Qedeq Logic v1 En

7/31/2019 Qedeq Logic v1 En

1/26

Elements of Mathematical Logic

Michael Meyling

August 15, 2010


2/26

2

The source for this document can be found here:

http://www.qedeq.org/0_03_12/doc/math/qedeq_logic_v1.xml

Copyright by the authors. All rights reserved.

If you have any questions, suggestions or want to add something to the list of modules that use this one, please send an email to the address [email protected]

The authors of this document are: Michael Meyling [email protected]
http://www.qedeq.org/0_03_12/doc/math/qedeq_logic_v1.xmlmailto:[email protected]:[email protected]:[email protected]:[email protected]://www.qedeq.org/0_03_12/doc/math/qedeq_logic_v1.xml


3/26

Contents

Summary 5

Foreword 7

Introduction 9

1 Language 111.1 Terms and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Axioms and Rules of Inference 152.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Derived Propositions 193.1 Propositional Calculus . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Predicate Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Derived Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Identity 234.1 Identity Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Restricted Quantiers . . . . . . . . . . . . . . . . . . . . . . . . 24Bibliography 25

3


4/26

4 CONTENTS


5/26


6/26

6 CONTENTS


7/26

Foreword

The whole mathematical universe can be unfolded by settheoretic means. Be-side the settheoretic axioms only logical axioms and rules are required. Theseelementary basics are sufficient to dene the most complex mathematical struc-tures and enable us to prove propositions for those structures. This approachcan be fully formalized and can be reduced to simple manipulations of characterstrings. The semantical interpretation of these character strings represent themathematical universum.

It is more than convenient to introduce abbreviations and use further derivationrules. But these comforts could be eliminated and replaced by the basic termsat any time 1 .

This project has its source in a childhood dream to undertake a formalization of mathematics. In the meantime the technical possibilities are highly developedso that a realization seems within reach.

Special thanks go to the professors W. Kerby and V. G unther of the universityof Hamburg for their inspiring lectures about logic and axiomatic set theory.Without these important impulses this project would not exist.

I am deeply grateful to my wife Gesine Dr ager and our son Lennart for theirsupport and patience.

Hamburg, August, 2010Michael Meyling

1 At least this is theoretically possible. This transformation is not in each case practicallyrealizable due to restrictions in time and space. For example it is not possible to write down

the natural number 1 , 000 , 000 , 000 completely in set notation.

7


8/26

8 CONTENTS


9/26

Introduction

At the beginning we quote D. Hilbert from the lecture The Logical Basis of Mathematics, September 1922 2 .

The fundamental idea of my proof theory is the following:All the propositions that constitute in mathematics are convertedinto formulas, so that mathematics proper becomes all inventory of

formulas. These differ from the ordinary formulas of mathematicsonly in that, besides the ordinary signs, the logical signs especially

implies ( ) and for not ( ) occur in them. Certain formulas,which serve as building blocks for the formal edice of mathematics,are called axioms. A proof is an array that must be given as suchto our perceptual intuition of it of inferences according to the schema

AA B

B

where each of the premises, that is, the formulae, A and A Bin the array either is an axiom or directly from an axiom by sub-stitution, or else coincides with the end formula B of an inferenceoccurring earlier in the proof or results from it by substitution. Aformula is said to be provable if it is either an axiom or the endformula of a proof.

At the beginning there is logic. Logic is the analysis of methods of reasoning.It helps to derive new propositions from already given ones. Logic is universallyapplicable.

In the 1928 published book Grundz uge der theoretischen Logik (Principles of Theoretical Logic) D. Hilbert and W. Ackermann formalized propositional cal-culus in a way that build the basis for the logical system used here. 1959P. S. Novikov specied a rened axiom and rule system for predicate calcu-lus.

In this text we present a rst order predicate calculus with identity and functorsthat is the starting point for the development of the mathematical theory. Onlythe results without any proofs and in short form are given in the following .3

2 Lecture given at the Deutsche Naturforscher-Gesellschaft, September 1922.3

If there is time proofs will be added.

