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217-471: Kripke made influential and original contributions to logic , especially modal logic. His principal contribution is a semantics for modal logic involving possible worlds , now called Kripke semantics . He received the 2001 Schock Prize in Logic and Philosophy. Kripke was also partly responsible for the revival of metaphysics and essentialism after the decline of logical positivism , claiming necessity

434-501: 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

651-474: A Bachelor of Science in Engineering (B.S.E.). Typically, A.B. students choose a major (called a concentration) at the end of sophomore year, while B.S.E. students declare at the end of their freshman year. Students must complete distribution requirements, departmental requirements, and independent work to graduate with either degree. A.B. students must complete distribution requirements in literature and

868-506: A bachelor's degree in mathematics. During his sophomore year at Harvard, he taught a graduate-level logic course at nearby MIT . Upon graduation he received a Fulbright Fellowship , and in 1963 was appointed to the Society of Fellows . Kripke later said, "I wish I could have skipped college. I got to know some interesting people but I can't say I learned anything. I probably would have learned it all anyway just reading on my own." His cousin

1085-444: 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 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

1302-587: A Collegiate Gothic project), Robert Venturi and Denise Scott Brown ( Frist Campus Center , among several others), Minoru Yamasaki (Robertson Hall), and Rafael Viñoly ( Carl Icahn Laboratory). A group of 20th-century sculptures scattered throughout the campus forms the Putnam Collection of Sculpture. It includes works by Alexander Calder ( Five Disks: One Empty ), Jacob Epstein ( Albert Einstein ), Henry Moore ( Oval with Points ), Isamu Noguchi ( White Sun ), and Pablo Picasso ( Head of

1519-611: A Woman ). Richard Serra 's The Hedgehog and The Fox is located between Peyton and Fine halls next to Princeton Stadium and the Lewis Library. At the southern edge of the campus is Lake Carnegie, an artificial lake named for Andrew Carnegie. Carnegie financed the lake's construction in 1906 at the behest of a friend and his brother who were both Princeton alumni. Carnegie hoped the opportunity to take up rowing would inspire Princeton students to forsake football , which he considered "not gentlemanly." The Shea Rowing Center on

1736-445: 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

1953-480: 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

2170-431: A class of finite frames. An application of this notion is the decidability question: it follows from Post's theorem that a recursively axiomatized modal logic L which has FMP is decidable, provided it is decidable whether a given finite frame is a model of L. In particular, every finitely axiomatizable logic with FMP is decidable. There are various methods for establishing FMP for a given logic. Refinements and extensions of

2387-533: A class of modal algebras, and a finite modal algebra can be transformed into a Kripke frame. As an example, Robert Bull proved using this method that every normal extension of S4.3 has FMP, and is Kripke complete. Kripke semantics has a straightforward generalization to logics with more than one modality. A Kripke frame for a language with { ◻ i ∣ i ∈ I } {\displaystyle \{\Box _{i}\mid \,i\in I\}} as

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2604-573: 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,

2821-439: A comprehensive collection of over 20,000 photographs. Approximately 750 works of African art are represented. The Museum oversees the outside John B. Putnam Jr., Memorial Collection of Sculpture . The Princeton University Chapel is located on the north side of campus near Nassau Street. It was built between 1924 and 1928 at a cost of $ 2.3 million, approximately $ 40.8 million adjusted for inflation in 2020. Ralph Adams Cram,

3038-425: 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

3255-455: A desire to provide direct access to works of art in a museum for a curriculum in the arts, an education system familiar to many European universities at the time. The museum took on the purposes of providing "exposure to original works of art and to teach the history of art through an encyclopedic collection of world art." Numbering over 112,000 objects, the collections range from ancient to contemporary art and come from Europe, Asia, Africa, and

3472-654: A fairly isolated dormitory; a security system was added, although the women deliberately broke it within a day. In 1971, Mary St. John Douglas and Susan Savage Speers became the first female trustees, and in 1974, quotas for men and women were eliminated. Following a 1979 lawsuit, the eating clubs were required to go coeducational in 1991, after an appeal to the U.S. Supreme Court was denied. In 2001, Princeton elected its first female president. The main campus consists of more than 200 buildings on 600 acres (2.4 km ) in Princeton, New Jersey. The James Forrestal Campus,

3689-510: 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

3906-686: 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)} "

4123-408: A frequent occurrence during Green's tenure, enrollment did increase under his administration. In 1823, James Carnahan became president, arriving as an unprepared and timid leader. With the college riven by conflicting views between students, faculty, and trustees, and enrollment hitting its lowest in years, Carnahan considered closing the university. Carnahan's successor, John Maclean Jr. , who

4340-415: 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

4557-551: 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

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4774-402: 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

4991-614: 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

5208-442: A major fundraising drive known as "A Campaign for Princeton" was conducted. President Harold T. Shapiro would succeed Bowen and remain president until 2001. Shapiro would continue to increase the endowment, expand academic programs, raise student diversity, and oversee the most renovations in Princeton's history. One of Shapiro's initiatives was the formation of the multidisciplinary Princeton Environmental Institute in 1994, renamed

5425-580: A paradox for skepticism about meaning . Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts. Saul Kripke was the oldest of three children born to Dorothy K. Kripke and Myer S. Kripke . His father was the leader of Beth El Synagogue, the only Conservative congregation in Omaha , Nebraska ; his mother wrote Jewish educational books for children. Saul and his two sisters, Madeline and Netta, attended Dundee Grade School and Omaha Central High School . Kripke

5642-456: A part of The New Plan of Study. He also brought about great expansion to the university, with the creation of the School of Architecture in 1919, the School of Engineering in 1921, and the School of Public and International Affairs in 1930. By the end of his presidency, the endowment had increased by 374 percent, the total area of the campus doubled, the faculty experienced impressive growth, and

5859-479: A program for servicemen, so they could resume their education once discharged. The post-war years saw scholars renewing broken bonds through numerous conventions, expansion of the campus, and the introduction of distribution requirements. The period saw the desegregation of Princeton, which was stimulated by changes to the New Jersey constitution. Princeton began undertaking a sharper focus towards research in

