================================================================================ UTF-8-file qedeq_logic_v1 Generated from http://www.qedeq.org/0_04_02/doc/math/qedeq_logic_v1.xml Generated at 2011-05-01 08:45:53.560 ================================================================================ If the characters of this document don't display correctly try opening this document within a webbrowser and if necessary change the encoding to Unicode (UTF-8) Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Elements of Mathematical Logic ══════════════════════════════ Michael Meyling The source for this document can be found here: http://www.qedeq.org/0_04_02/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 mime@qedeq.org The authors of this document are: Michael Meyling ___________________________________________________ Summary ═══════ The project Hilbert II deals with the formal presentation and documentation of mathematical knowledge. For this reason Hilbert II provides a program suite to accomplish that tasks. The concrete documentation of mathematical basics is also a purpose of this project. For further informations about the Hilbert II project see under http://www.qedeq.org . This document describes the logical axioms and the rules and meta rules that are used to derive new propositions. The presentation is axiomatic and in a formal form. A formal calculus is given that enables us to derive all true formulas. Additional derived rules, theorems, definitions, abbreviations and syntax extensions basically correspond with the mathematical practice. This document is also written in a formal language, the original text is a XML file with a syntax defined by the XSD http://www.qedeq.org/current/xml/qedeq.xsd . This document is work in progress and is updated from time to time. Especially at the locations marked by „+++” additions or changes will take place. ___________________________________________________ Foreword ════════ The whole mathematical universe can be unfolded by set‒theoretic means. Beside the set‒theoretic axioms only logical axioms and rules are required. These elementary basics are sufficient to define the most complex mathematical structures and enable us to prove propositions for those structures. This approach can be fully formalized and can be reduced to simple manipulations of character strings. The semantical interpretation of these character strings represent the mathematical universum. It is more than convenient to introduce abbreviations and use further derivation rules. But these comforts could be eliminated and replaced by the basic terms at any time ┌ │ At least this is theoretically possible. This transformation is not │ in each case practically realizable 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. └ . This project has its source in a childhood dream to undertake a formalization of mathematics. In the meantime the technical possibilities are highly developed so that a realization seems within reach. Special thanks go to the professors W .  K e r b y and V .  G ü n t h e r of the university of 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 G e s i n e  D r ä g e r and our son L e n n a r t for their support and patience. Hamburg, december, 2010 Michael Meyling ___________________________________________________ Introduction ════════════ At the beginning we quote D . H i l b e r t from the lecture „The Logical Basis of Mathematics”, September 1922 ┌ │ Lecture given at the Deutsche Naturforscher-Gesellschaft, September │ 1922. └ . „The fundamental idea of my proof theory is the following: All the propositions that constitute in mathematics are converted into formulas, so that mathematics proper becomes all inventory of formulas. These differ from the ordinary formulas of mathematics only 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 edifice of mathematics, are called axioms. A proof is an array that must be given as such to our perceptual intuition of it of inferences according to the schema A A → B _______________________________________ B where each of the premises, that is, the formulae, A and A → B in the array either is an axiom or directly from an axiom by substitution, or else coincides with the end formula B of an inference occurring earlier in the proof or results from it by substitution. A formula is said to be provable if it is either an axiom or the end formula 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 universally applicable. In the 1928 published book G r u n d z ü g e d e r t h e o r e t i s c h e n L o g i k (Principles of Theoretical Logic) D .  H i l b e r t and W .  A c k e r m a n n formalized propositional calculus in a way that build the basis for the logical system used here. 1959 P .  S .  