9


10/26

10 CONTENTS


11/26

Chapter 1

Language

In this chapter we dene a formal language to express mathematical proposi-tions in a very precise way. Although this document describes a very formalapproach to express mathematical content it is not sufficent to serve as a de-nition for an computer readable document format. Therefore such an extensivespecication has to be done elsewhere. The choosen format is the ExtensibleMarkup Language abbreviated XML. XML is a set of rules for encoding docu-ments electronically .1 The according formal syntax specication can be foundat http://www.qedeq.org/current/xml/qedeq.xsd . It species a completemathematical document format that enables the generation of L ATEXbooks andmakes automatic proof checking possible. Further syntax restrictions and someexplanations can be found at http://www.qedeq.org/current/doc/project/qedeq_logic_language_en.pdf .

Even this document is (or was generated) from an XML le that can be foundhere: http://wwww.qedeq.org/0_03_12/doc/math/qedeq_logic_v1.xml . Butnow we just follow the traditional mathematical way to present the elements of mathematical logic.

1.1 Terms and Formulas

We use the logical symbols L = { , , , , , , }, the predi-cate constants C = {cki | i, k }, the function variables 2 F = {f ki | i, k k > 0}, the function constants 3 H = {h ki | i, k }, the subject variablesV = {vi | i }, as well as predicate variables P = { pki | i, k }.4 Forthe arity or rank of an operator we take the upper index. The set of predicatevariables with zero arity is also called set of proposition variables or sentenceletters : A := { p0i | i }. For subject variables we write short hand certainlower letters: v1 = u, v2 = v, v3 = w, v4 = x, v5 = y, v5 = z. Further-more we use the following short notations: for the predicate variables pn1 = und pn2 = , where the appropriate arity n is calculated by counting the subse-quent parameters, for the proposition variables a 1 = A, a 2 = B and a 3 = C ,

1 See http://www.w3.org/XML/ for more information.2 Function variables are used for a shorter notation. For example writing an identity propo-

sition x = y f (x ) = f (y ). Also this introduction prepares for the syntax extension forfunctional classes.

3 Function constants are also introduced for convenience and are used for direct denedclass functions. For example to dene building of the power class operator, the union and in-tersection operator and the successor function. All these function constants can be interpretedas abbreviations.

4 By we understand the natural numbers including zero. All involved symbols are pairwise

disjoint. Therefore we can conclude for example: f ki = f

ki (k = k i = i ) and h

ki = v j .

11
http://www.qedeq.org/current/xml/qedeq.xsdhttp://www.qedeq.org/current/xml/qedeq.xsdhttp://www.qedeq.org/current/doc/project/qedeq_logic_language_en.pdfhttp://www.qedeq.org/current/doc/project/qedeq_logic_language_en.pdfhttp://www.qedeq.org/current/doc/project/qedeq_logic_language_en.pdfhttp://wwww.qedeq.org/0_03_12/doc/math/qedeq_logic_v1.xmlhttp://www.w3.org/XML/http://www.w3.org/XML/http://wwww.qedeq.org/0_03_12/doc/math/qedeq_logic_v1.xmlhttp://www.qedeq.org/current/doc/project/qedeq_logic_language_en.pdfhttp://www.qedeq.org/current/doc/project/qedeq_logic_language_en.pdfhttp://www.qedeq.org/current/xml/qedeq.xsd


12/26

12 CHAPTER 1. LANGUAGE

for the function variables: f n1 = f und f n2 = g, where again the appropriatearity n is calculated by counting the subsequent parameters. All binary propo-sitional operators are written in inx notation. Parentheses surrounding groupsof operands and operators are necessary to indicate the intended order in whichoperations are to be performed. E. g. for the operator with the parameters Aand B we write ( A B ).

In the absence of parentheses the usual precedence rules determine the orderof operations. Especially outermost parentheses are omitted. Also empty paren-theses are stripped.

The operators have the order of precedence described below (starting with thehighest).

, ,

,

The term term is dened recursively as follows:

1. Every subject variable is a term.

2. Let i, k and let t 1 , . . . , t k be terms. Then h ki (t 1 , . . . , t k ) is a term andif k > 0, so f ki (t 1 , . . . , t k ) is a term too.

