Zhegalkin polynomialFrom Wikipedia, the free encyclopedia

Contents

1 Warnier/Orr diagram 11.1 Basic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Using Warnier/Orr diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Constructs in Warnier/Orr diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3.1 Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.4 Alternation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.5 Concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.6 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Well-formed formula 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Atomic and open formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Closed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Properties applicable to formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 Usage of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 What the Tortoise Said to Achilles 93.1 Summary of the dialogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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ii CONTENTS

3.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Willard Van Orman Quine 134.1 Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1.1 Political beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2.1 Rejection of the analyticsynthetic distinction . . . . . . . . . . . . . . . . . . . . . . . . 144.2.2 Conrmation holism and ontological relativity . . . . . . . . . . . . . . . . . . . . . . . . 154.2.3 Existence and Its contrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.4 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.5 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2.6 Quines epistemology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 In popular culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.4.1 Selected books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.2 Important articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 William Kneale 225.1 Life and work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Window operator 236.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Wisdom of repugnance 247.1 Origin and usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Witness (mathematics) 268.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Henkin witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3 Relation to game semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9 Wolfram axiom 289.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

CONTENTS iii

9.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

10 Word sense 3010.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 Zhegalkin polynomial 3211.1 Boolean equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.3 Method of Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11.3.1 The method of Indeterminate Coecents . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.3.2 Using PDCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11.4 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 35

11.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Chapter 1

Warnier/Orr diagram

A Warnier/Orr diagram (also known as a logical construction of a program/system) is a kind of hierarchicalowchart that allows the description of the organization of data and procedures. They were initially developed inFrance by Jean-Dominique Warnier and in the United States by Kenneth Orr. This method aids the design of pro-gram structures by identifying the output and processing results and then working backwards to determine the stepsand combinations of input needed to produce them. The simple graphic method used inWarnier/Orr diagrams makesthe levels in the system evident and the movement of the data between them vivid.

1.1 Basic Elements

Warnier/Orr diagrams show the processes and sequences in which they are performed. Each process is dened ina hierarchical manner i.e. it consists of sets of subprocesses, that dene it. At each level, the process is shown inbracket that groups its components.Since a process can have many dierent subprocesses, Warnier/Orr diagram uses a set of brackets to show each levelof the system. Critical factors in s/w denition and development are iteration or repetition and alteration. Warnier/Orrdiagrams show this very well.

1.2 Using Warnier/Orr diagrams

To develop a Warnier/Orr diagram, the analyst works backwards, starting with systems output and using outputoriented analysis. On paper, the development moves from right to left . First, the intended output or results of theprocessing are dened. At the next level, shown by inclusion with a bracket, the steps needed to produce the outputare dened. Each step in turn is further dened. Additional brackets group the processes required to produce theresult on the next level.Warnier/Orr diagram oer some distinct advantages to systems experts. They are simple in appearance and easy tounderstand. Yet they are powerful design tools. They have advantage of showing groupings of processes and the datathat must be passed from level to level. In addition, the sequence of working backwards ensures that the system willbe result oriented. This method is useful for both data and process denition. It can be used for each independently,or both can be combined on the same diagram.

1.3 Constructs in Warnier/Orr diagrams

There are four basic constructs used onWarnier/Orr diagrams: hierarchy, sequence, repetition, and alternation. Thereare also two slightly more advanced concepts that are occasionally needed: concurrency and recursion.

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2 CHAPTER 1. WARNIER/ORR DIAGRAM

1.3.1 HierarchyHierarchy is the most fundamental of all of the Warnier/Orr constructs. It is simply a nested group of sets and subsetsshown as a set of nested brackets. Each bracket on the diagram (depending on how you represent it, the character isusually more like a brace "{" than a bracket "[", but we call them brackets) represents one level of hierarchy. Thehierarchy or structure that is represented on the diagram can show the organization of data or processing. However,both data and processing are never shown on the same diagram.

1.3.2 SequenceSequence is the simplest structure to show on a Warnier/Orr diagram. Within one level of hierarchy, the featureslisted are shown in the order in which they occur. In other words, the step listed rst is the rst that will be executed(if the diagram reects a process), while the step listed last is the last that will be executed. Similarly with data, thedata eld listed rst is the rst that is encountered when looking at the data, the data eld listed last is the nal oneencountered.

1.3.3 RepetitionRepetition is the representation of a classic loop in programming terms. It occurs whenever the same set of dataoccurs over and over again (for a data structure) or whenever the same group of actions is to occur over and overagain (for a processing structure). Repetition is indicated by placing a set of numbers inside parentheses beneath therepeating set.Typically there are two numbers listed in the parentheses, representing the fewest and the most number of times theset will repeat. By convention the rst letter of the repeating set is the letter chosen to represent the maximum.While the minimum bound and maximum bound can technically be anything, they are most often either "(1,n)" asin the example, or "(0,n). When used to depict processing, the "(1,n)" repetition is classically known as a DoUntilloop, while the "(0,n)" repetition is called a DoWhile loop. On the Warnier/Orr diagram, however, there is nodistinction between the two dierent types of repetition, other than the minimum bound value.On occasion, the minimum and maximum bound are predened and not likely to change: for instance the set Dayoccurs within the set Month from 28 to 31 times (since the smallest month has 28 days, the largest months, 31).This is not likely to change. And on occasion, the minimum and maximum are xed at the same number.In general, though, it is a bad idea to "hard code" a constant other than 0 or 1 in a number of times clausethe design should be exible enough to allow for changes in the number of times without changes to the design.For instance, if a company has 38 employees at the time a design is done, hard coding a 38 as the number ofemployees within company would certainly not be as exible as designing "(1,n)".The number of times clause is always an operator attached to some set (i.e., the name of some bracket), and is neverattached to an element (a diagram feature which does not decompose into smaller features). The reason for this willbecome more apparent as we continue to work with the diagrams. For now, you will have to accept this as a formationrule for a correct diagram.

1.3.4 AlternationAlternation, or selection, is the traditional decision process whereby a determination is made to execute one processor another. The Exclusive OR symbol (the plus sign inside the circle) indicates that the sets immediately above andbelow it are mutually exclusive (if one is present the other is not). This diagram indicates that an Employee is eitherManagement or Non-Management, one Employee cannot be both. It is also permissible to use a negation bar abovean alternative in a manner similar to engineering notation. The bar is read by simply using the word not.Alternatives do not have to be binary as in the previous examples, but may be many-way alternatives.

1.3.5 ConcurrencyConcurrency is one of the two advanced constructs used in themethodology. It is used whenever sequence is unimpor-tant. For instance, years and weeks operate concurrently (or at the same time) within our calendar. The concurrency

1.4. SEE ALSO 3

operator is rarely used in program design (since most languages do not support true concurrent processing anyway),but does come into play when resolving logical and physical data structure clashes.

1.3.6 RecursionRecursion is the least used of the constructs. It is used to indicate that a set contains an earlier or a less orderedversion of itself. In the classic bill of materials problem components contain parts and other sub-components.Sub-components also contain sub-sub-components, and so on. The doubled bracket indicates that the set is recursive.Data structures that are truly recursive are rather rare.

1.4 See also Structure chart

1.5 References

1.6 External links Warnier Dave Higgins Consulting website and original source for Wikipedia entry. James A. Senn, Analysis & Design of Information Systems, 2nd ed., McGraw-Hill Publishing Company Ken Orr Institute

Chapter 2

Well-formed formula

Symbols andstrings of symbols

Well-formed formulas

Theorems

This diagram shows the syntactic entities which may be constructed from formal languages. The symbols and strings of symbolsmay be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of itswell-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.

In mathematical logic, a well-formed formula, shortly w, often simply formula, is a word (i.e. a nite sequenceof symbols from a given alphabet) that is part of a formal language.[1] A formal language can be considered to beidentical to the set containing all and only its formulas.A formula is a syntactic formal object that can be given a semantic meaning by means of semantics.

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2.1. INTRODUCTION 5

2.1 IntroductionA key use of formulae is in propositional logic and predicate logics such as rst-order logic. In those contexts, aformula is a string of symbols for which it makes sense to ask is true?", once any free variables in have beeninstantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and thenal formula in the sequence is what is proven.Although the term formula may be used for written marks (for instance, on a piece of paper or chalkboard), it ismore precisely understood as the sequence being expressed, with the marks being a token instance of formula. It isnot necessary for the existence of a formula that there be any actual tokens of it. A formal language may thus have aninnite number of formulas regardless whether each formula has a token instance. Moreover, a single formula mayhave more than one token instance, if it is written more than once.Formulas are quite often interpreted as propositions (as, for instance, in propositional logic). However formulas aresyntactic entities, and as such must be specied in a formal language without regard to any interpretation of them.An interpreted formula may be the name of something, an adjective, an adverb, a preposition, a phrase, a clause, animperative sentence, a string of sentences, a string of names, etc.. A formula may even turn out to be nonsense, if thesymbols of the language are specied so that it does. Furthermore, a formula need not be given any interpretation.

