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That the system of propositional logic in PM was the result of an evolution of changes in choices of primitives is mirrored in the choice of theorems that are proved in the first chapters. While most are proved because they will be used later in PM, some remain simply as remants of the earlier systems. In particular PM contains several theorems that were primitive propositions in earlier systems, though not used in what follows.

The notion of truth-functional semantics for propositional logic, using the familiar truth tables, and the notion of completeness of an axiom system, was not developed until soon after the publication of PM by Bernays There is no explicit statement of a rule of substitution in PM. The free variables in the propositional logic of PM may be interpreted as schematic letters, and so the system will require a rule of substitution of formulas.

In this article they are be interpreted as real variables ranging over propositions, in which case instances would be derived by instantiation from generalizations over all propositions. The announcement in the Introduction that propositions are not necessary in what follows and so will be avoided suggests the schematic interpretation of the variables.

This interpretation of the letters as variables will also assist in the presentation of quantificational logic in PM below. As is standard for an axiomatic formulation of logic, a derivation of a formula of sentential logic in PM will consist of an instance of one of the six axioms, the result of a substitution in a preceding line, or the application of modus ponens to two preceding lines. Theorems of PM will be proved in order, allowing the use of instances of preceding theorems as lines in later derivations. The resulting system is complete, in the sense that all and only truth-functionally valid sentences are derivable in the system.

This despite the seeming defects of the system by modern standards, including the redundancy of one of the axioms, the use of defined symbols in expressions to which the rules of inference apply, and the use of defined symbols in the axioms. Theorems are proved primarily as needed in later numbers, but some were axioms, or important theorems of earlier versions of propositional logic, going back to The Principles of Mathematics. Aside from historical interest in their actual choices, however, the system of PM can be viewed as based on any standard system of propositional logic.

The theory of types in the initial chapters of PM is ramified , so that within a given type, of propositions, or of functions of individuals, and functions of functions of individuals, there will be finer subdivisions. The most prominent of these is the propositional Liar paradox created by the proposition that all propositions of a certain sort, say asserted by Epimenides, are false, when that very proposition is of that sort, that is the only proposition that Epimenides asserted.

The solution in the ramified theory of types requires that a proposition about a sort of first level propositions, say that they are all false, will itself be of the next order. The paradoxes of the theory of sets are resolved by reducing assertions about sets to assertions about propositional functions. The restriction that a function of one type cannot apply to a function of the same type is enough to block the paradoxes. In the Introduction to PM terminology is introduced for the two ways that variables may appear in formulas. The proper interpretation of higher-order variables in PM is the subject of contemporary dispute among scholars of PM.

Landini and Linsky offer two rival accounts. Then the more distinctive notions of PM that depend on the theory of types can be explained. The Axiom of Reducibility asserts that for an arbitrary function of any order there is an equivalent predicative function, that is, one true of exactly the same range of arguments. Although PM does not single out first order logic from the whole ramified theory of types, the actual deductive apparatus on the page looks exactly like a system of first order logic, and the complications of the logic of higher types can be expressed with an additional apparatus of type indices.

In what follows we will use the system of r-types in Church for type indices, and the use of lambda operators for propositional functions. Note that there are two kinds of variables, but they are all all assigned to an r-type. Individual variables behave as a special case of propositional function variables. The system of symbols for r -types and the assignment of r -types to variables for different entities individuals and functions is as follows:.

There are no predicate or individual names in this language. There are, however complex terms for propositional functions, defined together with formulas with the usual notion of bound and free variables :.

## Wit Humor America Volume Vii by Wilder Marshall

We then can define the well formed formulas wffs and terms of quantificational logic as follows:. The comprehension principle for a system of higher-order logic, or set theory, states which formulas express a property or set. The comprehension principle then is characterized by an infinite set of sentences of the form of:. Quine The offense comes from attributing orders r -types to propositional functions on the basis of the variables with which they are defined, but also to the functions themselves, as simply values of bound higher-order variables.