6076-584: A role similar to the Lindenbaum–Tarski algebra construction in algebraic semantics. A set of formulas is L - consistent if no contradiction can be derived from them using the axioms of L , and modus ponens . A maximal L-consistent set (an L - MCS for short) is an L -consistent set which has no proper L -consistent superset. The canonical model of L is a Kripke model ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } , where W

6293-537: A scholarly conference, a series of short plays, and an art project. In April 2018, university trustees announced that they would name two public spaces for James Collins Johnson and Betsey Stockton , enslaved people who lived and worked on Princeton's campus and whose stories were publicized by the project. In 2019, large-scale student activism again entered the mainstream concerning the school's implementation of federal Title IX policy relating to campus sexual assault . The activism consisted of sit-ins in response to

6510-664: 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

6727-457: A single accessibility relation R , and subsets D i  ⊆  W for each modality. Satisfaction is defined as: Carlson models are easier to visualize and to work with than usual polymodal Kripke models; there are, however, Kripke complete polymodal logics which are Carlson incomplete. In Semantical Considerations on Modal Logic , published in 1963, Kripke responded to a difficulty with classical quantification theory . The motivation for

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6944-407: A site for religious services and local celebrations. Measured on the exterior, the chapel is 277 feet (84 m) long, 76 feet (23 m) wide at its transepts , and 121 feet (37 m) high. The exterior is Pennsylvania sandstone , trimmed with Indiana limestone , and the interior is made of limestone and Aquia Creek sandstone . The design evokes characteristics of an English church of

7161-563: A smaller location designed mainly as a research and instruction complex, is split between nearby Plainsboro and South Brunswick . The campuses are situated about one hour from both New York City and Philadelphia on the train. The university also owns more than 520 acres (2.1 km ) of property in West Windsor Township , and is where Princeton is planning to construct a graduate student housing complex, which will be known as "Lake Campus North". The first building on campus

7378-602: A student's disciplinary sentence. In April 2024, students joined other campuses across the United States in protests and establishing encampments against the Israel–Hamas war and the alleged genocide of Palestinians in Gaza. The protestors called for divestment from Israel , started a hunger strike and were joined by faculty. The sit-in of Clio Hall led to arrests by police. Activism and protests continued in

7595-412: A system that uses the world-relative interpretation and preserves the classical rules. But the costs are severe. First, his language is artificially impoverished, and second, the rules for the propositional modal logic must be weakened. Kripke's possible worlds theory has been used by narratologists (beginning with Pavel and Dolezel) to understand "reader's manipulation of alternative plot developments, or

7812-772: A university focus on undergraduates. The university has ties with the Institute for Advanced Study, Princeton Theological Seminary , Rutgers University, and the Westminster Choir College of Rider University . Princeton is a member of the Association of American Universities , the Universities Research Association , and the National Association of Independent Colleges and Universities . The university

8029-465: Is Eric Kripke , known for creating the television show The Boys . After briefly teaching at Harvard, Kripke moved in 1968 to Rockefeller University in New York City, where he taught until 1976. In 1978 he took a chaired professorship at Princeton University . In 1988 he received the university's Behrman Award for distinguished achievement in the humanities. In 2002 Kripke began teaching at

8246-463: Is complete with respect to C if L  ⊇ Thm( C ). Semantics is useful for investigating a logic (i.e., a derivation system) only if the semantical entailment relation reflects its syntactical counterpart, the consequence relation ( derivability ). It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and for them, to determine which class it is. For any class C of Kripke frames, Thm( C )

8463-420: 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

8680-521: Is 3–4 hours a week, although this can vary depending on the course. The student to faculty ratio is 5 to 1, and a majority of classes have fewer than 20 students. In the Fiske Guide to Colleges , academic culture is considered as "tight-knit, extremely hardworking, highly cooperative, and supportive." Undergraduates agree to adhere to an academic integrity policy called the Honor Code. Under

8897-401: Is Christopher Eisgruber, who was appointed by the university's board of trustees in 2013. The board is responsible for the overall direction of the university. It consists of no fewer than 23 and no more than 40 members at any one time, with the president of the university and the governor of New Jersey serving as ex officio members . It approves the operating and capital budgets, supervises

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9114-467: Is a normal modal logic (in particular, theorems of the minimal normal modal logic, K , are valid in every Kripke model). However, the converse does not hold generally. There are Kripke incomplete normal modal logics, which is unproblematic, because most of the modal systems studied are complete of classes of frames described by simple conditions. A normal modal logic L corresponds to a class of frames C , if C  = Mod( L ). In other words, C

9331-803: Is a partially ordered Kripke frame, and ⊩ {\displaystyle \Vdash } satisfies the following conditions: Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the Finite Model Property. Intuitionistic first-order logic Let L be a first-order language. A Kripke model of L is a triple ⟨ W , ≤ , { M w } w ∈ W ⟩ {\displaystyle \langle W,\leq ,\{M_{w}\}_{w\in W}\rangle } , where ⟨ W , ≤ ⟩ {\displaystyle \langle W,\leq \rangle }

9548-481: Is a Kripke frame, and ⊩ {\displaystyle \Vdash } is a relation between nodes of W and modal formulas, such that: We read w ⊩ A {\displaystyle w\Vdash A} as " w satisfies A ", " A is satisfied in w ", or " w forces A ". The relation ⊩ {\displaystyle \Vdash } is called the satisfaction relation , evaluation , or forcing relation . The satisfaction relation

9765-467: Is a metaphysical notion distinct from the epistemic notion of a priori , and that there are necessary truths that are known a posteriori , such as that water is H 2 O. A 1970 Princeton lecture series, published in book form in 1980 as Naming and Necessity , is considered one of the most important philosophical works of the 20th century. It introduced the concept of names as rigid designators , designating (picking out, denoting, referring to)

9982-476: 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

10199-441: 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

10416-482: Is accredited by the Middle States Commission on Higher Education (MSCHE), with its last reaffirmation in 2014. Princeton University's endowment of $ 37 billion (per 2021 figures) was ranked as the fourth largest endowment in the United States, and it had the greatest per-student endowment in the world at over $ 4.4 million per student. The endowment is sustained through continued donations and