2.2 Propositional calculusThe formulas of propositional calculus, also called propositional formulas,[2] are expressions such as (A ^ (B _C)). Their denition begins with the arbitrary choice of a set V of propositional variables. The alphabet consists of theletters in V along with the symbols for the propositional connectives and parentheses "(" and ")", all of which areassumed to not be in V. The formulas will be certain expressions (that is, strings of symbols) over this alphabet.The formulas are inductively dened as follows:

Each propositional variable is, on its own, a formula. If is a formula, then : is a formula. If and are formulas, and is any binary connective, then ( ) is a formula. Here could be (but is notlimited to) the usual operators , , , or .

This denition can also be written as a formal grammar in BackusNaur form, provided the set of variables is nite:

::= p | q | r | s | t | u | ... (the arbitrary nite set of propositional variables) ::= | : | ( ^ ) | ( _ ) | (! ) |($ )

Using this grammar, the sequence of symbols

(((p! q) ^ (r! s)) _ ( : q ^ : s))

is a formula, because it is grammatically correct. The sequence of symbols

((p! q)! (qq))p))

is not a formula, because it does not conform to the grammar.A complex formula may be dicult to read, owing to, for example, the proliferation of parentheses. To alleviatethis last phenomenon, precedence rules (akin to the standard mathematical order of operations) are assumed amongthe operators, making some operators more binding than others. For example, assuming the precedence (from mostbinding to least binding) 1. : 2. ! 3. ^ 4. _ . Then the formula

(((p! q) ^ (r! s)) _ ( : q ^ : s))

6 CHAPTER 2. WELL-FORMED FORMULA

may be abbreviated as

p! q ^ r! s _ : q ^ : s

This is, however, only a convention used to simplify the written representation of a formula. If the precedence wasassumed, for example, to be left-right associative, in following order: 1. : 2. ^ 3. _ 4. ! , then the same formulaabove (without parentheses) would be rewritten as

(p! (q ^ r))! (s _ (( : q) ^ ( : s)))

2.3 Predicate logicThe denition of a formula in rst-order logic QS is relative to the signature of the theory at hand. This signaturespecies the constant symbols, relation symbols, and function symbols of the theory at hand, along with the arities ofthe function and relation symbols.The denition of a formula comes in several parts. First, the set of terms is dened recursively. Terms, informally,are expressions that represent objects from the domain of discourse.

1. Any variable is a term.2. Any constant symbol from the signature is a term3. an expression of the form f(t1,...,tn), where f is an n-ary function symbol, and t1,...,tn are terms, is again a

term.

The next step is to dene the atomic formulas.

1. If t1 and t2 are terms then t1=t2 is an atomic formula2. If R is an n-ary relation symbol, and t1,...,tn are terms, then R(t1,...,tn) is an atomic formula

Finally, the set of formulas is dened to be the smallest set containing the set of atomic formulas such that thefollowing holds:

1. : is a formula when is a formula2. ( ^ ) and ( _ ) are formulas when and are formulas;3. 9x is a formula when x is a variable and is a formula;4. 8x is a formula when x is a variable and is a formula (alternatively, 8x could be dened as an abbreviation

for :9x: ).

If a formula has no occurrences of 9x or 8x , for any variable x , then it is called quantier-free. An existentialformula is a formula starting with a sequence of existential quantication followed by a quantier-free formula.

2.4 Atomic and open formulasMain article: Atomic formula

An atomic formula is a formula that contains no logical connectives nor quantiers, or equivalently a formula that hasno strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; forpropositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atomsare predicate symbols together with their arguments, each argument being a term.According to some terminology, an open formula is formed by combining atomic formulas using only logical con-nectives, to the exclusion of quantiers.[3] This has not to be confused with a formula which is not closed.

2.5. CLOSED FORMULAS 7

2.5 Closed formulasMain article: Sentence (mathematical logic)

A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable.If A is a formula of a rst-order language in which the variables v1, ..., vn have free occurrences, then A preceded byv1 ... vn is a closure of A.

2.6 Properties applicable to formulas A formula A in a language Q is valid if it is true for every interpretation of Q . A formula A in a language Q is satisable if it is true for some interpretation of Q . A formula A of the language of arithmetic is decidable if it represents a decidable set, i.e. if there is aneective method which, given a substitution of the free variables of A, says that either the resulting instance ofA is provable or its negation is.

2.7 Usage of the terminologyIn earlier works on mathematical logic (e.g. by Church[4]), formulas referred to any strings of symbols and amongthese strings, well-formed formulas were the strings that followed the formation rules of (correct) formulas.Several authors simply say formula.[5][6][7][8] Modern usages (especially in the context of computer science withmathematical software such as model checkers, automated theorem provers, interactive theorem provers) tend toretain of the notion of formula only the algebraic concept and to leave the question of well-formedness, i.e. of theconcrete string representation of formulas (using this or that symbol for connectives and quantiers, using this or thatparenthesizing convention, using Polish or inx notation, etc.) as a mere notational problem.However, the expression well-formed formulas can still be found in various works,[9][10][11] these authors using thename well-formed formula without necessarily opposing it to the old sense of formula as arbitrary string of symbols sothat it is no longer common in mathematical logic to refer to arbitrary strings of symbols in the old sense of formulas.The expression well-formed formulas (WFF) also pervaded in popular culture. Indeed,WFF is part of an esotericpun used in the name of the academic game "WFF 'N PROOF: The Game of Modern Logic, by Layman Allen,[12]developed while he was at Yale Law School (he was later a professor at the University of Michigan). The suite ofgames is designed to teach the principles of symbolic logic to children (in Polish notation).[13] Its name is an echoof whienpoof, a nonsense word used as a cheer at Yale University made popular in The Whienpoof Song and TheWhienpoofs.[14]

2.8 See also Ground expression

2.9 Notes[1] Formulas are a standard topic in introductory logic, and are covered by all introductory textbooks, including Enderton

(2001), Gamut (1990), and Kleene (1967)

[2] First-order logic and automated theorem proving, Melvin Fitting, Springer, 1996

[3] Handbook of the history of logic, (Vol 5, Logic from Russell to Church), Tarskis logic by Keith Simmons, D. Gabbay andJ. Woods Eds, p568 .

[4] Alonzo Church, [1996] (1944), Introduction to mathematical logic, page 49

8 CHAPTER 2. WELL-FORMED FORMULA

[5] Hilbert, David; Ackermann, Wilhelm (1950) [1937], Principles of Mathematical Logic, New York: Chelsea

[6] Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6

[7] Barwise, Jon, ed. (1982), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Am-sterdam: North-Holland, ISBN 978-0-444-86388-1

[8] Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with Exercises, Oxford University Press, ISBN 978-0-19-850048-3

[9] Enderton, Herbert [2001] (1972), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN978-0-12-238452-3

[10] R. L. Simpson (1999), Essentials of Symbolic Logic, page 12

[11] Mendelson, Elliott [2010] (1964), An Introduction to Mathematical Logic (5th ed.), London: Chapman & Hall

[12] Ehrenburg 2002

[13] More technically, propositional logic using the Fitch-style calculus.

[14] Allen (1965) acknowledges the pun.

2.10 References Allen, Layman E. (1965), TowardAutotelic Learning ofMathematical Logic by theWFF 'N PROOFGames,

Mathematical Learning: Report of a Conference Sponsored by the Committee on Intellective Processes Researchof the Social Science Research Council, Monographs of the Society for Research in Child Development 30 (1):2941

Boolos, George; Burgess, John; Jerey, Richard (2002), Computability and Logic (4th ed.), Cambridge Uni-versity Press, ISBN 978-0-521-00758-0

Ehrenberg, Rachel (Spring 2002). Hes Positively Logical. Michigan Today (University of Michigan). Re-trieved 2007-08-19.

Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN978-0-12-238452-3

Gamut, L.T.F. (1990), Logic, Language, and Meaning, Volume 1: Introduction to Logic, University Of ChicagoPress, ISBN 0-226-28085-3

Hodges, Wilfrid (2001), Classical Logic I: First-Order Logic, in Goble, Lou, The Blackwell Guide to Philo-sophical Logic, Blackwell, ISBN 978-0-631-20692-7

Hofstadter, Douglas (1980), Gdel, Escher, Bach: An Eternal Golden Braid, Penguin Books, ISBN 978-0-14-005579-5

Kleene, Stephen Cole (2002) [1967], Mathematical logic, New York: Dover Publications, ISBN 978-0-486-42533-7, MR 1950307

Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: SpringerScience+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6

2.11 External links Well-Formed Formula for First Order Predicate Logic - includes a short Java quiz. Well-Formed Formula at ProvenMath WFF N PROOF game site

Chapter 3

What the Tortoise Said to Achilles

"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is abrief dialogue which problematises the foundations of logic. The title alludes to one of Zenos paradoxes of motion,in which Achilles could never overtake the tortoise in a race. In Carrolls dialogue, the tortoise challenges Achilles touse the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails,because the clever tortoise leads him into an innite regression.