In response, the defender of type theory must say that any semantic intrepretation of the notion of propositional function will have to attribute to functions these distinctions that are marked in linguistic expressions of some of them, and in particular, the variables involve in their definition. This shows the extent to which the earlier theory is indeed a theory of propositions , not an account of a fragment of quantificational logic allowing open sentences containing free variables. While of interest to scholars of PM, the upshot is the same for later uses of quantificational logic in PM.

Again, the reader interested in what distinguishes the logicist project in PM can skip this section, although passing attention may be paid to the system of higher-order logic that is used, as based here on the ramified theory of types. The extension to functions of more than one variable is obvious, and below, some applications will employ this extension. In what follows A is now an arbitrary possibly quantificational formula :. In part this is because of the application to an argument of lambda expressions for a propositional function, e.

Some that are often used in later numbers are:. In other words, this section introduces the logic of quantification, in a way that is familiar to contemporary logic. Consider the fundamental notion from the theory of real numbers of the least upper bound l. Consider the class of all real numbers whose square is less than or equal to 2, i.

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The resolution of this in the system of PM is to adopt an axiom which guarantees that any class defined in terms of another class will be of the same type. Thus impredicative definitions of classes are allowed, and do not introduce a class of a higher type.

More precisely, the Axiom of Reducibility asserts that for any function of any number of arguments of an arbitrary level, there is an equivalent function of level 1, ie. This notion of predicative functions is taken from the Introduction. See the accompanying entry on the notation in Principia Mathematica. It has seemed to some, beginning with Chwistek and continuing through Copi that the Axiom of Reducibility is technically faulty, leading to an inconsistency, or at least redundancy in the system of PM.

Ramsey early on argued that the supposed contradiction in fact demonstrated that certain predicative functions are indefinable. Church , confirms this assessment, and uses the presentation of r -types we describe here to show rigorously the limitations on what functions are definable in the system of PM. In PM the notion of identity is defined following Leibniz as indiscernibility, namely indiscernible objects are identical. But since the axiom of reducibility guarantees that if there is any type of function on which x and y differ, they will differ on some predicative function, PM uses the following definition of identity:.

In other words, identicals are indiscernible. The given definition of identity only suffices if it is not possible that entities x and y which share all predicative properties, cannot be distinguished by some property of a higher order. In the appendix B to the second edition of PM, which was written by Russell, there is a technical discussion of the consequences of abandoning the axiom of reducibility.

A faulty proof is proposed to show that the principle of Induction can be derived without using the axiom of reducibilty in a modified theory of types see Linsky The thesis that every class of reals with an upper bound has a real number as its least upper bound, discussed above, would not be provable. The theory of definite descriptions is essential for this argument.

The technical purpose, however, does indicate an important distinction between the logicism of Frege and Russell. Some logicians firmly in the tradition of mathematical logic do not find this to be an advance, but it does indicate a significant difference between the approaches of Frege and Russell see Linsky The latter is the reading on which it is not the case that there is one and only one present King of France and he is bald.

That may be true if there is not exactly one present King of France, as is actually the case, as France has no King. These expressions do not figure in theorems later in PM and only occasionally in the introductory material of some sections. This theorem is another indication of the way in which the philosophical basis of PM, with its propositional functions that are intensional is left behind as the mathematical content of PM is introduced with the definition of classes in the next sections.

The theory of sets classes in PM is based on a number of contextual definitions, similar in some ways to the theory of descriptions. The basic definition eliminates terms for classes from contexts in which they occur, just as the theory of definite descriptions eliminates descriptions occuring in the positions of terms:. After these foundational sections, all the individual variables that appear in PM should be seen as ranging over classes, and, as will be explained below, the relation symbols are to be interpreted as ranging over relations in extension.

The paradox arises when one asks whether that class is a member of iteself or not. A function must be of a higher order than its arguments. The definitions of existential and universal quantification are simple. Because formulas with Greek variables look and behave the same as individual variables with respect to quantificational logic, it is possible to overlook the interaction of the theory of classes with the theory of types.