10633-416: 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

10850-453: 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

11067-401: Is an intuitionistic Kripke frame, M w is a (classical) L -structure for each node w  ∈  W , and the following compatibility conditions hold whenever u  ≤  v : Given an evaluation e of variables by elements of M w , we define the satisfaction relation w ⊩ A [ e ] {\displaystyle w\Vdash A[e]} : Here e ( x → a ) is

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11284-415: 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

11501-417: Is canonical. In general, it is undecidable whether a given axiom is canonical. We know a nice sufficient condition: H. Sahlqvist identified a broad class of formulas (now called Sahlqvist formulas ) such that: This is a powerful criterion: for example, all axioms listed above as canonical are (equivalent to) Sahlqvist formulas. A logic has the finite model property (FMP) if it is complete with respect to

11718-421: 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

11935-407: 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

12152-453: 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

12369-566: 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. Princeton University This is an accepted version of this page Princeton University is a private Ivy League research university in Princeton, New Jersey , United States. Founded in 1746 in Elizabeth as

12586-504: Is governed by the Trustees of Princeton University and has an endowment of $ 37.7 billion, the largest endowment per student in the United States. Princeton provides undergraduate and graduate instruction in the humanities , social sciences , natural sciences , and engineering to approximately 8,500 students on its main campus spanning 600 acres (2.4 km ) within the borough of Princeton. It offers postgraduate degrees through

12803-610: 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

13020-432: 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

13237-415: 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

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13454-410: Is itself canonical. It follows from the preceding discussion that any logic axiomatized by a canonical set of formulas is Kripke complete, and compact . The axioms T, 4, D, B, 5, H, G (and thus any combination of them) are canonical. GL and Grz are not canonical, because they are not compact. The axiom M by itself is not canonical ( Goldblatt , 1991), but the combined logic S4.1 (in fact, even K4.1 )

13671-512: Is maintained by investment advisers. Princeton's operating budget is over $ 2 billion per year, with 50% going to academic departments and programs, 33% to administrative and student service departments, 10% to financial aid departments, and 7% to the Princeton Plasma Physics Laboratory . Princeton follows a liberal arts curriculum, and offers two bachelor's degrees to students: a Bachelor of Arts (A.B.) and

13888-447: 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,

14105-688: 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

14322-518: 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

14539-407: 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

14756-541: 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

14973-430: 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

15190-622: Is the largest class of frames such that L is sound wrt C . It follows that L is Kripke complete if and only if it is complete of its corresponding class. Consider the schema T  : ◻ A → A {\displaystyle \Box A\to A} . T is valid in any reflexive frame ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } : if w ⊩ ◻ A {\displaystyle w\Vdash \Box A} , then w ⊩ A {\displaystyle w\Vdash A} since w   R   w . On

15407-466: Is the set of all L - MCS , and the relations R and ⊩ {\displaystyle \Vdash } are as follows: The canonical model is a model of L , as every L - MCS contains all theorems of L . By Zorn's lemma , each L -consistent set is contained in an L - MCS , in particular every formula unprovable in L has a counterexample in the canonical model. The main application of canonical models are completeness proofs. Properties of

15624-410: 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

15841-432: 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

16058-540: 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

16275-416: Is uniquely determined by its value on propositional variables. A formula A is valid in: We define Thm( C ) to be the set of all formulas that are valid in C . Conversely, if X is a set of formulas, let Mod( X ) be the class of all frames which validate every formula from X . A modal logic (i.e., a set of formulas) L is sound with respect to a class of frames C , if L  ⊆ Thm( C ). L

16492-599: 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)

16709-431: 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

16926-549: The College of New Jersey , Princeton is the fourth-oldest institution of higher education in the United States and one of the nine colonial colleges chartered before the American Revolution . The institution moved to Newark in 1747 and then to its Mercer County campus in Princeton nine years later. It officially became a university in 1896 and was subsequently renamed Princeton University. The university

17143-726: The CUNY Graduate Center , and in 2003 he was appointed a distinguished professor of philosophy there. Kripke has received honorary degrees from the University of Nebraska , Omaha (1977), Johns Hopkins University (1997), University of Haifa , Israel (1998), and the University of Pennsylvania (2005). He was a member of the American Philosophical Society and an elected Fellow of the American Academy of Arts and Sciences , and in 1985

17360-625: The College of William & Mary in Williamsburg, Virginia ; it was also where some of the founders preached. Although their initial request was rejected by the Anglican governor Lewis Morrison , the acting governor after Morrison's death, John Hamilton , granted a charter for the College of New Jersey on October 22, 1746. In 1747, approximately five months after acquiring the charter,

17577-456: The GL - tautology ◻ A → ◻ ◻ A {\displaystyle \Box A\to \Box \Box A} . For any normal modal logic L , a Kripke model (called the canonical model ) can be constructed, which validates precisely the theorems of L , by an adaptation of the standard technique of using maximal consistent sets as models. Canonical Kripke models play

17794-689: The High Meadows Environmental Institute in 2020. In 2001, Princeton shifted the financial aid policy to a system that replaced all loans with grants. That same year, Princeton elected its first female president, Shirley M. Tilghman . Before retiring in 2012, Tilghman expanded financial aid offerings and conducted several major construction projects like the Lewis Center for the Arts and a sixth residential college. Tilghman also lead initiatives for more global programs,

18011-671: The Institute for Advanced Study was founded to provide a space for the influx of scientists, such as Albert Einstein . Many Princeton scientists would work on the Manhattan Project during the war, including the entire physics department. During World War II, Princeton offered an accelerated program for students to graduate early before entering the armed forces. Student enrollment fluctuated from month to month, and many faculty were forced to teach unfamiliar subjects. Still, Dodds maintained academic standards and would establish

18228-728: The Princeton School of Public and International Affairs , the School of Engineering and Applied Science , the School of Architecture and the Bendheim Center for Finance . The university also manages the Department of Energy's Princeton Plasma Physics Laboratory and is home to the NOAA's Geophysical Fluid Dynamics Laboratory . It is classified among "R1: Doctoral Universities – Very high research activity" and has one of