Therefore all zero arity function constants {h 0i | i } are terms. They arecalled individual constants .5

We dene a formula and the relations free and bound subject variable recursivlyas follows:

1. Every proposition variable is a formula. Such formulas contain no free orbound subject variables.

2. If pk is a predicate variable with arity k and ck is a predicate con-stant with arity k and t 1 , t 2 , . . . , t k are terms, then pk (t 1 , t 2 , . . . t k ) andck (t 1 , t 2 , . . . , t k ) are formulas. All subject variables that occur at least inone of t 1 , t 2 , . . . , t k are free subject variables. Bound subject variables doesnot occur .6

3. Let , be formulas in which no subject variables occur bound in oneformula and free in the other. Then , ( ), ( ), ( ) and( ) are also formulas. Subject variables which occur free (respectivelybound) in or stay free (respectively bound).

4. If in the formula the subject variable x 1 occurs not bound 7 , then alsox 1 and x 1 are formulas. The symbol is called universal quantier and as existential quantier .

Except for x 1 all free subject variables of stay free. All bound subjectvariables are still bound and additionally x 1 is bound too.

All formulas that are only built by usage of 1. and 3. are called formulas of thepropositional calculus .

5 In an analogous manner subject variables might be dened as function variables of zeroarity. Because subject variables play an important role they have their own notation.

6 This second item includes the rst one, which is only listed for clarity.7

This means that x 1 is free in the formula or does not occur at all.


13/26

1.1. TERMS AND FORMULAS 13

For each formula the following proposition holds: the set of free subject vari-ables is disjoint with the set of bound subject variables.. 8

If a formula has the form x 1 respectively x 1 then the formula is calledthe scope of the quantier respectively .

All formulas that are used to build up a formula by 1. to 4. are called part formulas .

8 Other formalizations allow for example x 1 also if x 1 occurs already bound within .Also propositions like (x ) (x 1 ) are allowed. In this formalizations free and bound are

dened for a single occurrence of a variable.


14/26

14 CHAPTER 1. LANGUAGE


15/26

Chapter 2

Axioms and Rules of Inference

We now state the system of axioms for the predicate calculus and present therules for obtaining new formulas from them.

2.1 Axioms

The language of our calculus bases on the formalizations of D. Hilbert , W. Ack-ermann [3], P. Bernays and P. S. Novikov [4]. New rules can be derived fromthe herein presented. Only these meta rules lead to a smooth owing logicalargumentation.

We want to present the axioms, denitions and rules in an syntactical manner

but to motivate the choosen form we already give some semantical interpreta-tions .

The logical operators of propositional calculus , , , und combinearbitrary propositions to new propositions. A proposition is a statement thataffirms or denies something and is either true or false (but not both) .

1

The binary operator (logical disjunction) combines the two propositions and into the new proposition . It results in true if at least one of itsoperands is true.

The unary operator (logical negation) changes the truth value of a propo-sition . has a value of true when its operand is false and a value of falsewhen its operand is true.

The logical implication (if ) the, logical conjunction (and ) and the logical equiv-alence (biconditional ) are dened as abbreviations. 2

The logical implication (if) could be dened as follows.

Denition 1 (Implication) . [definition:implication]

:

The logical conjunction (and) could be dened with de Morgan.1 Later on we will dene the symbols and as truth values.2 Actually the symbols , , are dened later on and are a syntax extension. But for

convenience these symbols are already part of the logical language.

15


16/26

16 CHAPTER 2. AXIOMS AND RULES OF INFERENCE

Denition 2 (Conjunction) . [definition:conjunction]

: ( )

The logical equivalence (iff) is dened as usual.

Denition 3 (Equivalence) . [definition:equivalence]

: ( ) ( )

Now we come to the rst axiom of propositional calculus. This axiom enablesus to get rid of an unnecessary disjunction.

Axiom 1 (Disjunction Idempotence) . [axiom:disjunction_idempotence]

(A A) A

If a proposition is true, any alternative may be added without making it false.

Axiom 2 (Axiom of Weakening) . [axiom:disjunction_weakening]

A (A B )

The disjunction should be commutative.

Axiom 3 (Commutativity of the Disjunction) . [axiom:disjunction_commutative]

(A B ) (B A)

An disjunction could be added at both side of an implication.

Axiom 4 (Disjunctive Addition) . [axiom:disjunction_addition]

(A B ) ((C A) (C B ))

If something is true for all x , it is true for any specic y.

Axiom 5 (Universal Instantiation) . [axiom:universalInstantiation]

x (x) (y)

If a predicate holds for some particular y, then there is an x for which thepredicate holds.