3.1 Summary of the dialogueThe discussion begins by considering the following logical argument:

A: Things that are equal to the same are equal to each other (Euclidean relation, a weakened form of thetransitive property)

B: The two sides of this triangle are things that are equal to the same Therefore Z: The two sides of this triangle are equal to each other

The Tortoise asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that itobviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid who grants that theargument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a readermight exist, and that he would hold that if A and B are true, then Z must be true, while not yet accepting that A andB are true. (A reader who denies the premises.)The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and B are true, butwho does not yet accept the principle that if A and B are both true, then Z must be true. Achilles grants the Tortoisethat this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat the Tortoise as a reader ofthis second kind. Achilles must now logically compel the Tortoise to accept that Z must be true. (The tortoise is areader who denies the argument itself; the syllogism's conclusion, structure, or validity.)After writing down A, B, and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:

C: If A and B are true, Z must be true

The Tortoise agrees to accept C, if Achilles will write down what it has to accept in his notebook, making the newargument:

A: Things that are equal to the same are equal to each other B: The two sides of this triangle are things that are equal to the same C: If A and B are true, Z must be true Therefore Z: The two sides of this triangle are equal to each other

9

10 CHAPTER 3. WHAT THE TORTOISE SAID TO ACHILLES

But now that the Tortoise accepts premise C, it still refuses to accept the expanded argument. When Achilles de-mands that If you accept A and B and C, you must accept Z, the Tortoise remarks that thats another hypotheticalproposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:

D: If A and B and C are true, Z must be true

The Tortoise continues to accept each hypothetical premise onceAchilles writes it down, but denies that the conclusionnecessarily follows, since each time it denies the hypothetical that if all the premises written down so far are true, Zmust be true:

And at last we've got to the end of this ideal racecourse! Now that you accept A and B and C and D, ofcourse you accept Z.Do I?" said the Tortoise innocently. Lets make that quite clear. I acceptA and B and C andD. SupposeI still refused to accept Z?"Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. Logicwould tell you, 'You can't help yourself. Now that you've accepted A and B and C and D, you mustaccept Z!' So you've no choice, you see.Whatever Logic is good enough to tell me is worth writing down, said the Tortoise. So enter it in yournotebook, please. We will call it

(E) If A and B and C and D are true, Z must be true.Until I've granted that, of course I needn't grant Z. So its quite a necessary step, you see?"I see, said Achilles; and there was a touch of sadness in his tone.

Thus, the list of premises continues to grow without end, leaving the argument always in the form:

(1): Things that are equal to the same are equal to each other (2): The two sides of this triangle are things that are equal to the same (3): (1) and (2) (Z) (4): (1) and (2) and (3) (Z) ... (n): (1) and (2) and (3) and (4) and ... and (n 1) (Z) Therefore (Z): The two sides of this triangle are equal to each other

At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there issome further premise (that if all of (1)(n) are true, then (Z) must be true) that it still needs to accept before it iscompelled to accept that (Z) is true.

3.2 ExplanationLewis Carroll was showing that there is a regress problem that arises from modus ponens deductions.

P ! Q; P) Q

The regress problem arises because a prior principle is required to explain logical principles, here modus ponens, andonce that principle is explained, another principle is required to explain that principle. Thus, if the causal chain isto continue, the argument falls into innite regress. However, if a formal system is introduced where modus ponensis simply a rule of inference dened by the system, then it can be abided by simply because it is so. For example,chess has particular rules that simply go without question and players must abide by them because they form the very

3.3. DISCUSSION 11

framework of the game. Likewise, a formal system of logic is dened by rules that are to be followed, by denition,without question. Having a dened formal system of logic stops the innite regression---that is, the regression stopsat the axioms or rules, per se, of the given game, system, etc.However, the story also hints at problems with this solution because, within the system, no proposition or variablecarries any semantic content. The moment any proposition or variable takes on semantic content, the problem arisesagain because semantic content runs outside the system. Thus, if the solution is to be said to work, then it is to besaid to work solely within the given formal system, and not otherwise.Some logicians (Kenneth Ross, Charles Wright) draw a rm distinction between the conditional connective (thesyntactic sign ""), and the implication relation (the formal object denoted by the double arrow symbol ""). Theselogicians use the phrase not p or q for the conditional connective and the term implies for the implication relation.Some explain the dierence by saying that the conditional is the contemplated relation while the implication is theasserted relation. In most elds of mathematics, it is treated as a variation in the usage of the single sign ", notrequiring two separate signs. Not all of those who use the sign "" for the conditional connective regard it as asign that denotes any kind of object but instead treat it as a so-called syncategorematic sign, that is, a sign with apurely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use thetwo-sign notation, but allow the sign "" to denote the boolean function that is associated with the truth table of thematerial conditional.These considerations result in the following scheme of notation.p! q p) q

not p or q p implies qThe paradox ceases to exist the moment informal logic is replaced with propositional logic. The Tortoise and Achillesdon't agree on any denition of logical implication. In propositional logic the logical implication is dened as follows:P Q if and only if the proposition P Q is a tautology.Hence de modus ponens, [P (P Q)] Q, is a valid logical implication according to the denition of logicalimplication just stated. There is no need to recurse since the logical implication can be translated into symbols andpropositional operators such as . Demonstrating the logical implication simply translates into verifying that thecompound truth table produces a tautology.

3.3 DiscussionSeveral philosophers have tried to resolve Carrolls paradox. Bertrand Russell discussed the paradox briey in 38of The Principles of Mathematics (1903), distinguishing between implication (associated with the form if p, then q"),which he held to be a relation between unasserted propositions, and inference (associated with the form "p, thereforeq"), which he held to be a relation between asserted propositions; having made this distinction, Russell could deny thatthe Tortoises attempt to treat inferring Z from A and B is equivalent to, or dependent on, agreeing to the hypotheticalIf A and B are true, then Z is true.The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relationto Philosophy (1958), where he argued that the paradox showed that the actual process of drawing an inference,which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learningto infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to dosomething (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson,to the eect that the proper application of rules governing a form of human activity cannot itself be summed up witha set of further rules, and so that a form of human activity can never be summed up in a set of explicit precepts (p.53).According to PenelopeMaddy,[1] Carrolls dialogue is apparently the rst description of an obstacle to Conventionalismabout logical truth, then reworked in more sober philosophical terms by W. O. Quine.[2]

3.4 See also Deduction theorem Homunculus argument

12 CHAPTER 3. WHAT THE TORTOISE SAID TO ACHILLES

Mnchhausen trilemma Paradox Regress argument Rule of inference

3.5 References[1] Maddy, P. (December 2012). The philosophy of logic. Bulletin of Symbolic Logic 18 (4): 481504. doi:10.2178/bsl.1804010.

JSTOR 23316289.

[2] Quine, W.V.O. (1976). The ways of paradox, and other essays. Cambridge, MA: Havard University Press. ISBN9780674948358. OCLC 185411480.

3.6 Sources Carroll, Lewis (1895). What the Tortoise Said toAchilles. Mind 104 (416): 691693. doi:10.1093/mind/104.416.691.JSTOR 2254477. Reprinted in The Penguin Complete Lewis Carroll (Harmondsworth, Penguin, 1982), pp1104-1108.

Hofstadter, Douglas. Gdel, Escher, Bach: an Eternal Golden Braid. See the second dialogue, entitled Two-Part Invention. Hofstadter appropriated the characters of Achilles and the Tortoise for other, original, di-alogues in the book which alternate contrapuntally with prose chapters. Hofstadters Tortoise is of the malesex, though the Tortoises sex is never specied by Carroll. The French translation of the book rendered theTortoises name as Madame Tortue.

A number of websites, including What the Tortoise Said to Achilles at the Lewis Carroll Society of NorthAmerica, What the Tortoise Said to Achilles at Digital Text International, and What the Tortoise Said toAchilles at Fair Use Repository.