18445-752: The Princeton Tigers , has won the most titles in its conference and has sent many students and alumni to the Olympics . As of October 2021, 75 Nobel laureates , 16 Fields Medalists and 16 Turing Award laureates have been affiliated with Princeton University as alumni, faculty members, or researchers. In addition, Princeton has been associated with 21 National Medal of Science awardees, 5 Abel Prize awardees, 11 National Humanities Medal recipients, 217 Rhodes Scholars , 137 Marshall Scholars , and 62 Gates Cambridge Scholars . Two U.S. presidents , twelve U.S. Supreme Court justices (three of whom serve on

18662-720: The Rutgers-Princeton Cannon War . A compromise between the presidents of Princeton and Rutgers ended the war and forced the return of the Little Cannon to Princeton. The protruding cannons are occasionally painted scarlet by Rutgers students who continue the traditional dispute. Though art collection at the university dates back to its very founding, the Princeton University Art Museum was not officially established until 1882 by President McCosh. Its establishment arose from

18879-463: The School of Architecture , the School of Engineering and Applied Science , and the School of Public and International Affairs . Additionally, the school's Bendheim Center for Finance provides education for the area of money and finance in lieu of a business school . Princeton did host a Princeton Law School for a short period, before eventually closing in 1852 due to poor income. Princeton's lack of other professional schools can be attributed to

19096-469: The accessibility relation . Depending on the properties of the accessibility relation ( transitivity , reflexivity, etc.), the corresponding frame is described, by extension, as being transitive, reflexive, etc. A Kripke model is a triple ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } , where ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle }

19313-451: The 2008 plan, Princeton has aimed at reducing its carbon dioxide emissions to 1990 levels without the purchase of market offsets and predicts to meet the goal by 2026 (the former goal was by 2020 but COVID-19 requirements delayed this). Princeton released its second Sustainability Action Plan in 2019 on Earth Day with its main goal being reducing campus greenhouse gases to net zero by 2046 as well as other objectives building on those in

19530-734: The 2008 plan. In 2021, the university agreed to divest from thermal coal and tar sand segments of the fossil fuel industry and from companies that are involved in climate disinformation after student protest. Princeton is a member of the Ivy Plus Sustainability Consortium, through which it has committed to best-practice sharing and the ongoing exchange of campus sustainability solutions along with other member institutions. Princeton's Sustainability Action Plan also aims to have zero waste through recycling programs, sustainable purchasing, and behavioral and operational strategies. Princeton's 20th and current president

19747-630: The Americas. The museum's art is divided into ten extensive curatorial areas. There is a collection of Greek and Roman antiquities , including ceramics, marbles, bronzes, and Roman mosaics from faculty excavations in Antioch , as well as other art from the ancient Egyptian, Byzantium , and Islamic worlds. The non-Islamic coins were catalogued by Dorothy B. Waage . Medieval Europe is represented by sculpture, metalwork, and stained glass. The collection of Western European paintings includes examples from

19964-494: The College of New Jersey initially suggested that Nassau Hall be named in recognition of Belcher because of his interest in the institution; the governor vetoed the request. Burr, who would die in 1757, devised a curriculum for the school and enlarged the student body. Following the untimely death of Burr and the college's next three presidents , John Witherspoon became president in 1768 and remained in that post until his death in 1794. With his presidency, Witherspoon focused

20181-469: 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

20398-484: The Honor Code, faculty do not proctor examinations; instead, the students proctor one another and must report any suspected violation to an Honor Committee made up of undergraduates. The Committee investigates reported violations and holds a hearing if it is warranted. An acquittal at such a hearing results in the destruction of all records of the hearing; a conviction results in the student's suspension or expulsion. Violations pertaining to all other academic work fall under

20615-599: The Middle Ages . The extensive iconography, in stained glass, stonework, and wood carvings, has the common theme of connecting religion and scholarship. Published in 2008, the Sustainability Action Plan was the first formal plan for sustainability enacted by the university. It focused on reducing greenhouse gas emissions, conservation of resources, and research, education, and civic engagement for sustainability through 10 year objectives. Since

20832-654: The Synod and devised a plan to establish a new college, for they were disappointed with Harvard and Yale 's opposition to the Great Awakening and dissatisfied with the limited instruction at the Log College. They convinced three other Presbyterians to join them and decided on New Jersey as the location for the college, as at the time, there was no institution between Yale College in New Haven, Connecticut , and

21049-623: The Third World Center (now known as the Carl A. Fields Center) was dedicated in 1971. Goheen also oversaw great expansion for the university, with square footage increasing by 80 percentage. Throughout the 1960s and 1970s, Princeton experienced unprecedented activism, with most of it centered on the Vietnam War . While Princeton activism initially remained relatively timid compared to other institutions, protests began to grow with

21266-622: The United States) from June 30, 1783, to November 4, 1783. Since 1911, the front entrance has been flanked by two bronze tigers, a gift of the Princeton Class of 1879, which replaced two lions previously given in 1889. Starting in 1922, commencement has been held on the front lawn of Nassau Hall when there is good weather. In 1966, Nassau Hall was added to the National Register of Historic Places . Nowadays, it houses

21483-733: The administration commissioned a report on admitting women. The final report was issued in January 1969, supporting the idea. That same month, Princeton's trustees voted 24–8 in favor of coeducation and began preparing the institution for the transition. The university finished these plans in April 1969 and announced there would be coeducation in September. Ultimately, 101 female freshman and 70 female transfer students enrolled at Princeton in September 1969. Those admitted were housed in Pyne Hall,

21700-418: 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

21917-412: The armed forces, with 151 dying during the war. After the war, enrollment spiked and the trustees established the system of selective admission in 1922. From the 1920s to the 1930s, the student body featured many students from preparatory schools, zero Black students, and dwindling Jewish enrollment because of quotas. Aside from managing Princeton during WWI, Hibben introduced the senior thesis in 1923 as

22134-445: The arts, science and engineering, social analysis, cultural difference, epistemology and cognition, ethical thought and moral values, historical analysis, and quantitative and computational reasoning; they must also have satisfactory ability in a foreign language. Additionally, they must complete two papers of independent work during their junior year—known as the junior papers—and craft a senior thesis to graduate; both revolve around