Axiom 6 (Existential Generalization) . [axiom:existencialGeneralization]

(y) x (x)

2.2 Rules of Inference

The following rules of inference enable us to obtain new true formulas from theaxioms that are assumed to be true. From these new formulas we derive furtherformulas. So we can successively extend the set of true formulas.

Rule 1 (Modus Ponens) . [rule:modusPonens] If both formulas and are true,

then we can conclude that is true as well.


17/26

2.2. RULES OF INFERENCE 17

Rule 2 (Replace Free Subject Variable) . [rule:replaceFree] We start with a true for-mula. A free subject variable may be replaced by an arbitrary term, provided that the substituted term contains no subject variable that have a bound occurrence in the original formula. All occurrences of the free variable must be simultaneously replaced.

The prohibition to use subject variables within the term that occur bound inthe original formula has two reasons. First it ensures that the resulting formulais well-formed. Secondly it preserves the validity of the formula. Let us look atthe following derivation.

x y (x, y ) y (z, y ) with axiom 5x y (x, y ) y (y, y ) forbidden replacement: z in y, despite y is

already boundx y x = y y = y replace = for

This last proposition is not valid in many models.

Rule 3 (Rename Bound Subject Variable) . [rule:renameBound] We may replace a bound subject variable occurring in a formula by any other subject variable, provided

that the new variable occurs not free in the original formula. If the variable to bereplaced occurs in more than one scope, then the replacement needs to be madein one scope only.

Rule 4 (Replace Predicate Variable) . [rule:replacePred] Let be a true formula that contains a predicate variable p of arity n , let x 1 , . . . , x n be subject variablesand let (x 1 , . . . , x n ) be a formula where x 1 , . . . , x n are not bound. The for-mula (x 1 , . . . , x n ) must not contain all x 1 , . . . , x n as free subject variables.Furthermore it can also have other subject variables either free or bound.

If the following conditions are fullled, then a replacement of all occurrences of p(t 1 , . . . , t n ) each with appropriate terms t 1 , . . . , t n in by (t 1 , . . . , t n ) resultsin another true formula.

the free variables of (x 1 , . . . , x n ) without x 1 , . . . , x n do not occur asbound variables in

each occurrence of p(t 1 , . . . , t n ) in contains no bound variable of (x 1 , . . . , x n )

the result of the substitution is a well-formed formula

See III 5 in [3].

The prohibition to use additional subject variables within the replacement for-mula that occur bound in the original formula assurs that the resulting formulais well-formed. Furthermore it preserves the validity of the formla. Take a lookat the following derivation.

(x) y (y) with axiom 6(y y = y) (x) y (y)y (y = y (x)) y (y)y (y = y x = y) y y = y forbidden replacment: (x) by x = y,

despite y is already boundy x = y y y = y

The last proposition is not valid in many models.

Analogous to rule 4 we can replace function variables too.

Rule 5 (Replace Function Variable) . [rule:replaceFunct] Let be an already proved

formula that contains a function variable of arity n , let x 1 , . . . , x n be subject


18/26

18 CHAPTER 2. AXIOMS AND RULES OF INFERENCE

variables and let (x 1 , . . . , x n ) be an arbitrary term where x 1 , . . . , x n are not bound. The term (x 1 , . . . , x n ) must not contain all x 1 , . . . , x n . as free subject variables. Furthermore it can also have other subject variables either free or bound.

If the following conditions are fullled we can obtain a new true formula by replacing each occurrence of (t 1 , . . . , t n ) with appropriate terms t 1 , . . . , t n in by (t 1 , . . . , t n ).

the free variables of (x 1 , . . . , x n ) without x 1 , . . . , x n do not occur asbound variables in

each occurrence of (x 1 , . . . , x n ) in contains no bound variable of (x 1 , . . . , x n )

the result of the substitution is a well-formed formula

Rule 6 (Universal Quantier Introduction) . [rule:universalIntroduction] If (x 1 )is a true formula and does not contain the subject variable x 1 , then (x 1 ( (x 1 ))) is a true formula too.

Rule 7 (Existential Quantier Introduction) . [rule:existentialIntroduction] If (x 1 ) is already proved to be true and does not contain the subject variable x 1 , then (x 1 (x 1 )) is also a true formula.