Linsky argues that PM has no notation for classes of propositional functions to distinguish them from classes of classes, although one could be added. This means that variables and terms for classes will obey the simple theory of types. It should be noted that the every s -type is also an r -type, namely one that is hereditarily predicative. Thus it might seem that the expressions of the theory of classes are all simply a special case of formulas of the full system of the ramified theory of types. This is the step comparable to the proof that a definite description is proper, i. It is widely thought that the system of PM offers a very different approach to the solution of the paradoxes than that of axiomatic set theory as formulated in the Zermelo-Fraenkel system ZF.

This view has been forcefully expressed by Quine:. Whatever the inconveniences of type theory, contradictions such as [the Russell paradox] show clearly enough that the previous naive logic needs reforming. None has the backing of common sense. Common sense is bankrupt, for it wound up in contradiction. At least hitherto only one solution which meets these two requirements [of avoiding the paradoxes while retaining mathematics and the theory of aggregates] has been found. It is the same intuition that underlies the hierarchy of types. Strictly as presented in PM, however, the no-classes theory differs significantly from ZF.

The sentences of the PM theory are expressed in the theory of types, as opposed to the first order theory of ZF. ZF and PM cannot simply be compared in terms of their theorems. Not only are there different axioms in the two theories, but the very languages in which they are expressed differ in logical power.

As Quine remarks in his study of the logic of Whitehead and Russell, it would seem that after a certain point the body of PM makes use of extensional higher-order logic in a simple theory of types:. In any case there are no specific attributes [propositional functions] that can be proved in Principia to be true of just the same things and yet to differ from one another. The theory of attributes receives no application, therefore, for which the theory of classes would not have served. Once classes have been introduced, attributes are scarcely mentioned again in the course of the three volumes.

Quine here hints at the view of PM that is widely shared among mathematical logicians, who see the ramified theory of types, with its accompanying Axiom or Reducibility, as a digression taking logic into a realm of obscure intensional notions, when instead logic, even if expressed in a theory of types, is extensional and is comparable to axiomatic set theory presented with a simple hierarchy of sets of individuals, sets of sets individuals, and so on.

It is certainly true that the the remainder of PM is devoted to the theory of individuals, classes, and relations in extension between those entities. Thus the ontology of these later portions is a hierarchy of predicative functions arranged in a simple theory of types. This has led one interpreter, Gregory Landini , to argue that only predicative functions are values of bound variables in PM. The only bound variables in PM, he asserts, range over predicative functions. This is a strong version of a view that others such as Kanamori have expressed, going back to Ramsey , namely that the introduction of the Axiom of Reducibility has the effect of undoing the ramification of the theory of types, at least for a theory of classes, and so a higher-order logic used for the foundations of mathematics ought to have only a simple type structure.

In the summary of the later sections of PM that follows below, it will appear that in fact the symbolic development follows very closely that of PoM from ten years earlier.

## Humour - Wikipedia

To remind the reader of the change from talking of propositional functions to relations in extension, two further notational alterations are introduced. The obvious limitation of this notation is that it is not readily extended to three place relations, adding a third variable, say z. The notions of the subset relation and the intersection and union of sets are defined in PM exactly as they are now albeit with different terminology. The complement of a set of a given type is the set of all entities of that type that are not in the set.

There is no class of all classes of whatever type. This is in common with axiomatic set theory which holds that there is no set of all sets. If there is binary relation which has a unique second argument for each first argument, i. The definition of a monadic functional term then is:. The diligent reader will find that this presentation does not follow PM exactly.

The practice of reading the argument of a relational function as the x and the value as the y is so well established that we have taken a liberty with the actual definitions in PM. A series of notions are defined in a way quite familiar to the modern treatment of relations as sets of n -tuples:. The notions of the domain and range of a relation are also given a contemporary definition and so also the notions of the domain and range of a function. Note that it is possible that a relation can its domain in one type and range in another. This adds complications in the theory of cardinal numbers when a relation of similarity equinumerousity holds between classes of different types.