22351-399: 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

22568-495: 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

22785-487: 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

23002-574: The board of trustees and dissatisfaction with his administration, he was forced to resign in 1902. Following Patton's resignation, Woodrow Wilson , an alumnus and popular professor, was elected the 13th president of the university. Noticing falling academic standards, Wilson orchestrated significant changes to the curriculum, where freshman and sophomores followed a unified curriculum while juniors and seniors concentrated study in one discipline. Ambitious seniors were allowed to undertake independent work, which would eventually shape Princeton's emphasis on

23219-648: The campus R.O.T.C. chapter in a game of touch football . As the years went on, the protests' agenda broadened to investments in South Africa, environmental issues, and women's rights. In response to these broadening protests, the Council of the Princeton University Community (CPUC) was founded to serve as a method for greater student voice in governance. Activism culminated in 1970 with a student, faculty, and staff member strike , so

23436-494: The cannon was returned to Princeton and placed at the eastern end of town. Two years later, it was moved to the campus under cover of night by Princeton students, and in 1840, it was buried in its current location. A second "Little Cannon" is buried in the lawn in front of nearby Whig Hall. The cannon, which may also have been captured in the Battle of Princeton, was stolen by students of Rutgers University in 1875. The theft ignited

23653-407: The canonical model construction often work, using tools such as filtration or unravelling. As another possibility, completeness proofs based on cut-free sequent calculi usually produce finite models directly. Most of the modal systems used in practice (including all listed above) have FMP. In some cases, we can use FMP to prove Kripke completeness of a logic: every normal modal logic is complete wrt

23870-437: The canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames. This argument does not work for arbitrary L , because there is no guarantee that the underlying frame of the canonical model satisfies the frame conditions of L . We say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if A union of canonical sets of formulas

24087-403: The certificates effectively serve as one. Course structure is determined by the instructor and department. Classes vary in their format, ranging from small seminars to medium-sized lecture courses to large lecture courses. The latter two typically have precepts , which are extra weekly discussion sessions that are led by either the professor or a graduate student. The average class meeting time

24304-597: The characters' planned or fantasized alternative action series." This application has become especially useful in the analysis of hyperfiction . Kripke semantics for intuitionistic logic follows the same principles as the semantics of modal logic, but uses a different definition of satisfaction. An intuitionistic Kripke model is a triple ⟨ W , ≤ , ⊩ ⟩ {\displaystyle \langle W,\leq ,\Vdash \rangle } , where ⟨ W , ≤ ⟩ {\displaystyle \langle W,\leq \rangle }

24521-645: 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

24738-592: The college became the site for the Battle of Princeton . During the battle, British soldiers briefly occupied Nassau Hall before eventually surrendering to American forces led by General George Washington . During the summer and fall of 1783, the Continental Congress and Washington met in Nassau Hall, making Princeton the country's capital for four months; Nassau Hall is where Congress learned of

24955-473: The college on preparing a new generation of both educated clergy and secular leadership in the new American nation. To this end, he tightened academic standards, broadened the curriculum, solicited investment for the college, and grew its size. A signatory of the Declaration of Independence , Witherspoon and his leadership led the college to becoming influential to the American Revolution . In 1777,

25172-473: The college's president in 1868, and lifted the institution out of a low period that had been brought about by the war. During his two decades of service, he overhauled the curriculum, oversaw an expansion of inquiry into the sciences, recruited distinguished faculty, and supervised the addition of a number of buildings in the High Victorian Gothic style to the campus. McCosh's tenure also saw

25389-495: 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,

25606-628: The concentration they are pursuing. B.S.E. majors complete fewer courses in the humanities and social sciences and instead fulfill requirements in mathematics, physics, chemistry, and computer programming. They likewise must complete independent work, which typically involves a design project or senior thesis, but not the junior papers. A.B. majors must complete 31 courses, whereas B.S.E. majors must complete 36 courses. Students can choose from either 36 concentrations or create their own. They can also participate in 55 interdisciplinary certificate programs; since Princeton does not offer an academic minor ,

25823-502: 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

26040-511: 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

26257-458: 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

26474-591: 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

26691-460: The construction of two new buildings. In 1807, a large student riot occurred at Nassau Hall, spurred by underlying distrust of educational reforms by Smith away from the Church. Following Smith's mishandling of the situation, falling enrollment, and faculty resignations, the trustees of the university offered resignation to Smith, which he accepted. In 1812, Ashbel Green was unanimously elected by

26908-438: 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

27125-470: The court as of 2010 ) and numerous living industry and media tycoons and foreign heads of state are all counted among Princeton's alumni body. Princeton has graduated many members of the U.S. Congress and the U.S. Cabinet , including eight secretaries of state , three secretaries of defense and two chairmen of the Joint Chiefs of Staff . Princeton University, founded as the College of New Jersey,

27342-589: The creation and rise of many extracurricular activities, like the Princeton Glee Club , the Triangle Club , the first intercollegiate football team, and the first permanent eating club , as well as the elimination of fraternities and sororities. In 1879, Princeton conferred its first doctorates on James F. Williamson and William Libby, both members of the Class of 1877. Francis Patton took

27559-454: The creation of an office of sustainability, and investments into the sciences. Princeton's 20th and current president, Christopher Eisgruber , was elected in 2013. In 2017, Princeton University unveiled a large-scale public history and digital humanities investigation into its historical involvement with slavery called the Princeton & Slavery Project. The project saw the publication of hundreds of primary sources, 80 scholarly essays,

27776-453: The definition of ⊩ {\displaystyle \Vdash } . T corresponds to the class of reflexive Kripke frames. It is often much easier to characterize the corresponding class of L than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose L 1  ⊆  L 2 are normal modal logics that correspond to