The usage and elimination of abbreviations and constants is also an inferencerule. In many texts about mathematical logic these rules are not explicitly statedand this text is no exception. But in the exact QEDEQ format correspondingrules exist.


19/26

Chapter 3

Derived Propositions

Now we derive elementary propositions with the axioms and rules of inferenceof chapter 2.

3.1 Propositional Calculus

At rst we look at the propositional calculus.

To dene the predicate true we just combine a predicate and its negation.

Denition 4 (True) . [definition:True]

: A A

For a precise denition we should have written something like p00 = and: (A A). In the formal language this predicate has the name TRUE and

zero arguments. So we just have to map names to natural numbers to fulll theformer denition. In the future we only write the symbol itself. Its arity shouldbe evident from the formula.

For the predicate false we just negate true .

Denition 5 (False) . [definition:False]

:

We have the following basic propositions.

Proposition 1 (Basic Propositions) . [theorem:propositionalCalculus]

(aa) (ab)

A A (ac)A A (ad)

(A B ) (B A) (ae)(A B ) (B A) (af)

(A B ) A (ag)(A B ) (B A) (ah)

(A (B C )) ((A B ) C ) (ai)(A (B C )) ((A B ) C ) (aj)

A (A A) (ak)A (A A) (al)

A A (am)

19


20/26

20 CHAPTER 3. DERIVED PROPOSITIONS

(A B ) (B A) (an)(A B ) (A B ) (ao)

(A (B C )) (B (A C )) (ap)(A B ) (A B ) (aq)(A B ) (A B ) (ar)

(A (B C )) ((A B ) (A C )) (as)

(A (B C )) ((A B ) (A C )) (at)(A ) A (au)(A ) (av)(A ) (aw)(A ) A (ax)

(A A) (ay)(A A) (az)( A) A (ba)( A) (bb)

(A ) A (bc)(A ) (bd)(A ) A (be)

((A B ) (B C )) (A C ) (bf)((A B ) (C B )) (A C ) (bg)

((A B ) (A C )) (A (B C )) (bh)((A B ) (A B )) A (bi)(A (A B )) (A B ) (bj)

(A B ) ((A C ) (B C )) (bk)(A B ) ((A C ) (B C )) (bl)

3.2 Predicate Calculus

For predicate calculus we achieve the following propositions.

We have the following basic propositions.Proposition 2 (Basic Propositions) . [theorem:predicateCalculus]

x ((x ) (x)) (x (x) x (x)) (a)x ((x ) (x)) (x (x) x (x)) (b)x ((x) (x)) (x (x) x (x)) (c)(x (x ) x (x)) x ((x) (x)) (d)x ((x) (x)) (x (x) x (x)) (e)x ((x) (x)) (x (x) x (x)) (f)

x y (x, y ) y x (x, y ) (g)x y (x, y ) y x (x, y ) (h)

x ((x) A) (x (x) A) (i)

x (A (x)) (A x (x)) (j)x ((x) A) (x (x) A) (k)x ((x) A) (x (x) A) (l)x ((x) A) (x (x) A) (m)

3.3 Derived Rules

Beginning with the logical basis logical propositions and metarules can be de-rived an enable a convenient argumentation. Only with these metarules andadditional denitions and abbreviations the mathematical world is unfolded.Every additional syntax is conservative . That means: within extended system

no formulas can be derived, that are written in the old syntax but can not


21/26

3.3. DERIVED RULES 21

be derived in the old system. In the following such conservative extensions areintroduced.

Rule 8 (Replace by Logical Equivalent Formula) . [rule:replaceEquiFormula] Let the for-mula be true. If in a formula we replace an arbitrary occurence of by and the result is also a formula 1 and contains all the free subject variablesof , then is a true formula.

