle \exists xB(x)} . First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer ∃ x B ( x ) {\displaystyle \exists xB(x)} from B ( r ) {\displaystyle B(r)} . Extended logics are logical systems that accept
17568-474: The smallest units, propositional logic takes full propositions with truth values as its most basic component. Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition. First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates
17712-507: The sort of Boolean algebra employed in electrical engineering , and with Edward J. McCluskey , devised the Quine–McCluskey algorithm of reducing Boolean equations to a minimum covering sum of prime implicants . While his contributions to logic include elegant expositions and a number of technical results, it is in set theory that Quine was most innovative. He always maintained that mathematics required set theory and that set theory
17856-418: The streets are wet ( q {\displaystyle q} ). The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false. Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in
18000-437: The strict sense. When understood in a wide sense, logic encompasses both formal and informal logic. Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse. Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments. In this regard, it considers problems that formal logic on its own
18144-550: The strongest form of support: if their premises are true then their conclusion must also be true. This is not the case for ampliative arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into inductive and abductive arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens. Abductive arguments are inferences to
18288-438: The study of logical truths. Truth tables can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for
18432-700: The subject in two ways: either by affirming it or by denying it. For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one predicate in each case. Thus, these three propositions contain three predicates, referred to as major term , minor term , and middle term . The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how
18576-417: The subject to the predicate. For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is". The subject and the predicate are the terms of the proposition. Aristotelian logic does not contain complex propositions made up of simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using
18720-423: The symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted. Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism . It
18864-409: The table. This conclusion is justified because it is the best explanation of the current state of the kitchen. For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it
19008-399: The term ampliative or inductive reasoning is used. Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic. A deductively valid argument is one whose premises guarantee the truth of its conclusion. For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs"
19152-502: The term " abstract object ". He also, in his famous essay On What There is , coined the term " Plato's beard " to refer to the problem of empty names : Suppose now that two philosophers, McX and I, differ over ontology . Suppose McX maintains there is something which I maintain there is not. McX can, quite consistently with his own point of view, describe our difference of opinion by saying that I refuse to recognize certain entities...When I try to formulate our difference of opinion, on
19296-443: The traditional analytic-synthetic distinction and reductionism, undermining the then-popular logical positivism , advocating instead a form of semantic holism and ontological relativity . They also include the books The Web of Belief (1970), which advocates a kind of coherentism , and Word and Object (1960), which further developed these positions and introduced Quine's famous indeterminacy of translation thesis, advocating
19440-479: The truth values "true" and "false". The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression " p ∧ q {\displaystyle p\land q} " uses the logical connective ∧ {\displaystyle \land } ( and ). It could be used to express
19584-516: The ultimate reason is to be found in neurology and not in behavior. For him, behavioral criteria establish only the terms of the problem, the solution of which, however, lies in neurology . Like other analytic philosophers before him, Quine accepted the definition of "analytic" as "true in virtue of meaning alone". Unlike them, however, he concluded that ultimately the definition was circular . In other words, Quine accepted that analytic statements are those that are true by definition, then argued that
19728-485: The view that philosophy is not conceptual analysis , but continuous with science; it is the abstract branch of the empirical sciences. This led to his famous quip that " philosophy of science is philosophy enough". He led a "systematic attempt to understand science from within the resources of science itself" and developed an influential naturalized epistemology that tried to provide "an improved scientific explanation of how we have developed elaborate scientific theories on
19872-406: The whole sentences". For example, 'The author of Waverly was a poet' becomes 'some thing is such that it is the author of Waverly and was a poet and nothing else is such that it is the author of Waverly' . Using this sort of analysis with the word ' Pegasus ' (that which Quine is wanting to assert does not exist), he turns Pegasus into a description. Turning the word 'Pegasus' into a description
20016-480: Was a mythological winged horse we make sense, and moreover we speak the truth! If we speak the truth, this must be truth about something . So we cannot be speaking of nothing. Quine resists the temptation to say that non-referring terms are meaningless for reasons made clear above. Instead he tells us that we must first determine whether our terms refer or not before we know the proper way to understand them. However, Czesław Lejewski criticizes this belief for reducing
20160-405: Was discussed at length around the turn of the 20th century but it is not widely accepted today. Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex. A complex proposition has other propositions as its constituents, which are linked to each other through propositional connectives like "and" or "if...then". Simple propositions, on
20304-399: Was famous for his position that first order logic is the only kind worthy of the name, and developed his own system of mathematics and set theory, known as New Foundations . In the philosophy of mathematics , he and his Harvard colleague Hilary Putnam developed the Quine–Putnam indispensability argument , an argument for the reality of mathematical entities . He was the main proponent of
20448-441: Was on variants of his predicate functor logic , one of several ways that have been proposed for doing logic without quantifiers . For a comprehensive treatment of predicate functor logic and its history, see Quine (1976). For an introduction, see ch. 45 of his Methods of Logic . Quine was very warm to the possibility that formal logic would eventually be applied outside of philosophy and mathematics. He wrote several papers on
20592-481: Was quite distinct from logic. He flirted with Nelson Goodman 's nominalism for a while but backed away when he failed to find a nominalist grounding of mathematics. Over the course of his career, Quine proposed three axiomatic set theories. All three set theories admit a universal class, but since they are free of any hierarchy of types , they have no need for a distinct universal class at each type level. Quine's set theory and its background logic were driven by
20736-601: Was so severe that he struggled to continue following arguments. Quine also had considerable difficulty in his project to make the desired revisions to Word and Object . Before passing away, Quine noted to Morton White : "I do not remember what my illness is called, Althusser or Alzheimer , but since I cannot remember it, it must be Alzheimer." He died from the illness on Christmas Day in 2000. Quine's Ph.D. thesis and early publications were on formal logic and set theory . Only after World War II did he, by virtue of seminal papers on ontology , epistemology and language, emerge as
#431568