In his survey of PM, Quine complains that this last pages of Part I is occupied with proving theorems relating redundant definitions of the same notions. Thus PM defines the notion of domain and range and then introduces notions that again define the same classes, which are proved to be equivalent. PM, In contemporary logic with the notation of set theory used above, there is no need for a special symbol for this notion, as it is written as:.

So the cardinal number 1 is the class of all singletons. There will be a different number 1 for each type of x. Frege, by contrast, defines the natural number 1 as the extension of a certain concept, namely being identical with the number 0, which itself is the extension of the empty concept of not being self identical. This construction is named the von Neumann ordinals. Similarly, the number 2 is the class of all pairs, rather than a particular pair. In the type theory of PM there will be distinct couples for the types of y and x. Even with homogenous pairs there will be distinct classes of pairs for each type, and thus a different number 2 for each type.

The same notion applies to relations. It is a relation in extension, which is the analogue of a property in extension or class. A relation in extension has a distinction between the first and second elements due to the order of the defining relation. The closest in contemporary language would be:. Given the definition of extensions of relations this is the version of the no-classes theory for relations.

After attending classes of Russell the year before, and having several discussions, Norbert Wiener proposed the following definition in modern notation :. This section is little used in Volume I. The special consequences for this notion when dealing with relative types of cardinal numbers is the topic of the Preface to Volume II, which was added after the first volume was already in print.

The delay due to working out these details partially explains the three year gap between the publication of Volume I in , and the remaining volumes II and III in Difficulties arise with respect to the definition of cardinal numbers when the relation of similarity they involve is one that has a domain and range in different types.

The proof here explicitly follows the proof by Ernst Zermelo from The cardiovascular benefits of laughter also seem to be just a figment of imagination as a study that was designed to test oxygen saturation levels produced by laughter, showed that even though laughter creates sporadic episodes of deep breathing, oxygen saturation levels are not affected. As humour is often used to ease tension, it might make sense that the same would be true for anxiety. The study subject were told that they would be given to an electric shock after a certain period of time.

One group was exposed to humorous content, while the other was not. The anxiety levels were measured through self-report measures as well as the heart rate. Subjects which rated high on sense of humour reported less anxiety in both groups, while subjects which rated lower on sense of humour reported less anxiety in the group which was exposed to the humorous material.

However, there was not a significant difference in the heart rate between the subjects. Humour is a ubiquitous, highly ingrained, and largely meaningful aspect of human experience and is therefore decidedly relevant in organisational contexts, such as the workplace. The significant role that laughter and fun play in organisational life has been seen as a sociological phenomenon and has increasingly been recognised as also creating a sense of involvement among workers. Humour may also be used to offset negative feelings about a workplace task or to mitigate the use of profanity, or other coping strategies, that may not be otherwise tolerated.

Managers may use self-deprecating humour as a way to be perceived as more human and "real" by their employees. Laughter and play can unleash creativity , thus raising morale , so in the interest of encouraging employee consent to the rigours of the labour process, management often ignore, tolerate and even actively encourage playful practices, with the purpose of furthering organisational goals.

One of the main focuses of modern psychological humour theory and research is to establish and clarify the correlation between humour and laughter.

### EDITED BY MARSHALL P. WILDER

The major empirical findings here are that laughter and humour do not always have a one-to-one association. While most previous theories assumed the connection between the two almost to the point of them being synonymous, psychology has been able to scientifically and empirically investigate the supposed connection, its implications, and significance.

In , Diana Szameitat conducted a study to examine the differentiation of emotions in laughter. They hired actors and told them to laugh with one of four different emotional associations by using auto-induction, where they would focus exclusively on the internal emotion and not on the expression of laughter itself. This brings into question the definition of humour, then. If it is to be defined by the cognitive processes which display laughter, then humour itself can encompass a variety of negative as well as positive emotions.