27993-451: 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

28210-662: The early Renaissance through the 19th century, with pieces by Monet , Cézanne , and Van Gogh , and features a growing collection of 20th-century and contemporary art, including paintings such as Andy Warhol 's Blue Marilyn. The museum features a collection of Chinese and Japanese art, with holdings in bronzes, tomb figurines, painting, and calligraphy , as well as collections of Korean, Southeast, and Central Asian art. Its collection of pre-Columbian art includes examples of Mayan and Olmec art, and its indigenous art ranges from Chile to Alaska to Greenland. The museum has collections of old master prints and drawings, and it has

28427-603: The end of the 19th century, much of Princeton's architecture was designed by the Cope and Stewardson firm (led by the same University of Pennsylvania professors of architecture who designed a large part of Washington University in St. Louis and University of Pennsylvania ) resulting in the Collegiate Gothic style for which Princeton is known for today. Implemented initially by William Appleton Potter , and later enforced by

28644-648: The enrollment doubled. Hibben's successor, Harold Willis Dodds would lead the university through the Great Depression , World War II , and the Korean Conflict . With the Great Depression, many students were forced to withdraw due to financial reasons. At the same time, Princeton's reputation in physics and mathematics surged as many European scientists left for the United States due to uneasy tension caused by Nazi Germany . In 1930,

28861-457: The evaluation which gives x the value a , and otherwise agrees with e . The three lectures that form Naming and Necessity constitute an attack on the descriptivist theory of names . Kripke attributes variants of descriptivist theories to Frege , Russell , Wittgenstein , and John Searle , among others. According to descriptivist theories, proper names either are synonymous with descriptions, or have their reference determined by virtue of

29078-409: 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

29295-434: 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

29512-507: The first half of the 20th century. Due to the tightening of academic standards, enrollment declined severely until 1907. In 1906, the reservoir Lake Carnegie was created by Andrew Carnegie , and the university officially became nonsectarian . Before leaving office, Wilson strengthened the science program to focus on "pure" research and broke the Presbyterian lock on the board of trustees. However, he did fail in winning support for

29729-486: The following year in the Graduate School. In 1964, T'sai-ying Cheng became the first woman at Princeton to receive a Ph.D. In 1963, five women came to Princeton for one year to study "critical languages" as undergraduates, but were not candidates for a Princeton degree. Following abortive discussions with Sarah Lawrence College to relocate the women's college to Princeton and merge it with the university in 1967,

29946-485: 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

30163-513: 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

30380-523: 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

30597-419: 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

30814-521: The former written when he was a teenager, were on modal logic . The most familiar logics in the modal family are constructed from a weak logic called K, named after Kripke. Kripke introduced the now-standard Kripke semantics (also known as relational semantics or frame semantics) for modal logics. Kripke semantics is a formal semantics for non-classical logic systems. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The discovery of Kripke semantics

31031-639: The founding of a local chapter of Students for a Democratic Society (SDS) in 1965, which organized many of the later Princeton protests. In 1966, the SDS gained prominence on campus following picketing against a speech by President Lyndon B. Johnson , which gained frontpage coverage by the New York Times. A notable point of contention on campus was the Institute for Defense Analyses (IDA) and would feature multiple protests, some of which required police action. In 1967, SDS members and sympathizers beat

31248-410: The future as special cases, but does not plan to make general admissions of women graduate students." The student-run Daily Princetonian ran four articles about Meservey in one issue, including an editorial lamenting the potential "far reaching implications" of Meservey's admission which concluded: "Princeton is unique as an undergraduate men's college and must remain so." Eight more women enrolled

31465-421: 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

31682-494: The honor system was established, allowing for unproctored exams. In 1896, the college officially became a university, and as a result, it officially changed its name to Princeton University. In 1900, the Graduate School was formally established. Even with such accomplishments, Patton's administration remained lackluster with its administrative structure and towards its educational standards. Due to profile changes in

31899-406: 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

32116-407: 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

32333-474: The investment of the university's endowment, and oversees campus real estate and long-range physical planning. The trustees also exercise prior review and approval concerning changes in major policies such as those in instructional programs and admission as well as tuition and fees and the hiring of faculty members. The university is composed of the Undergraduate College, the Graduate School ,

32550-505: The jurisdiction of the Faculty-Student Committee on Discipline. Undergraduates are expected to sign a pledge on their written work affirming that they have not plagiarized the work. The first focus on issues of grade inflation by the Princeton administration began in 1998 when a university report was released showcasing a steady rise in undergraduate grades from 1973 to 1997. Subsequent reports and discussion from

32767-486: The lake's shore continues to serve as the headquarters for Princeton rowing. Princeton's grounds were designed by Beatrix Farrand between 1912 and 1943. Her contributions were most recently recognized with the naming of a courtyard for her. Subsequent changes to the landscape were introduced by Quennell Rothschild & Partners in 2000. In 2005, Michael Van Valkenburgh was hired as the new consulting landscape architect for Princeton's 2016 Campus Plan. Lynden B. Miller

32984-453: The land was converted to commercial and residential spaces. Robert Goheen would succeed Dodds by unanimous vote and serve as president until 1972. Goheen's presidency was characterized as being more liberal than previous presidents, and his presidency would see a rise in Black applicants, as well as the eventual coeducation of the university in 1969. During this period of rising diversity,

33201-490: The largest libraries in the colonies. In 1756, the college moved again to its present home in Princeton, New Jersey , because Newark was felt to be too close to New York. Princeton was chosen for its location in central New Jersey and by strong recommendation by Belcher. The college's home in Princeton was Nassau Hall , named for the royal William III of England , a member of the House of Orange-Nassau . The trustees of

33418-460: The largest university libraries in the world. Princeton uses a residential college system and is known for its eating clubs for juniors and seniors. The university has over 500 student organizations. Princeton students embrace a wide variety of traditions from both the past and present. The university is an NCAA Division I school and competes in the Ivy League. The school's athletic team,

33635-447: The library. In 1961, Princeton admitted its first female graduate student, Sabra Follett Meservey, who would go on to be the first woman to earn a master's degree at Princeton. Meservey was, at the time of her admission, already a member of the faculty at Douglass College within Princeton. The dean of the graduate school issued a statement clarifying that Meservey's admission was an exception, and that "Princeton may permit other women in