3.7 External links Works related to What the Tortoise Said to Achilles at Wikisource

Chapter 4

Willard Van Orman Quine

Willard Van Orman Quine (/kwan/; June 25, 1908 December 25, 2000) (known to intimates as Van)[1] was anAmerican philosopher and logician in the analytic tradition, recognized as one of the most inuential philosophersof the twentieth century.[2] From 1930 until his death 70 years later, Quine was continually aliated with HarvardUniversity in one way or another, rst as a student, then as a professor of philosophy and a teacher of logic and settheory, and nally as a professor emeritus who published or revised several books in retirement. He lled the EdgarPierce Chair of Philosophy at Harvard from 1956 to 1978. A recent poll conducted among analytic philosophersnamed Quine as the fth most important philosopher of the past two centuries.[3][4] He won the rst Schock Prizein Logic and Philosophy in 1993 for his systematical and penetrating discussions of how learning of language andcommunication are based on socially available evidence and of the consequences of this for theories on knowledgeand linguistic meaning.[5] In 1996 he was awarded the Kyoto Prize in Arts and Philosophy for his outstandingcontributions to the progress of philosophy in the 20th century by proposing numerous theories based on keen insightsin logic, epistemology, philosophy of science and philosophy of language.[6]

Quine falls squarely into the analytic philosophy tradition while also being the main proponent of the view thatphilosophy is not conceptual analysis but the abstract branch of the empirical sciences. His major writings include"Two Dogmas of Empiricism" (1951), which attacked the distinction between analytic and synthetic propositionsand advocated a form of semantic holism, andWord and Object (1960), which further developed these positions andintroduced Quines famous indeterminacy of translation thesis, advocating a behaviorist theory of meaning. He alsodeveloped an inuential naturalized epistemology that tried to provide an improved scientic explanation of how wehave developed elaborate scientic theories on the basis ofmeager sensory input.[7] He is also important in philosophyof science for his systematic attempt to understand science from within the resources of science itself[7] and forhis conception of philosophy as continuous with science. This led to his famous quip that philosophy of scienceis philosophy enough.[8] In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the"QuinePutnam indispensability thesis, an argument for the reality of mathematical entities.[9]

4.1 BiographyAccording to his autobiography, The Time of My Life (1986), Quine grew up in Akron, Ohio, where he lived with hisparents and older brother Robert C. His father, Cloyd R., was a manufacturing entrepreneur and his mother, HarriettE. (also known as Hattie according to the 1920 census), was a schoolteacher and later a housewife.[1] He receivedhis B.A. in mathematics from Oberlin College in 1930, and his Ph.D. in philosophy from Harvard University in 1932.His thesis supervisor was Alfred North Whitehead. He was then appointed a Harvard Junior Fellow, which excusedhim from having to teach for four years. During the academic year 193233, he travelled in Europe thanks to aSheldon fellowship, meeting Polish logicians (including Alfred Tarski) and members of the Vienna Circle (includingRudolf Carnap), as well as the logical positivist A.J. Ayer.[1]

It was through Quines good oces that Alfred Tarski was invited to attend the September 1939 Unity of ScienceCongress in Cambridge. To attend that Congress, Tarski sailed for the USA on the last ship to leave Danzig beforethe Third Reich invaded Poland. Tarski survived the war and worked another 44 years in the USA.During World War II, Quine lectured on logic in Brazil, in Portuguese, and served in the United States Navy ina military intelligence role, deciphering messages from German submarines, and reaching the rank of Lieutenant

13

14 CHAPTER 4. WILLARD VAN ORMAN QUINE

Commander.[1]

At Harvard, Quine helped supervise the Harvard theses of, among others, Donald Davidson, Hubert Dreyfus, DavidLewis, Daniel Dennett, Gilbert Harman, Dagnn Fllesdal, Hao Wang, Hugues LeBlanc and Henry Hiz. For theacademic year 19641965, Quine was a Fellow on the faculty in the Center for Advanced Studies at Wesleyan Uni-versity.[10]

Quine was an atheist.[11]

Quine had four children by two marriages.[1] Guitarist Robert Quine was his nephew.

4.1.1 Political beliefs

Quinewas politically conservative, but the bulk of his writing was in technical areas of philosophy removed from directpolitical issues.[12] He did, however, write in defense of several conservative positions: for example, in Quiddities: AnIntermittently Philosophical Dictionary, he wrote a defense of moral censorship;[13] while, in his autobiography, hemade some criticisms of American postwar academic culture.[14][15]

4.2 WorkQuines Ph.D. thesis and early publications were on formal logic and set theory. Only after World War II did he, byvirtue of seminal papers on ontology, epistemology and language, emerge as a major philosopher. By the 1960s, hehad worked out his naturalized epistemology whose aim was to answer all substantive questions of knowledge andmeaning using the methods and tools of the natural sciences. Quine roundly rejected the notion that there should be arst philosophy, a theoretical standpoint somehow prior to natural science and capable of justifying it. These viewsare intrinsic to his naturalism.Quine could lecture in French, Spanish, Portuguese and German, as well as his native English.But like the logical positivists, he evinced little interest in the philosophical canon: only once did he teach a coursein the history of philosophy, on Hume.Quine has an Erds number of 3.[16]

4.2.1 Rejection of the analyticsynthetic distinction

See also: Two Dogmas of Empiricism

In the 1930s and 1940s, discussions with Rudolf Carnap, Nelson Goodman and Alfred Tarski, among others, ledQuine to doubt the tenability of the distinction between analytic statementsthose true simply by the meaningsof their words, such as All bachelors are unmarriedand synthetic statements, those true or false by virtue offacts about the world, such as There is a cat on the mat. This distinction was central to logical positivism. AlthoughQuine is not normally associated with vericationism, some philosophers believe the tenet is not incompatible withhis general philosophy of language, citing his Harvard colleague B. F. Skinner, and his analysis of language in VerbalBehavior.[17]

Like other Analytic philosophers before him, Quine accepted the denition of analytic as true in virtue of meaningalone. Unlike them, however, he concluded that ultimately the denitionwas circular. In other words, Quine acceptedthat analytic statements are those that are true by denition, then argued that the notion of truth by denition wasunsatisfactory. This criticism of Kants epistemology was similar to that of the 18th century writer Johann GottfriedHerder, as both individuals found fault in the Kantian system for not suciently accounting for the dependence ofreasoning on language.Quines chief objection to analyticity is with the notion of synonymy (sameness of meaning), a sentence being analytic,just in case it substitutes a synonym for one black in a proposition like All black things are black (or any otherlogical truth). The objection to synonymy hinges upon the problem of collateral information. We intuitively feel thatthere is a distinction between All unmarried men are bachelors and There have been black dogs, but a competentEnglish speaker will assent to both sentences under all conditions since such speakers also have access to collateralinformation bearing on the historical existence of black dogs. Quine maintains that there is no distinction between

4.2. WORK 15

universally known collateral information and conceptual or analytic truths.Another approach to Quines objection to analyticity and synonymy emerges from the modal notion of logical pos-sibility. A traditional Wittgensteinian view of meaning held that each meaningful sentence was associated with aregion in the space of possible worlds. Quine nds the notion of such a space problematic, arguing that there is nodistinction between those truths which are universally and condently believed and those which are necessarily true.

4.2.2 Conrmation holism and ontological relativityThe central theses underlying the indeterminacy of translation and other extensions of Quines work are ontologicalrelativity and the related doctrine of conrmation holism. The premise of conrmation holism is that all theories(and the propositions derived from them) are under-determined by empirical data (data, sensory-data, evidence);although some theories are not justiable, failing to t with the data or being unworkably complex, there are manyequally justiable alternatives. While the Greeks assumption that (unobservable) Homeric gods exist is false, and oursupposition of (unobservable) electromagnetic waves is true, both are to be justied solely by their ability to explainour observations.Quine concluded his "Two Dogmas of Empiricism" as follows:

As an empiricist I continue to think of the conceptual scheme of science as a tool, ultimately, forpredicting future experience in the light of past experience. Physical objects are conceptually importedinto the situation as convenient intermediaries not by denition in terms of experience, but simply asirreducible posits comparable, epistemologically, to the gods of Homer . . . For my part I do, qua layphysicist, believe in physical objects and not in Homers gods; and I consider it a scientic error to believeotherwise. But in point of epistemological footing, the physical objects and the gods dier only in degreeand not in kind. Both sorts of entities enter our conceptions only as cultural posits.

Quines ontological relativism (evident in the passage above) led him to agreewith PierreDuhem that for any collectionof empirical evidence, there would always be many theories able to account for it. However, Duhems holism is muchmore restricted and limited than Quines. For Duhem, underdetermination applies only to physics or possibly tonatural science, while for Quine it applies to all of human knowledge. Thus, while it is possible to verify or falsifywhole theories, it is not possible to verify or falsify individual statements. Almost any particular statement can besaved, given suciently radical modications of the containing theory. For Quine, scientic thought forms a coherentweb in which any part could be altered in the light of empirical evidence, and in which no empirical evidence couldforce the revision of a given part.Quines writings have led to the wide acceptance of instrumentalism in the philosophy of science.