However, if humour is limited to positive emotions and things which cause positive affect, it must be delimited from laughter and their relationship should be further defined. Humour has shown to be effective for increasing resilience in dealing with distress and also effective in undoing negative affects. Madeljin Strick, Rob Holland, Rick van Baaren, and Ad van Knippenberg of Radboud University conducted a study that showed the distracting nature of a joke on bereaved individuals.

Their findings showed that humorous therapy attenuated the negative emotions elicited after negative pictures and sentences were presented. In addition, the humour therapy was more effective in reducing negative affect as the degree of affect increased in intensity.

## Wit and Humor of America 4

The escapist nature of humour as a coping mechanism suggests that it is most useful in dealing with momentary stresses. Stronger negative stimuli requires a different therapeutic approach. Humour is an underlying character trait associated with the positive emotions used in the broaden-and-build theory of cognitive development.

Studies, such as those testing the undoing hypothesis , [40] : have shown several positive outcomes of humour as an underlying positive trait in amusement and playfulness. Several studies have shown that positive emotions can restore autonomic quiescence after negative affect. Using humour judiciously can have a positive influence on cancer treatment. Humour can serve as a strong distancing mechanism in coping with adversity.

In Kelter and Bonanno found that Duchenne laughter correlated with reduced awareness of distress. A distancing of thought leads to a distancing of the unilateral responses people often have to negative arousal. In parallel with the distancing role plays in coping with distress, it supports the broaden and build theory that positive emotions lead to increased multilateral cognitive pathway and social resource building. Humour has been shown to improve and help the ageing process in three areas. The areas are improving physical health, improving social communications, and helping to achieve a sense of satisfaction in life.

Studies have shown that constant humour in the ageing process gives health benefits to individuals. Such benefits as higher self-esteem , lower levels of depression , anxiety , and perceived stress , and a more positive self-concept as well as other health benefits which have been recorded and acknowledged through various studies.

Another way that research indicates that humour helps with the ageing process, is through helping the individual to create and maintain strong social relationship during transitory periods in their lives. With this transition certain social interactions with friend and family may be limited forcing the individual to look else where for these social interactions. Humour has been shown to make transitions easier, as humour is shown reduce stress and facilitate socialisation and serves as a social bonding function.

These new social interactions can be critical for these transitions in their lives and humour will help these new social interactions to take place making these transitions easier. Humour can also help ageing individuals maintain a sense of satisfaction in their lives. Through the ageing process many changes will occur, such as losing the right to drive a car. This can cause a decrease in satisfaction in the lives of the individual.

Humour helps to alleviate this decrease of satisfaction by allowing the humour to release stress and anxiety caused by changes in the individuals life. In an article published in Nature Reviews Neuroscience , it is reported that a study's results indicate that humour is rooted in the frontal lobe of the cerebral cortex. The study states, in part:. Humour can be verbal, visual, or physical. Non-verbal forms of communication—for example, music or visual art—can also be humorous.

Rowan Atkinson explains in his lecture in the documentary Funny Business [48] that an object or a person can become funny in three ways:. Most sight gags fit into one or more of these categories. Some theoreticians of the comic consider exaggeration to be a universal comic device. There are many taxonomies of humor; the following is used to classify humorous tweets in Rayz Different cultures have different typical expectations of humour so comedy shows are not always successful when transplanted into another culture.

For example, a BBC News article discusses a stereotype among British comedians that Americans and Germans do not understand irony , and therefore UK sitcoms are not appreciated by them. From Wikipedia, the free encyclopedia. This is the latest accepted revision , reviewed on 25 June However, if humour is limited to positive emotions and things which cause positive affect, it must be delimited from laughter and their relationship should be further defined. Humour has shown to be effective for increasing resilience in dealing with distress and also effective in undoing negative affects.

Madeljin Strick, Rob Holland, Rick van Baaren, and Ad van Knippenberg of Radboud University conducted a study that showed the distracting nature of a joke on bereaved individuals. Their findings showed that humorous therapy attenuated the negative emotions elicited after negative pictures and sentences were presented.