Rule 9 (Replacement of by already derived formula) . [rule:replaceTrueByTrueFormula] Let be an already derived true formula and a formula that contains . If weget a well formed formula by replacing an arbitray occurence of in with then the following formula is also true:

Rule 10 (Replacement of already derived formula by ). [rule:replaceTrueFormulaByTrue]Let be an already derived true formula and a formula that contains . If weget a well formed formula by replacing an arbitray occurence of in by then the following formula is also true:

Rule 11 (Derived Quantication) . [rule:derivedQuantification] If we have already derived the true formula (x) and x is not bound in (x) then the formula x (x) is

also true.Rule 12 (General Associativity) . [rule:generalAssociativity] If an operator of arity two fullls the associative law it also fullls the general associative law. The opera-tor can be extended to an operator of arbitrary arity greater one. For example:instead of (a + b) + ( c + d) we simply write a + b + c + d.2

Rule 13 (General Commutativity) . [rule:generalCommutativity] If an operator fullls thegeneral associative law and is commutative then all permutations of parametersare equal or equivalent .3 For example we have: a + b + c + d = c + a + d + b.

Rule 14 (Deducible from Formula) . [rule:definitionDeductionFromFormula] We shall say that the formula is deducible from the formula if the formula from the totality of all true formulas of the predicate calculus and the formula by means of application of all the rules of the predicate calculus, in which connection both rules for binding by a quantier, the rules for substitution in place of predicatevariables and in place of free subject variables must be applied only to predicatevariables or subject variables which do not occur in the formula and isa formula.

Notation: .

That a formula is deducible from th formula must be strictly distinguishedfrom the deduction of a true formula from the axioms of the predicate calculus.In the second case more derivation rules are available. For example if A is addedto the axioms then the formula B can be derived. But B is not deducible fromA .

Rule 15 (Deduction Theorem) . [rule:deductionTheorem] If the formula is deducible from the formula , then the formula can be derived from the predicatecalculus.

1 During that substitution it might be necessary to rename bound variables of .2 The operator of arity n is dened with a certain bracketing, but every other bracketing

gives the same result.3

That depends on the operator type: term or formula operator.


22/26

22 CHAPTER 3. DERIVED PROPOSITIONS


23/26

Chapter 4

Identity

Everything that exists has a specic nature. Each entity exists as somethingin particular and it has characteristics that are a part of what it is. Identity iswhatever makes an entity denable and recognizable, in terms of possessing aset of qualities or characteristics that distinguish it from entities of a differenttype. An entity can have more than one characteristic, but any characteristic ithas is a part of its identity.

4.1 Identity Axioms

We start with the identy axioms.

We dene a predicate constant of arity two that shall stand for the identity of subjects.

Initial Denition 6 (Identity) . [definition:identity]

x = y

For convenience we also dene the negation of the identity a predicate constant.

Denition 7 (Not Identical) . [definition:notEqual]

x = y : x = y

Axiom 7 (Reexivity of Identity) . [axiom:identityIsReflexive]

x = x

Axiom 8 (Leibniz replacement) . [axiom:leibnizReplacement]

x = y ((x) (y))

Axiom 9 (Symmetrie of identity) . [axiom:symmetryOfIdentity]

x = y y = x

Axiom 10 (Transitivity of identity) . [axiom:transitivityOfIdentity]

(x = y y = z) x = z

We can reverse the second implication in the Leibniz replacement.

23


24/26

24 CHAPTER 4. IDENTITY

Proposition 3. [theorem:leibnizEquivalence]

x = y ((x) (y))

Proposition 4. [theorem:identyImpliesFunctionalEquality]

x = y f (x) = f (y)

4.2 Restricted Quantiers

Every quantication involves one specic subject variable and a domain of dis-course or range of quantication of that variable. Until now we assumed a xeddomain of discourse for every quantication. Specication of the range of quan-tication allows us to express that a predicate holds only for a restricted domain.

At the following denition the replacement formula for (x) must reveal itsquantication subject variable. This is usually the rst following subject vari-able. 1 In the exact syntax of the QEDEQ format 2 the quantication subject

variable is always given.Denition 8 (Restricted Universal Quantier) . [definition:restrictedUniversalQuantifier]

(x) ( (x)) : x ( (x) (x))

A matching deniton for the restricted existential quantier is the following .3

Denition 9 (Restricted Existential Quantier) . [definition:restrictedExistentialQuantifier]

(x) ( (x)) : x ( (x) (x))

For restricted quantiers we nd formulas according to Proposition 2.+++To express the existence of only one individuum with a certain property weintroduce a new quantier.