4.2.3 Existence and Its contraryThe problem of non-referring names is an old puzzle in philosophy, which Quine captured when he wrote,

A curious thing about the ontological problem is its simplicity. It can be put into three Anglo-Saxonmonosyllables: 'What is there?' It can be answered, moreover, in a word'Everything'and everyonewill accept this answer as true.[18]

More directly, the controversy goes,

How can we talk about Pegasus? To what does the word 'Pegasus refer? If our answer is, 'Something,'then we seem to believe in mystical entities; if our answer is, 'nothing', then we seem to talk about nothingand what sense can be made of this? Certainly when we said that Pegasus was a mythological wingedhorse we make sense, and moreover we speak the truth! If we speak the truth, this must be truth aboutsomething. So we cannot be speaking of nothing.

Quine resists the temptation to say that non-referring terms are meaningless for reasons made clear above. Insteadhe tells us that we must rst determine whether our terms refer or not before we know the proper way to understandthem. However, Czesaw Lejewski criticizes this belief for reducing the matter to empirical discovery when it seemswe should have a formal distinction between referring and non-referring terms or elements of our domain. Lejewskiwrites further,

16 CHAPTER 4. WILLARD VAN ORMAN QUINE

This state of aairs does not seem to be very satisfactory. The idea that some of our rules of inferenceshould depend on empirical information, which may not be forthcoming, is so foreign to the characterof logical inquiry that a thorough re-examination of the two inferences [existential generalization anduniversal instantiation] may prove worth our while.

Lejewski then goes on to oer a description of free logic, which he claims accommodates an answer to the problem.Lejewski also points out that free logic additionally can handle the problem of the empty set for statements like8xFx! 9xFx . Quine had considered the problem of the empty set unrealistic, which left Lejewski unsatised.[19]

4.2.4 LogicOver the course of his career, Quine published numerous technical and expository papers on formal logic, some ofwhich are reprinted in his Selected Logic Papers and in The Ways of Paradox.Quine conned logic to classical bivalent rst-order logic, hence to truth and falsity under any (nonempty) universeof discourse. Hence the following were not logic for Quine:

Higher order logic and set theory. He referred to higher order logic as set theory in disguise"; Much of what Principia Mathematica included in logic was not logic for Quine. Formal systems involving intensional notions, especially modality. Quine was especially hostile to modal logicwith quantication, a battle he largely lost when Saul Kripke's relational semantics became canonical for modallogics.

Quine wrote three undergraduate texts on formal logic:

Elementary Logic. While teaching an introductory course in 1940, Quine discovered that extant texts forphilosophy students did not do justice to quantication theory or rst-order predicate logic. Quine wrote thisbook in 6 weeks as an ad hoc solution to his teaching needs.

Methods of Logic. The four editions of this book resulted from a more advanced undergraduate course in logicQuine taught from the end of World War II until his 1978 retirement.

Philosophy of Logic. A concise and witty undergraduate treatment of a number of Quinian themes, such as theprevalence of use-mention confusions, the dubiousness of quantied modal logic, and the non-logical characterof higher-order logic.

Mathematical Logic is based on Quines graduate teaching during the 1930s and 40s. It shows that much of whatPrincipia Mathematica took more than 1000 pages to say can be said in 250 pages. The proofs are concise, evencryptic. The last chapter, on Gdels incompleteness theorem and Tarskis indenability theorem, along with thearticle Quine (1946), became a launching point for Raymond Smullyan's later lucid exposition of these and relatedresults.Quines work in logic gradually became dated in some respects. Techniques he did not teach and discuss includeanalytic tableaux, recursive functions, and model theory. His treatment of metalogic left something to be desired.For example, Mathematical Logic does not include any proofs of soundness and completeness. Early in his career,the notation of his writings on logic was often idiosyncratic. His later writings nearly always employed the now-datednotation of Principia Mathematica. Set against all this are the simplicity of his preferred method (as exposited inhis Methods of Logic) for determining the satisability of quantied formulas, the richness of his philosophical andlinguistic insights, and the ne prose in which he expressed them.Most of Quines original work in formal logic from 1960 onwards was on variants of his predicate functor logic,one of several ways that have been proposed for doing logic without quantiers. For a comprehensive treatment ofpredicate functor logic and its history, see Quine (1976). For an introduction, see chpt. 45 of his Methods of Logic.Quine was very warm to the possibility that formal logic would eventually be applied outside of philosophy and math-ematics. He wrote several papers on the sort of Boolean algebra employed in electrical engineering, and with EdwardJ. McCluskey, devised the QuineMcCluskey algorithm of reducing Boolean equations to a minimum covering sumof prime implicants.

4.2. WORK 17

4.2.5 Set theory

While his contributions to logic include elegant expositions and a number of technical results, it is in set theory thatQuine was most innovative. He always maintained that mathematics required set theory and that set theory was quitedistinct from logic. He irted with Nelson Goodman's nominalism for a while, but backed away when he failed tond a nominalist grounding of mathematics.Over the course of his career, Quine proposed three variants of axiomatic set theory, each including the axiom ofextensionality:

New Foundations, NF, creates and manipulates sets using a single axiom schema for set admissibility, namelyan axiom schema of stratied comprehension, whereby all individuals satisfying a stratied formula composea set. A stratied formula is one that type theory would allow, were the ontology to include types. However,Quines set theory does not feature types. The metamathematics of NF are curious. NF allows many largesets the now-canonical ZFC set theory does not allow, even sets for which the axiom of choice does not hold.Since the axiom of choice holds for all nite sets, the failure of this axiom in NF proves that NF includesinnite sets. The (relative) consistency of NF is an open question. A modication of NF, NFU, due to R. B.Jensen and admitting urelements (entities that can be members of sets but that lack elements), turns out to beconsistent relative to Peano arithmetic, thus vindicating the intuition behind NF. NF and NFU are the onlyQuinian set theories with a following. For a derivation of foundational mathematics in NF, see Rosser (1952);

The set theory ofMathematical Logic is NF augmented by the proper classes of Von NeumannBernaysGdelset theory, except axiomatized in a much simpler way;

The set theory of Set Theory and Its Logic does away with stratication and is almost entirely derived froma single axiom schema. Quine derived the foundations of mathematics once again. This book includes thedenitive exposition of Quines theory of virtual sets and relations, and surveyed axiomatic set theory as itstood circa 1960. However, Fraenkel, Bar-Hillel and Levy (1973) do a better job of surveying set theory as itstood at mid-century.

All three set theories admit a universal class, but since they are free of any hierarchy of types, they have no need fora distinct universal class at each type level.Quines set theory and its background logic were driven by a desire to minimize posits; each innovation is pushed asfar as it can be pushed before further innovations are introduced. For Quine, there is but one connective, the Sheerstroke, and one quantier, the universal quantier. All polyadic predicates can be reduced to one dyadic predicate,interpretable as set membership. His rules of proof were limited to modus ponens and substitution. He preferredconjunction to either disjunction or the conditional, because conjunction has the least semantic ambiguity. He wasdelighted to discover early in his career that all of rst order logic and set theory could be grounded in a mere twoprimitive notions: abstraction and inclusion. For an elegant introduction to the parsimony of Quines approach tologic, see his New Foundations for Mathematical Logic, ch. 5 in his From a Logical Point of View.

4.2.6 Quines epistemology

Just as he challenged the dominant analyticsynthetic distinction, Quine also took aim at traditional normativeepistemology. According to Quine, traditional epistemology tried to justify the sciences, but this eort (as ex-emplied by Rudolf Carnap) failed, and so we should replace traditional epistemology with an empirical study ofwhat sensory inputs produce what theoretical outputs.:[20] Epistemology, or something like it, simply falls into placeas a chapter of psychology and hence of natural science. It studies a natural phenomenon, viz., a physical humansubject. This human subject is accorded a certain experimentally controlled input certain patterns of irradiationin assorted frequencies, for instance and in the fullness of time the subject delivers as output a description of thethree-dimensional external world and its history. The relation between the meager input and the torrential output is arelation that we are prompted to study for somewhat the same reasons that always prompted epistemology: namely,in order to see how evidence relates to theory, and in what ways ones theory of nature transcends any availableevidence...But a conspicuous dierence between old epistemology and the epistemological enterprise in this newpsychological setting is that we can now make free use of empirical psychology. (Quine, 1969: 823)Quines proposal is extremely controversial among contemporary philosophers and has several important critics, withJaegwon Kim the most prominent among them.[21]

18 CHAPTER 4. WILLARD VAN ORMAN QUINE

4.3 In popular culture A computer program whose output is its own source code is called a "quine" after W.V. Quine. This usagewas introduced by Douglas Hofstadter in his 1979 book, Gdel, Escher, Bach: An Eternal Golden Braid.