In addition, the humour therapy was more effective in reducing negative affect as the degree of affect increased in intensity. The escapist nature of humour as a coping mechanism suggests that it is most useful in dealing with momentary stresses. Stronger negative stimuli requires a different therapeutic approach.

Humour is an underlying character trait associated with the positive emotions used in the broaden-and-build theory of cognitive development. Studies, such as those testing the undoing hypothesis , [40] : have shown several positive outcomes of humour as an underlying positive trait in amusement and playfulness.

Several studies have shown that positive emotions can restore autonomic quiescence after negative affect. Using humour judiciously can have a positive influence on cancer treatment. Humour can serve as a strong distancing mechanism in coping with adversity. In Kelter and Bonanno found that Duchenne laughter correlated with reduced awareness of distress. A distancing of thought leads to a distancing of the unilateral responses people often have to negative arousal. In parallel with the distancing role plays in coping with distress, it supports the broaden and build theory that positive emotions lead to increased multilateral cognitive pathway and social resource building.

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Humour has been shown to improve and help the ageing process in three areas. The areas are improving physical health, improving social communications, and helping to achieve a sense of satisfaction in life. Studies have shown that constant humour in the ageing process gives health benefits to individuals. Such benefits as higher self-esteem , lower levels of depression , anxiety , and perceived stress , and a more positive self-concept as well as other health benefits which have been recorded and acknowledged through various studies.

Another way that research indicates that humour helps with the ageing process, is through helping the individual to create and maintain strong social relationship during transitory periods in their lives. With this transition certain social interactions with friend and family may be limited forcing the individual to look else where for these social interactions. Humour has been shown to make transitions easier, as humour is shown reduce stress and facilitate socialisation and serves as a social bonding function.

These new social interactions can be critical for these transitions in their lives and humour will help these new social interactions to take place making these transitions easier. Humour can also help ageing individuals maintain a sense of satisfaction in their lives. Through the ageing process many changes will occur, such as losing the right to drive a car. This can cause a decrease in satisfaction in the lives of the individual. Humour helps to alleviate this decrease of satisfaction by allowing the humour to release stress and anxiety caused by changes in the individuals life.

In an article published in Nature Reviews Neuroscience , it is reported that a study's results indicate that humour is rooted in the frontal lobe of the cerebral cortex. The study states, in part:. Humour can be verbal, visual, or physical. Non-verbal forms of communication—for example, music or visual art—can also be humorous. Rowan Atkinson explains in his lecture in the documentary Funny Business [48] that an object or a person can become funny in three ways:.

Most sight gags fit into one or more of these categories. Some theoreticians of the comic consider exaggeration to be a universal comic device. There are many taxonomies of humor; the following is used to classify humorous tweets in Rayz Different cultures have different typical expectations of humour so comedy shows are not always successful when transplanted into another culture. For example, a BBC News article discusses a stereotype among British comedians that Americans and Germans do not understand irony , and therefore UK sitcoms are not appreciated by them.

From Wikipedia, the free encyclopedia. This is the latest accepted revision , reviewed on 25 June For other uses, see Humour disambiguation. For the stand-up special by Louis C. From top-left to bottom-right or from top to bottom mobile : various people laughing from Afghanistan , Tibet , Brazil , and Malaysia. Main article: Theories of humour. Main article: Humor research. Main article: Exaggeration. British humour Deadpan Form-versus-content humour Gelotology , the study of laughing and laughter Humour in translation Humour styles List of humorists Surreal humour Theories of humour.

Retrieved 26 August Cambridge: Cambridge University Press. Rabelais and His World [, ]. Bloomington: Indiana University Press p. Retrieved 14 April Harbsmeier, "Confucius-Ridens, Humor in the Analects. Archived from the original on Retrieved Personal Relationships. Higher Education Studies. Marriage and Family Living. Sex Roles.