33852-522: 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

34069-456: The name's being associated with a description or cluster of descriptions that an object uniquely satisfies. Kripke rejects both these kinds of descriptivism. He gives several examples purporting to render descriptivism implausible as a theory of how names get their references determined (e.g., surely Aristotle could have died at age two and so not satisfied any of the descriptions we associate with his name, but it would seem wrong to deny that he

34286-438: The new academic year starting September 2024 with administrators facing calls for resignation from faculty. Princeton explicitly prohibited the admission of women from its founding in 1746 until 1969. Since it lacked an affiliated women's college, it was often referred to as a "monastery", both lovingly and derisively, by members of the Princeton community. For about a decade, from 1887 to 1897, nearby Evelyn College for Women

34503-449: The office of the university president and other administrative offices. To the south of Nassau Hall lies a courtyard that is known as Cannon Green. Buried in the ground at the center is the "Big Cannon", which was left in Princeton by British troops as they fled following the Battle of Princeton. It remained in Princeton until the War of 1812 , when it was taken to New Brunswick . In 1836,

34720-488: The other hand, a frame which validates T has to be reflexive: fix w  ∈  W , and define satisfaction of a propositional variable p as follows: u ⊩ p {\displaystyle u\Vdash p} if and only if w   R   u . Then w ⊩ ◻ p {\displaystyle w\Vdash \Box p} , thus w ⊩ p {\displaystyle w\Vdash p} by T , which means w   R   w using

34937-409: 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

35154-444: 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

35371-496: The peace treaty between the colonies and the British. The college did suffer from the revolution, with a depreciated endowment and hefty repair bills for Nassau Hall. In 1795, President Samuel Stanhope Smith took office, the first alumnus to become president. Nassau Hall suffered a large fire that destroyed its interior in 1802, which Smith blamed on rebellious students. The college raised funds for reconstruction, as well as

35588-486: The permanent location of the Graduate School and the elimination of the eating clubs, which he proposed replacing with quadrangles, a precursor to the residential college system. Wilson also continued to keep Princeton closed off from accepting Black students. When an aspiring Black student wrote a letter to Wilson, he got his secretary to reply telling him to attend a university where he would be more welcome. John Grier Hibben became president in 1912, and would remain in

35805-512: The post for two decades. On October 2, 1913, the Princeton University Graduate College was dedicated. When the United States entered World War I in 1917, Hibben allocated all available University resources to the government. As a result, military training schools opened on campus and laboratories and other facilities were used for research and operational programs. Overall, more than 6,000 students served in

36022-463: The practice for the future. Wilson further reformed the educational system by introducing the preceptorial system in 1905, a then-unique concept in the United States that augmented the standard lecture method of teaching with a more personal form in which small groups of students, or precepts, could interact with a single instructor, or preceptor, in their field of interest. The changes brought about many new faculty and cemented Princeton's academics for

36239-634: The predecessor of the university, the idea has been rebuked by Princeton historians. The founding of the university itself originated from a split in the Presbyterian church following the Great Awakening . In 1741, New Light Presbyterians were expelled from the Synod of Philadelphia in defense of how the Log College ordained ministers. The four founders of the College of New Jersey, who were New Lights, were either expelled or withdrew from

36456-485: 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,

36673-413: 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

36890-470: 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

37107-408: The presidency in 1888, and although his election was not met by unanimous enthusiasm, he was well received by undergraduates. Patton's administration was marked by great change, for Princeton's enrollment and faculty had doubled. At the same time, the college underwent large expansion and social life was changing in reflection of the rise in eating clubs and burgeoning interest in athletics. In 1893,

37324-485: 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

37541-406: 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

37758-417: 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

37975-564: The report culminated to when in 2004, Nancy Weiss Malkiel , the dean of the college, implemented a grade deflation policy to address the findings. Malkiel's reason for the policy was that an A was becoming devalued as a larger percentage of the student body received one. Following its introduction, the number of A's and average GPA on campus dropped, although A's and B's were still the most frequent grades awarded. The policy received mixed approval from both faculty and students when first instituted. Criticism for grade deflation continued through

38192-436: 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

38409-456: The same class of frames, but L 1 does not prove all theorems of L 2 . Then L 1 is Kripke incomplete. For example, the schema ◻ ( A ≡ ◻ A ) → ◻ A {\displaystyle \Box (A\equiv \Box A)\to \Box A} generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse well-founded frames), but does not prove

38626-701: The same object in every possible world, as contrasted with descriptions . It also established Kripke's causal theory of reference , disputing the descriptivist theory found in Gottlob Frege 's concept of sense and Bertrand Russell 's theory of descriptions . Kripke is often seen in opposition to the other great late-20th-century philosopher to eschew logical positivism: W. V. O. Quine . Quine rejected essentialism and modal logic. Kripke also gave an original reading of Ludwig Wittgenstein , known as " Kripkenstein ", in his Wittgenstein on Rules and Private Language . The book contains his rule-following argument,

38843-410: The school was open to those of any religious denomination, with many of the founders being of Presbyterian faith, the college became the educational and religious capital of Scotch-Irish Presbyterian America. In 1747, following the death of then President Jonathan Dickinson, the college moved from Elizabeth to Newark, New Jersey , as that was where presidential successor Aaron Burr Sr. 's parsonage

39060-492: 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

39277-418: 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

39494-604: 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

39711-518: 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

39928-599: The set of its necessity operators consists of a non-empty set W equipped with binary relations R i for each i  ∈  I . The definition of a satisfaction relation is modified as follows: A simplified semantics, discovered by Tim Carlson, is often used for polymodal provability logics . A Carlson model is a structure ⟨ W , R , { D i } i ∈ I , ⊩ ⟩ {\displaystyle \langle W,R,\{D_{i}\}_{i\in I},\Vdash \rangle } with

40145-404: 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

40362-415: 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

40579-870: 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

40796-474: 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

41013-418: 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

41230-437: 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

41447-550: 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

41664-438: 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

41881-700: 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

42098-417: 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

42315-423: 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

42532-409: 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

42749-399: 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"