N o v i k o v specified a refined axiom and rule system for predicate calculus. In this text we present a first order predicate calculus with identity and functors that is the starting point for the development of the mathematical theory. Only the results without any proofs and in short form are given in the following. ┌ │ If there is time proofs will be added. └ ___________________________________________________ Chapter 1 Language ════════ In this chapter we define a formal language to express mathematical propositions in a very precise way. Although this document describes a very formal approach to express mathematical content it is not sufficent to serve as a definition for an computer readable document format. Therefore such an extensive specification has to be done elsewhere. The choosen format is the E x t e n s i b l e M a r k u p L a n g u a g e abbreviated X M L . XML is a set of rules for encoding documents electronically. ┌ │ See http://www.w3.org/XML/ for more information. └ The according formal syntax specification can be found at http://www.qedeq.org/current/xml/qedeq.xsd . It specifies a complete mathematical document format that enables the generation of LaTeX books and makes automatic proof checking possible. Further syntax restrictions and some explanations 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 file that can be found here: http://wwww.qedeq.org/0_04_02/doc/math/qedeq_logic_v1.xml . But now we just follow the traditional mathematical way to present the elements of mathematical logic. 1.1 Terms and Formulas ―――――――――――――――――――――― We use the l o g i c a l s y m b o l s L = { ‘¬’, ‘∧’, ‘∨’, ‘↔’, ‘→’, ‘∀’, ∃’ }, the p r e d i c a t e c o n s t a n t s C = {c^k_i | i, k ∈ ω}, the f u n c t i o n v a r i a b l e s ┌ │ Function variables are used for a shorter notation. For example │ writing an identity proposition x = y → f(x) = f(y). Also this │ introduction prepares for the syntax extension for functional │ classes. └ F = {f^k_i | i, k ∈ ω ∧ k > 0}, the f u n c t i o n c o n s t a n t s ┌ │ Function constants are also introduced for convenience and are used │ for direct defined class functions. For example to define building │ of the power class operator, the union and intersection operator and │ the successor function. All these function constants can be │ interpreted as abbreviations. └ H = {h^k_i | i, k ∈ ω}, the s u b j e c t v a r i a b l e s V = {v_i | i ∈ ω}, as well as p r e d i c a t e v a r i a b l e s P = {p^k_i | i, k ∈ ω}. ┌ │ By ω we understand the natural numbers including zero. All involved │ symbols are pairwise disjoint. Therefore we can conclude for │ example: f^k_i = f^(k')_(i') → (k = k’ ∧ i = i’) and h^k_i ≠ v_j. └ For the a r i t y or r a n k of an operator we take the upper index. The set of predicate variables with zero arity is also called set of p r o p o s i t i o n v a r i a b l e s or s e n t e n c e l e t t e r s : A := {p_i⁰ | i ∈ ω }. For subject variables we write short hand certain lower letters: v₁ = ‘u’, v₂ = ‘v’, v₃ = ‘w’, v₄ = ‘x’, v₅ = ‘y’, v₅ = ‘z’. Furthermore we use the following short notations: for the predicate variables pⁿ₁ = ‘φ’ und pⁿ₂ = ‘ψ’, where the appropriate arity n is calculated by counting the subsequent parameters, for the proposition variables a₁ = ‘A’, a₂ = ‘B’ and a₃ = C’, for the function variables: fⁿ₁ = ‘f’ und fⁿ₂ = ‘g’, where again the appropriate arity n is calculated by counting the subsequent parameters. All binary propositional operators are written in infix notation. Parentheses surrounding groups of operands and operators are necessary to indicate the intended order in which operations are to be performed. E. g. for the operator ∧ with the parameters A and B we write (A ∧ B). In the absence of parentheses the usual precedence rules determine the order of operations. Especially outermost parentheses are omitted. Also empty parentheses are stripped. The operators have the order of precedence described below (starting with the highest). ¬, ∀, ∃ ∧ ∨ →, ↔ The term t e r m is defined recursively as follows: 1. Every subject variable is a term. 2. Let i, k ∈ ω and let t₁, ..., t_k be terms. Then h^k_i(t₁, ..., t_k) is a term and if k > 0, so f^k_i(t₁, ..., t_k) is a term too. Therefore all zero arity function constants {h⁰_i | i ∈ ω} are terms. They are called i n d i v i d u a l c o n s t a n t s . ┌ │ In an analogous manner subject variables might be defined as │ function variables of zero arity. Because subject variables play an │ important role they have their own notation. └ We define a f o r m u l a and the relations f r e e and b o u n d subject variable recursivly as follows: 1. Every proposition variable is a formula. Such formulas contain no free or bound subject variables. 2. If p^k is a predicate variable with arity k and c^k is a predicate constant with arity k and t₁, t₂, ..., t_k are terms, then p^k(t₁, t₂, ... t_k) and c^k(t₁, t₂, ..., t_k) are formulas. All subject variables that occur at least in one of t₁, t₂, ..., t_k are free subject variables. Bound subject variables does not occur. ┌ │ This second item includes the first one, which is only listed for │ clarity. └ 3. Let α, β be formulas in which no subject variables occur bound in one formula and free in the other. Then ¬ α, (α ∧ β), (α ∨ β), (α → β) and (α ↔ β) are also formulas. Subject variables which occur free (respectively bound) in α or β stay free (respectively bound). 4. If in the formula α the subject variable x₁ occurs not bound ┌ │ This means that x₁ is free in the formula or does not occur at │ all. └ , then also ∀ x₁ α and ∃ x₁ α are formulas. The symbol ∀ is called u n i v e r s a l q u a n t i f i e r and ∃ as e x i s t e n t i a l q u a n t i f i e r . Except for x₁ all free subject variables of α stay free. All bound subject variables are still bound and additionally x₁ is bound too. All formulas that are only built by usage of 1. and 3. are called formulas of the p r o p o s i t i o n a l c a l c u l u s . For each formula α the following proposition holds: the set of free subject variables is disjoint with the set of bound subject variables.. ┌ │ Other formalizations allow for example ∀ x₁ α also if x₁ occurs │ already bound within α. Also propositions like α(x₁) ∧ (∀ x₁ β) are │ allowed. In this formalizations free and bound are defined for a │ single occurrence of a variable. └ If a formula has the form ∀ x₁   α respectively ∃ x₁   α then the formula α is called the s c o p e of the quantifier ∀ respectively ∃. All formulas that are used to build up a formula by 1. to 4. are called p a r t f o r m u l a s . ___________________________________________________ Chapter 2 Axioms and Rules of Inference ═════════════════════════════ We now state the system of axioms for the predicate calculus and present the rules for obtaining new formulas from them. 2.1 Axioms ―――――――――― The language of our calculus bases on the formalizations of D .  