Quine is a recurring character in the webcomic Existential Comics.[22]

4.4 Bibliography

4.4.1 Selected books 1934 A System of Logistic. Harvard Univ. Press.[23]

1951 (1940). Mathematical Logic. Harvard Univ. Press. ISBN 0-674-55451-5. 1966. Selected Logic Papers. New York: Random House. 1970 (2nd ed., 1978). With J. S. Ullian. The Web of Belief. New York: Random House. 1980 (1941). Elementary Logic. Harvard Univ. Press. ISBN 0-674-24451-6. 1982 (1950). Methods of Logic. Harvard Univ. Press. 1980 (1953). From a Logical Point of View. Harvard Univ. Press. ISBN 0-674-32351-3. Contains "Twodogmas of Empiricism."

1960 Word and Object. MIT Press; ISBN 0-262-67001-1. The closest thing Quine wrote to a philosophicaltreatise. Chpt. 2 sets out the indeterminacy of translation thesis.

1974 (1971) The Roots of Reference. Open Court Publishing Company ISBN 0-8126-9101-6 (developed fromQuines Carus Lectures)

1976 (1966). The Ways of Paradox. Harvard Univ. Press. 1969 Ontological Relativity and Other Essays. Columbia Univ. Press. ISBN 0-231-08357-2. Contains chapterson ontological relativity, naturalized epistemology, and natural kinds.

1969 (1963). Set Theory and Its Logic. Harvard Univ. Press. 1985 The Time of My Life An Autobiography. Cambridge, The MIT Press. ISBN 0-262-17003-5. 1986:Harvard Univ. Press.

1986 (1970). The Philosophy of Logic. Harvard Univ. Press. 1987 Quiddities: An Intermittently Philosophical Dictionary. Harvard Univ. Press. ISBN 0-14-012522-1. Awork of essays, many subtly humorous, for lay readers, very revealing of the breadth of his interests.

1992 (1990). Pursuit of Truth. Harvard Univ. Press. A short, lively synthesis of his thought for advancedstudents and general readers not fooled by its simplicity. ISBN 0-674-73951-5.

1995, From Stimulus to Science. Harvard Univ. Press. ISBN 0-674-32635-0.

4.4.2 Important articles 1946, Concatenation as a basis for arithmetic. Reprinted in his Selected Logic Papers. Harvard Univ. Press. 1948, "On What There Is", Review of Metaphysics. Reprinted in his 1953 From a Logical Point of View.Harvard University Press.

1951, "Two Dogmas of Empiricism", The Philosophical Review 60: 2043. Reprinted in his 1953 From aLogical Point of View. Harvard University Press.

4.5. SEE ALSO 19

1956, Quantiers and Propositional Attitudes, Journal of Philosophy 53. Reprinted in his 1976 Ways ofParadox. Harvard Univ. Press: 18596.

1969, "Epistemology Naturalized" in Ontological Relativity and Other Essays. New York: Columbia UniversityPress: 6990.

"Truth by Convention, rst published in 1936. Reprinted in the book, Readings in Philosophical Analysis,edited by Herbert Feigl and Wilfrid Sellars, pp. 250273, Appleton-Century-Crofts, 1949.

4.5 See also Hold come what may List of American philosophers Platos beard

4.6 Notes[1] O'Connor, John J.; Robertson, Edmund F. (October 2003), Willard Van Orman Quine,MacTutor History of Mathematics

archive, University of St Andrews.

[2] http://www.nytimes.com/2000/12/29/arts/29QUIN.html

[3] So who *is* the most important philosopher of the past 200 years?" Leiter Reports. Leiterreports.typepad.com. 11March2009. Accessed 8 March 2010.

[4] Poll Results: Who is the most important philosopher of the past 200 years? Brian Leiter. 11 March 2009. Accessed 24Oct 2014.

[5] Prize winner page. The Royal Swedish Academy of Sciences. Kva.se. Retrieved 29 August 2010.

[6] Willard Van Orman Quine. Inamori Foundation. Retrieved 15 December 2012.

[7] Quines Philosophy of Science. Internet Encyclopedia of Philosophy. Iep.utm.edu. 27 July 2009. Accessed 8 March2010.

[8] Mr Strawson on Logical Theory. WV Quine. Mind Vol. 62 No. 248. Oct. 1953.

[9] Colyvan, Mark, Indispensability Arguments in the Philosophy ofMathematics, The Stanford Encyclopedia of Philosophy(Fall 2004 Edition), Edward N. Zalta (ed.)

[10] Guide to the Center for Advanced Studies Records, 19581969. Weselyan University. Wesleyan.edu. Accessed 8 March2010.

[11] Quine, Willard Van Orman; Hahn, Lewis Edwin (1986). The Philosophy of W.V. Quine. Open Court. p. 6. ISBN9780812690101. In my third year of high school I walked often with my new Jamaican friends, Fred and Harold Cassidy,trying to convert them from their Episcopalian faith to atheism.

[12] Wall Street Journal obituary for W V Quine Jan 4 2001

[13] Quiddities: An Intermittently Philosophical Dictionary, entry for Tolerance (pp. 2068)

[14] Paradoxes of Plenty in Theories and Things p.197

[15] The Time of My Life: An Autobiography, pp. 3523

[16] MR: Collaboration Distance. American Mathematical Society. Ams.org. Retrieved 29 August 2010.

[17] Prawitz, Dag. 'Quine and Vericationism.' In Inquiry, Stockholm, 1994, pp 487494

[18] W.V.O. Quine, On What There Is The Review of Metaphysics, New Haven 1948, 2, 21

[19] Czeslaw Lejewski, Logic and Existence British Journal for the Philosophy of Science Vol. 5 (19545), pp. 104119

[20] http://plato.stanford.edu/entries/epistemology-naturalized/#2

20 CHAPTER 4. WILLARD VAN ORMAN QUINE

[21] Naturalized Epistemology. Stanford Encyclopedia of Philosophy. Plato.stanford.edu. 5 July 2001. Accessed 8 March2010.

[22] . Existential Comics. Accessed 24 November 2014

[23] Church, Alonzo (1935). Review: A System of Logistic by Willard Van Orman Quine (PDF). Bull. Amer. Math. Soc. 41(9): 598603. doi:10.1090/s0002-9904-1935-06146-4.

4.7 Further reading Roger F Gibson, ed. (2004). The Cambridge companion to Quine. Cambridge University Press. ISBN0521639492.

, 1988. The Philosophy of W.V. Quine: An Expository Essay. Tampa: University of South Florida. , 1988. Enlightened Empiricism: An Examination of W. V. Quines Theory of Knowledge Tampa:University of South Florida.

, 2004. Quintessence: Basic Readings from the Philosophy of W. V. Quine. Harvard Univ. Press. and Barrett, R., eds., 1990. Perspectives on Quine. Oxford: Blackwell. Gochet, Paul, 1978. Quine en perspective, Paris, Flammarion. Godfrey-Smith, Peter, 2003. Theory and Reality: An Introduction to the Philosophy of Science. Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 18701940. Princeton University Press. Grice, Paul and Peter Strawson. In Defense of a Dogma. The Philosophical Review 65 (1965). Hahn, L. E., and Schilpp, P. A., eds., 1986. The Philosophy of W. V. O. Quine (The Library of Living Philoso-phers). Open Court.

Khler, Dieter, 1999/2003. Sinnesreize, Sprache und Erfahrung: eine Studie zur Quineschen Erkenntnistheorie.Ph.D. thesis, Univ. of Heidelberg.

Murray Murphey, The Development of Quines Philosophy (Heidelberg, Springer, 2012) (Boston Studies in thePhilosophy of Science, 291).

Orenstein, Alex (2002). W.V. Quine. Princeton University Press. Putnam, Hilary. TheGreatest Logical Positivist. Reprinted inRealismwith aHuman Face, ed. James Conant.Cambridge, MA: Harvard University Press, 1990.

Rosser, John Barkley, The axiom of innity in Quines new foundations, Journal of Symbolic Logic 17(4):238242, 1952.

Valore, Paolo, 2001. Questioni di ontologia quineana, Milano: Cusi.

4.8 External links Willard Van Orman QuinePhilosopher and Mathematician Willard Van Orman Quine at the Stanford Encyclopedia of Philosophy Quines Philosophy of Science at the Internet Encyclopedia of Philosophy Quines New Foundations at the Stanford Encyclopedia of Philosophy Willard Van Orman Quine at the Mathematics Genealogy Project Obituary from The Guardian On What There Is

4.8. EXTERNAL LINKS 21

Two Dogmas of Empiricism On Simple Theories Of A Complex World What is Quines Ontology?