Denition 10 (Restricted Uniqueness Quantier) . [definition:restrictedUniquenessQuantifier]

! (x) ( (x)) : (x) ( (x) (y) ( (y) x = y))

Rule 16 (Term Denition by Formula) . If the formula !x (x) holds, wecan expand the term syntax by D (x, (x)) . May the formula alpha (x ) doesnt contain the variable y and let (y) be a formula that doesnt contain the variablex . Then we dene a new formula (D (x, (x))) by (y)!x ( (x)x = y). Alsoin this abbreviate notation the subject variable x counts as bound, the subject variable y is arbitrary (if it fullls the given conditions) and will be ignored in theabbreviation. Changes in that lead to another formula because of variablecollision with must also be done in the abbreviation. All term building rulesare extended accordingly. The expression is also replaceble by !y ( (y) (y)or by (y) (y).

1 For example: in the following formula we identify the subject variable m for the secondquantication: n N m n m < n .

2 Again see http://www.qedeq.org/current/xml/qedeq/ .3 Matching because of (x ) ( (x )) x ( (x ) (x )) x ( (x ) (x ))

(x ) ( (x )).
http://www.qedeq.org/current/xml/qedeq/http://www.qedeq.org/current/xml/qedeq/http://www.qedeq.org/current/xml/qedeq/


25/26

Bibliography

[1] A.N. Whitehead, B. Russell , Principia Mathematica, Cambridge UniversityPress, London 1910

[2] P. Bernays , Axiomatische Untersuchung des Aussagen-Kalkuls der Prin-cipia Mathematica, Math. Zeitschr. 25 (1926), 305-320

[3] D. Hilbert, W. Ackermann , Grundz uge der theoretischen Logik, 2nd ed.,Berlin: Springer, 1938. English version: Principles of Mathematical Logic,Chelsea, New York 1950, ed. by R. E. Luce. See also http://www.math.uwaterloo.ca/ ~snburris/htdocs/scav/hilbert/hilbert.html 15, 17

[4] P.S. Novikov , Elements of Mathematical Logic, Edinburgh: Oliver andBoyd, 1964. 15

[5] E. Mendelson , Introduction to Mathematical Logic, 3rd. ed., Belmont, CA:Wadsworth, 1987.

[6] V. G unther , Lecture Mathematik und Logik, given at the University of Hamburg, 1994/1995.

[7] M. Meyling , Hilbert II, Presentation of Formal Correct Mathemati-

cal Knowledge, Basic Concept, http://www.qedeq.org/current/doc/project/qedeq_basic_concept_en.pdf .

[8] qedeq set theory v1 http://www.qedeq.org/0_03_12/doc/math/qedeq_set_theory_v1.xml

25
http://www.math.uwaterloo.ca/~snburris/htdocs/scav/hilbert/hilbert.htmlhttp://www.math.uwaterloo.ca/~snburris/htdocs/scav/hilbert/hilbert.htmlhttp://www.math.uwaterloo.ca/~snburris/htdocs/scav/hilbert/hilbert.htmlhttp://www.math.uwaterloo.ca/~snburris/htdocs/scav/hilbert/hilbert.htmlhttp://www.qedeq.org/current/doc/project/qedeq_basic_concept_en.pdfhttp://www.qedeq.org/current/doc/project/qedeq_basic_concept_en.pdfhttp://www.qedeq.org/current/doc/project/qedeq_basic_concept_en.pdfhttp://www.qedeq.org/0_03_12/doc/math/qedeq_set_theory_v1.xmlhttp://www.qedeq.org/0_03_12/doc/math/qedeq_set_theory_v1.xmlhttp://www.qedeq.org/0_03_12/doc/math/qedeq_set_theory_v1.xmlhttp://www.qedeq.org/0_03_12/doc/math/qedeq_set_theory_v1.xmlhttp://www.qedeq.org/current/doc/project/qedeq_basic_concept_en.pdfhttp://www.qedeq.org/current/doc/project/qedeq_basic_concept_en.pdfhttp://www.math.uwaterloo.ca/~snburris/htdocs/scav/hilbert/hilbert.htmlhttp://www.math.uwaterloo.ca/~snburris/htdocs/scav/hilbert/hilbert.html


26/26

Date post:	05-Apr-2018
Category:	Documents
Upload:	abdullah-uenal
View:	224 times
Download:	0 times

Qedeq Logic v1 En

Documents