42966-532: The trustees elected Jonathan Dickinson as president and opened in Elizabeth, New Jersey , where classes were held in Dickinson's parsonage . With its founding, it became the fourth-oldest institution of higher education in the United States, and one of nine colonial colleges chartered before the American Revolution . The founders aimed for the college to have an expansive curriculum to teach people of various professions, not solely ministerial work. Though

43183-483: The trustees of the college to become the eighth president. After the liberal tenure of Smith, Green represented the conservative "Old Side", in which he introduced rigorous disciplinary rules and heavily embraced religion. Even so, believing the college was not religious enough, he took a prominent role in establishing the Princeton Theological Seminary next door. While student riots were

43400-479: 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

43617-473: The university could become an "institution against expansion of the war." Princeton's protests would taper off later that year, with The Daily Princetonian saying that, "Princeton 1970–71 was an emotionally burned out university." In 1982, the residential college system was officially established under Goheen's successor William G. Bowen , who would serve until 1988. During his presidency, Princeton's endowment increased from $ 625 million to $ 2 billion, and

43834-712: The university through the American Civil War . When Nassau Hall burned down again in 1855, Maclean raised funds and used the money to rebuild Nassau Hall and run the university on an austerity budget during the war years. With a third of students from the college being from the South, enrollment fell. Once many of the Southerners left, the campus became a sharp proponent for the Union , even bestowing an honorary degree to President Lincoln . James McCosh became

44051-586: The university's supervising architect, Ralph Adams Cram , the Collegiate Gothic style remained the standard for all new building on the Princeton campus until 1960. A flurry of construction projects in the 1960s produced a number of new buildings on the south side of the main campus, many of which have been poorly received. Several prominent architects have contributed some more recent additions, including Frank Gehry (Lewis Library), I. M. Pei (Spelman Halls), Demetri Porphyrios ( Whitman College ,

44268-422: The university's supervising architect, designed the chapel, which he viewed as the crown jewel for the Collegiate Gothic motif he had championed for the campus. At the time of its construction, it was the second largest university chapel in the world, after King's College Chapel, Cambridge . It underwent a two-year, $ 10 million restoration campaign between 2000 and 2002. The Chapel seats around 2,000 and serves as

44485-429: The world-relative approach was to represent the possibility that objects in one world may fail to exist in another. But if standard quantifier rules are used, every term must refer to something that exists in all the possible worlds. This seems incompatible with our ordinary practice of using terms to refer to things that exist contingently. Kripke's response to this difficulty was to eliminate terms. He gave an example of

44702-584: The years after the war, with the construction of Firestone Library in 1948 and the establishment of the Forrestal Research Center in nearby Plainsboro Township in the 1950s. Government sponsored research increased sharply, particularly in the physics and engineering departments, with much of it occurring at the new Forrestal campus. Though, as the years progressed, scientific research at the Forrestal campus declined, and in 1973, some of

44919-430: Was Nassau Hall, completed in 1756, and situated on the northern edge of the campus facing Nassau Street. The campus expanded steadily around Nassau Hall during the early and middle 19th century. The McCosh presidency (1868–88) saw the construction of a number of buildings in the High Victorian Gothic and Romanesque Revival styles, although many of them are now gone, leaving the remaining few to appear out of place. At

45136-804: Was a Corresponding Fellow of the British Academy . He won the Schock Prize in Logic and Philosophy in 2001. Kripke was married to philosopher Margaret Gilbert . Kripke died of pancreatic cancer on September 15, 2022, in Plainsboro, New Jersey, at the age of 81. Kripke's contributions to philosophy include: He has also contributed to recursion theory (see admissible ordinal and Kripke–Platek set theory ). Two of Kripke's earlier works, "A Completeness Theorem in Modal Logic" (1959) and "Semantical Considerations on Modal Logic" (1963),

45353-407: Was a breakthrough in the making of non-classical logics, because the model theory of such logics was absent before Kripke. A Kripke frame or modal frame is a pair ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } , where W is a non-empty set, and R is a binary relation on W . Elements of W are called nodes or worlds , and R is known as

45570-405: 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

45787-616: Was invited to work with him as Princeton's consulting gardening architect, focusing on the 17 gardens that are distributed throughout the campus. Nassau Hall is the oldest building on campus. Begun in 1754 and completed in 1756, it was the first seat of the New Jersey Legislature in 1776, was involved in the Battle of Princeton in 1777, and was the seat of the Congress of the Confederation (and thus capitol of

46004-460: Was labeled a prodigy , teaching himself Ancient Hebrew by the age of six, reading Shakespeare 's complete works by nine, and mastering the works of Descartes and complex mathematical problems before finishing elementary school. He wrote his first completeness theorem in modal logic at 17, and had it published a year later. After graduating from high school in 1958, Kripke attended Harvard University and graduated summa cum laude in 1962 with

46221-471: Was largely composed of daughters of professors and sisters of Princeton undergraduates. While no legal connection existed, many Princeton professors taught there and several Princeton administrators, such as Francis Patton, were on its board of trustees. It closed in 1897 following the death of its founder, Joshua McIlvaine . In 1947, three female members of the library staff enrolled in beginning Russian courses to deal with an increase in Russian literature in

46438-427: Was located. That same year, Princeton's first charter came under dispute by Anglicans, but on September 14, 1748, the recently appointed governor Jonathan Belcher granted a second charter. Belcher, a Congregationalist , had become alienated from his alma mater , Harvard, and decided to "adopt" the infant college. Belcher would go on to raise funds for the college and donate his 474-volume library, making it one of

46655-406: Was only a professor at the time, recommended saving the university with the help of alumni; as a result, Princeton's alumni association, led by James Madison , was created and began raising funds. With Carnahan and Maclean, now vice-president, working as partners, enrollment and faculty increased, tensions decreased, and the college campus expanded. Maclean took over the presidency in 1854, and led

46872-401: Was shaped much in its formative years by the " Log College ", a seminary founded by the Reverend William Tennent at Neshaminy, Pennsylvania , in about 1726. While no legal connection ever existed, many of the pupils and adherents from the Log College would go on to financially support and become substantially involved in the early years of the university. While early writers considered it as

47089-579: Was still Aristotle). Logic 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 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

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