Chapter 5

William Kneale

William Calvert Kneale (22 June 1906 24 June 1990) was an English logician best known for his 1962 book TheDevelopment of Logic, a history of logic from its beginnings in Ancient Greece written with his wife Martha. Knealewas also known as a philosopher of science and the author of a book on probability and induction. He was a Fellowof Exeter College, Oxford, and in 1960 succeeded to the Whites Professor of Moral Philosophy previously occupiedby the linguistic philosopher J. L. Austin. He retired in 1966.

5.1 Life and workKneales interest in the history of logic began in the 1940s. The focus of much of Kneales early work was the legacyof the work of the 19th century logician George Boole. His rst major publication in the history of logic was hispaper Boole and the Revival of Logic, published in the philosophy journal Mind in 1948. He was also the authorof a number of papers in Philosophical logic, particularly on the nature of truth for natural languages, and the rolewhich linguistic concepts play in the treatment of logical paradoxes.Kneale worked on his great history of logic from 1947 to 1957 together with his wife Martha (who was responsiblefor the chapters on the ancient Greeks). The result was the 800-page The Development of Logic, rst published in1962 which went through ve impressions before going into a second, paperback, edition in 1984.The 'History' is commonly referred to in the academic world simply as Kneale and Kneale. It was the only majorhistory of logic available in English in the mid-twentieth century, and the rst major history of logic in English sinceThe Development of Symbolic Logic published in 1906 by A.T. Shearman. The treatise has been a standard work inthe history of logic for decades.In 1938 he married Martha Hurst; they had two children, George (born 1942) and Jane (married name Heal); born1946).

5.2 References Thomas Drucker and Irving H. Anellis, 'William Kneale' memorial notice, Modern Logic Volume 3, Number2 (1993), 158161.

22

Chapter 6

Window operator

In modal logic, the window operator4 is a modal operator with the following semantic denition:M;w j= 4 () 8u;M; u j= ) RwuforM = (W;R; f) a Kripke model andw; u 2W . Informally, it says thatw sees every -world (or every -worldis seen by w). This operator is not denable in the basic modal logic (i.e. some propositional non-modal languagetogether with a single primitive necessity (universal) operator, often denoted by ' ', or its existential dual, oftendenoted by ' '). Notice that its truth condition is the converse of the truth condition for the standard necessityoperator.For references to some of its applications, see the References section.

6.1 References Blackburn, P; de Rijke, M; Venema, Y (2002). Modal Logic. Cambridge University Press.

23

Chapter 7

Wisdom of repugnance

Thewisdom of repugnance, or the yuck factor,[1] also known informally as "appeal to disgust",[2] is the belief thatan intuitive (or deep-seated) negative response to some thing, idea or practice should be interpreted as evidence forthe intrinsically harmful or evil character of that thing. Furthermore, it refers to the notion that wisdom may manifestitself in feelings of disgust towards anything which lacks goodness or wisdom, though the feelings or the reasoningof such 'wisdom' may not be immediately explicable through reason.

7.1 Origin and usageThe term wisdom of repugnance was coined in 1997 by Leon Kass, chairman (20012005) of the PresidentsCouncil on Bioethics, in an article in The New Republic,[3] which was later expanded into a further (2001) articlein the same magazine,[4] and also incorporated into his 2002 book Life, Liberty, and the Defense of Dignity.[5] Kassstated that disgust was not an argument per se, but went on to say that in crucial cases...repugnance is the emotionalexpression of deep wisdom, beyond reasons power fully to articulate it.The term remains largely conned to discussions of bioethics, and is somewhat related to the term yuck factor.However, unlike the latter, it is used almost exclusively by those who accept its underlying premise; i.e., that repug-nance does, in fact, indicate wisdom. It is thus often viewed as loaded language, and is primarily used by certainbioconservatives to justify their position.The term has since migrated to other controversies, such as same-sex marriage, pornography, marijuana legalization,alternative sexualities and, in some cases, legalization of early abortion. In all cases, it expresses the view that ones"gut reaction" might justify objecting to some practice even in the absence of a persuasive rational case against thatpractice.

7.2 CriticismThe wisdom of repugnance has been criticized, both as an example of a fallacious appeal to emotion and for anunderlying premise which seems to reject rationalism. Although mainstream science concedes that a sense of disgustmost likely evolved as a useful defense mechanism (e.g. in that it tends to prevent or prohibit potentially harmfulbehaviour such as incest, cannibalism, and coprophagia), social psychologists question whether the instinct can serveany moral or logical value when removed from the context in which it was originally acquired.Martha Nussbaum explicitly opposes the concept of a disgust-based morality. Nussbaum notes that disgust has beenused throughout history as a justication for persecution. For example, at various times racism, antisemitism, sexism,and homophobia have all been driven by popular repulsion.[6]

Stephen Jay Gould has remarked that our prejudices often overwhelm our limited information. [They] are so vener-able, so reexive, so much a part of our second nature, that we never stop to recognize their status as social decisionswith radical alternatives and we view them instead as given and obvious truths.[7]

British bioethicist John Harris replied to Kasss view by arguing that, there is no necessary connection betweenphenomena, attitudes, or actions that make us uneasy, or even those that disgust us, and those phenomena, attitudes,

24

7.3. SEE ALSO 25

and actions that there are good reasons for judging unethical. Nor does it follow that those things we are condentare unethical must be prohibited by legislation or regulation.[8]

Theword squickwas created within BDSM subculture in reaction to this sort of reasoning, and denotes a gut reactionof disgust without the implication of any sort of actual moral judgment.[9]

7.3 See also Anti-intellectualism Appeal to emotion DunningKruger eect Ethical intuitionism Emotivism, which claims that all statements like X is morally wrong only express repugnance, not moral facts Irrationality Moral panic Repugnancy costs Repugnant market Truthiness Uncanny valley

7.4 References[1] Cohen, Patricia (Jan 31, 2008). Economists Dissect the Yuck Factor. The New York Times.[2] Nussbaum, Martha (July 15, 2004). Discussing Disgust. Interview with Reason. Retrieved September 5, 2012.[3] Kass, Leon R. (June 2, 1997). The Wisdom of Repugnance. The New Republic 216 (22) (Washington, DC: CanWest).

pp. 1726.[4] Kass, Leon R (May 21, 2001). Preventing a Brave New World: Why We Should Ban Human Cloning Now. The New

Republic 224 (21). pp. 3039.[5] Kass, Leon R. (2002). Life, Liberty, and the Defense of Dignity. Encounter Books. ISBN 1-893554-55-4.[6] Nussbaum, Martha C. (August 6, 2004). Danger to Human Dignity: The Revival of Disgust and Shame in the Law. The

Chronicle of Higher Education. Washington, DC. Retrieved 2007-11-24.[7] Gould, Stephen Jay (1997). Full House: The Spread of Excellence From Plato to Darwin. Harmony. ISBN 0-517-70849-3.[8] Harris, John (1998). Clones, Genes, and Immortality: Ethics and the Genetic Revolution. Oxford: Oxford University Press.

p. 37. ISBN 0-19-288080-2.[9] Barrett, Grant (ed.) (June 23, 2005). Squick. Double-Tongued Dictionary. Brooklyn, NY: Grant Barrett. Retrieved

2007-11-24.

Hughes, James. Repugnance Isn't Wisdom. McGee, Glenn (2000). ""Playing God: Fears About Genetic Engineering"". The Perfect Baby: A Pragmatic

Approach to Genetics (2nd ed. ed.). New York, NY: Rowman & Littleeld. ISBN 0-8476-9759-2. Reviewedin The Journal of the American Medical Association (subscription required; access date November 24, 2007)

Parisi, Mike (March 26, 2004). The Wisdom of Repugnance: A Critique of Leon Kass. Orions Arm. pp.Book Reviews section. Archived from the original on 2007-09-27. Retrieved 2007-11-24.

Bloom, Paul (July 22, 2004). To Urgh Is Human. The Guardian. London and Manchester: Guardian MediaGroup. pp. Science section. Retrieved 2007-11-24.

Chapter 8

Witness (mathematics)

In mathematical logic, a witness is a specic value t to be substituted for variable x of an existential statement of theform x (x) such that (t) is true.

8.1 ExamplesFor example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula 0=1. Theformula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T isa particular proof of 0 = 1 in T.Boolos, Burgess, and Jerey (2002:81) dene the notion of a witness with the example, in which S is an n-placerelation on natural numbers, R is an n-place recursive relation, and indicates logical equivalence (if and only if):

" S(x1, ..., xn) y R(x1, . . ., xn, y)" A y such that R holds of the xi may be called a 'witness to the relation S holding of the xi (providedwe understand that when the witness is a number rather than a person, a witness only testies to whatis true). In this particular example, B-B-J have dened s to be (positively) recursively semidecidable, orsimply semirecursive.

8.2 Henkin witnessesIn predicate calculus, a Henkin witness for a sentence 9x(x) in a theory T is a term c such that T proves (c)(Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gdels completeness theorempresented by Leon Henkin in 1949.

8.3 Relation to game semanticsThe notion of witness leads to the more general idea of game semantics. In the case of sentence 9x(x) the winningstrategy for the verier is to pick a witness for . For more complex formulas involving universal quantiers, theexistence of a winning strategy for the verier depends on the existence of appropriate Skolem functions. For example,if S denotes 8x9y (x; y) then an equisatisable statement for S is 9f8x(x; f(x)) . The Skolem function f (if itexists) actually codies a winning strategy for the verier of S by returning a witness for the existential sub-formulafor every choice of x the falsier might make.

8.4 See also Certicate (complexity), an analogous concept in computational complexity theory

26

8.5. REFERENCES 27

8.5 References George S. Boolos, John P. Burgess, and Richard C. Jerey, 2002, Computability and Logic: Fourth Edition,Cambridge University Press, ISBN 0-521-00758-5.

Leon Henkin, 1949, The completeness of the rst-order functional calculus, Journal of Symbolic Logic v. 14n. 3, pp. 159166.

Peter G. Hinman, 2005, Fundamentals of mathematical logic, A.K. Peters, ISBN 1-56881-262-0. J. Hintikka and G. Sandu, 2009, Game-Theoretical Semantics in Keith Allan (ed.) Concise Encyclopedia of

Semantics, Elsevier, ISBN 0-08095-968-7, pp. 341343

Chapter 9

Wolfram axiom

TheWolfram axiom is the result of a computer exploration undertaken by StephenWolfram[1] in his A New Kind ofScience looking for the shortest single axiom equivalent to the axioms of Boolean algebra (or propositional calculus).The result[2] of his search was an axiom with six Nands and three variables equivalent to Boolean algebra:

((a.b).c).(a.((a.c).a)) = c

With the dot representing the Nand logical operation (also known as the Sheer stroke), with the following meaning:p Nand q is true if and only if not both p and q are true. It is named for Henry M. Sheer, who proved that all theusual operators of Boolean algebra (Not, And, Or, Implies) could be expressed in terms of Nand. This means thatlogic can be set up using a single operator.Wolframs 25 candidates are precisely the set of Sheer identities of length less or equal to 15 elements (excludingmirror images) that have no noncommutative models of size less or equal to 4 (variables).[3]

Researchers have known for some time that single equational axioms (i.e., 1-bases) exist for Boolean algebra, includ-ing representation in terms of disjunction and negation and in terms of the Sheer stroke. Wolfram proved that therewere no smaller 1-bases candidates than the axiom he found using the techniques described in his NKS book. Theproof is given in two pages (in 4-point type) in Wolframs book. Wolframs axiom is therefore the single simplestaxiom by number of operators and variables needed to reproduce Boolean algebra.Sheer identities were independently obtained by dierent means and reported in a technical memorandum[4] in June2000 acknowledging correspondence with Wolfram in February 2000 in which Wolfram discloses to have found theaxiom in 1999 while preparing his book. In[5] is also shown that a pair of equations (conjectured by StephenWolfram)are equivalent to Boolean algebra.

9.1 See also Boolean algebra

9.2 References[1] Stephen Wolfram, A New Kind of Science, 2002, p. 808811 and 1174.

[2] Rudy Rucker, A review of NKS, The Mathematical Association of America, Monthly 110, 2003.

[3] William Mccune, Robert Vero, Branden Fitelson, Kenneth Harris, Andrew Feist and Larry Wos, Short Single Axiomsfor Boolean algebra, J. Automated Reasoning, 2002.

[4] Robert Vero and William McCune, A Short Sheer Axiom for Boolean algebra, Technical Memorandum No. 244

[5] Robert Vero, Short 2-Bases for Boolean algebra in Terms of the Sheer stroke. Tech. Report TR-CS-2000-25, ComputerScience Department, University of New Mexico, Albuquerque, NM

28

9.3. EXTERNAL LINKS 29

9.3 External links Stephen Wolfram, 2002, "A New Kind of Science, online. Weisstein, Eric W., Wolfram Axiom, MathWorld. http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html Weisstein, Eric W., Boolean algebra, MathWorld. Weisstein, Eric W., Robbins Axiom, MathWorld. Weisstein, Eric W., Huntington Axiom, MathWorld.

Chapter 10

Word sense

In linguistics, a word sense is one of the meanings of a word. A dictionary may have over 50 dierent senses ofthe word play, each of these having a dierent meaning based on the context of the words usage in a sentence. Forexample:

We went to see the play Romeo and Juliet at the theater.

The coach devised a great play that put the visiting team on the defensive.

The children went out to play in the park.

In each sentence we associate a dierent meaning of the word play based on hints the rest of the sentence gives us.People and computers, as they read words, must use a process called word-sense disambiguation[1][2] to nd thecorrect meaning of a word. This process uses context to narrow the possible senses down to the probable ones. Thecontext includes such things as the ideas conveyed by adjacent words and nearby phrases, the known or probablepurpose and register of the conversation or document, and the orientation (time and place) implied or expressed. Thedisambiguation is thus context-sensitive.A word sense may correspond to either a seme (the smallest unit of meaning) or a sememe (the next larger unit ofmeaning), and polysemy is the property of having multiple semes or sememes and thus multiple senses.

10.1 Related termsPolysemy diers from homonymy, where two dierent words (lexemes) happen to have the same spelling andpronunciation.

10.2 See also semantics - study of meaning lexical semantics - the study of what the words of a language denote and how it is that they do this word sense induction - the task of automatically acquiring the senses of a target word word sense disambiguation - the task of automatically associating a sense with a word in context lexical substitution - the task of replacing a word in context with a lexical substitute sememe - unit of meaning linguistics - the scientic study of language, which can be theoretical or applied. sense and reference

30

10.3. REFERENCES 31

10.3 References[1] N. Ide and J. VronisWord Sense Disambiguation: The State of the Art, Computational Linguistics, 24, 1998, pp. 1-40.

[2] R. Navigli. Word Sense Disambiguation: A Survey, ACM Computing Surveys, 41(2), 2009, pp. 1-69.

10.4 External links I dont believe in word senses -- Adam Kilgarri (1997) WordNet(R) - A large lexical database of English words and their meanings maintained by the PrincetonCognitive Science Laboratory.

Chapter 11

Zhegalkin polynomial

Zhegalkin (also Zegalkin or Gegalkine) polynomials form one of many possible representations of the operationsof boolean algebra. Introduced by the Russian mathematician I. I. Zhegalkin in 1927, they are the polynomials ofordinary high school algebra interpreted over the integers mod 2. The resulting degeneracies of modular arithmeticresult in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coecients nor exponents.Coecients are redundant because 1 is the only nonzero coecient. Exponents are redundant because in arithmeticmod 2, x2 = x. Hence a polynomial such as 3x2y5z is congruent to, and can therefore be rewritten as, xyz.

11.1 Boolean equivalentPrior to 1927 boolean algebra had been considered a calculus of logical values with logical operations of conjunction,disjunction, negation, etc. Zhegalkin showed that all boolean operations could be written as ordinary numeric poly-nomials, thinking of the logical constants 0 and 1 as integers mod 2. The logical operation of conjunction is realizedas the arithmetic operation of multiplication xy, and logical exclusive-or as arithmetic addition mod 2, (written hereas xy to avoid confusion with the common use of + as a synonym for inclusive-or ). Logical complement x isthen derived from 1 and as x1. Since and form a sucient basis for the whole of boolean algebra, meaningthat all other logical operations are obtainable as composites of these basic operations, it follows that the polynomialsof ordinary algebra can represent all boolean operations, allowing boolean reasoning to be performed reliably by ap-pealing to the familiar laws of high school algebra without the distraction of the dierences from high school algebrathat arise with disjunction in place of addition mod 2.An example application is the representation of the boolean 2-out-of-3 threshold or median operation as the Zhegalkinpolynomial xyyzzx, which is 1 when at least two of the variables are 1 and 0 otherwise.

11.2 Formal propertiesFormally a Zhegalkin monomial is the product of a nite set of distinct variables (hence square-free), includingthe empty set whose product is denoted 1. There are 2n possible Zhegalkin monomials in n variables, since eachmonomial is fully specied by the presence or absence of each variable. A Zhegalkin polynomial is the sum (exclusive-or) of a set of Zhegalkin monomials, with the empty set denoted by 0. A given monomials presence or absencein a polynomial corresponds to that monomials coecient being 1 or 0 respectively. The Zhegalkin monomials,being linearly independent, span a 2n-dimensional vector space ove