H i l b e r t , W .  A c k e r m a n n [hilback], P .  B e r n a y s and P .  S .  N o v i k o v [novikov]. New rules can be derived from the herein presented. Only these meta rules lead to a smooth flowing logical argumentation. We want to present the axioms, definitions and rules in an syntactical manner but to motivate the choosen form we already give some semantical i n t e r p r e t a t i o n s . The logical operators of propositional calculus ‘¬’, ‘∧’, ‘∨’, ‘→’ and ‘↔’ combine arbitrary p r o p o s i t i o n s to new propositions. A proposition is a statement that affirms or denies something and is either „true” or „false” (but not both). ┌ │ Later on we will define the symbols ⊤ and ⊥ as truth values. └ The binary operator ‘∧’ (logical disjunction) combines the two propositions α and β into the new proposition α ∧ β. It results in true if at least one of its operands is true. The unary operator ‘¬’ (logical negation) changes the truth value of a proposition α. ¬ α has a value of true when its operand is false and a value of false when its operand is true. The l o g i c a l i m p l i c a t i o n ( i f ... t h e n ) the, l o g i c a l c o n j u n c t i o n ( a n d ) and the l o g i c a l e q u i v a l e n c e ( l o g i c a l b i c o n d i t i o n a l ) are defined as abbreviations. ┌ │ Actually the symbols ∨, →, ↔ are defined later on and are a syntax │ extension. But for convenience these symbols are already part of the │ logical language. └ The logical implication (‘if ... then’) could be defined as follows. α → β :↔ ¬ α ∨ β The logical conjunction (‘and’) could be defined with de Morgan. α ∧ β :↔ ¬ (¬ α ∨ ¬ β) The logical equivalence (‘iff’) is defined as usual. α ↔ β :↔ (α → β) ∧ (β → α) Now we come to the first axiom of propositional calculus. This axiom enables us 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 specific y. ☉ Axiom 5 (Universal Instantiation) [axiom:universalInstantiation] ∀ x φ(x) → φ(y) If a predicate holds for some particular y, then there is an x for which the predicate holds. ☉ Axiom 6 (Existential Generalization) [axiom:existencialGeneralization] φ(y) → ∃ x φ(x) 2.2 Rules of Inference ―――――――――――――――――――――― The following rules of inference enable us to obtain new true formulas from the axioms that are assumed to be true. From these new formulas we derive further formulas. So we can successively extend the set of true formulas. ☉ Rule 1 (Modus Ponens) [rule:modusPonens] If both formulas α and α → β are true, then we can conclude that β is true as well. ☉ Rule 2 (Replace Free Subject Variable) [rule:replaceFree] We start with a true formula. 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 in the original formula has two reasons. First it ensures that the resulting formula is well-formed. Secondly it preserves the validity of the formula. Let us look at the following derivation. ∀ x ∃ y φ(x, y) → ∃ y φ(z,y) with axiom 5 ∀ x ∃ y φ(x, y) → ∃ y φ(y,y) forbidden replacement: z in y, despite y is already bound ∀ x ∃ y x ≠ y → ∃ y ≠ y replace ≠ for φ 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 be replaced occurs in more than one scope, then the replacement needs to be made in one scope only. ☉ Rule 4 (Replace Predicate Variable) [rule:replacePred] Let α be a true formula that contains a predicate variable p of arity n, let x₁, ..., x_n be pairwise different subject variables and let β(x₁, ..., x_n) be a formula where x₁, ..., x_n are not bound. The formula β(x₁, ..., x_n) must not contain all x₁, ..., x_n as free subject variables. Furthermore it can also have other subject variables either free or bound. If the following conditions are fulfilled, then a replacement of all occurrences of p(t₁, ..., t_n) each with appropriate terms t₁, ..., t_n in α by β(t₁, ..., t_n) results in another true formula. 1. the free variables of β(x₁, ..., x_n) without x₁, ..., x_n do not occur as bound variables in α 2. each occurrence of p(t₁, ..., t_n) in α contains no bound variable of β(x₁, ..., x_n) 3. the result of the substitution is a well-formed formula See III § 5 in [hilback]. The prohibition to use additional subject variables within the replacement formula that occur bound in the original formula assurs that the resulting formula is well-formed. Furthermore it preserves the validity of the formla. Take a look at the following derivation. φ(x) → ∃ y φ(y) with axiom 6 (∃ y y = y) ∧ φ(x) → ∃ y φ(y) ∃ y (y = y ∧ φ(x)) → ∃ y φ(y) ∃ y (y = y ∧ x ≠ y) → ∃ y y ≠ y forbidden replacment: φ(x) by x ≠ y, despite y is already bound ∃ y 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 α be an already proved formula that contains a function variable σ of arity n, let x₁, ..., x_n be pairwise different subject variables and let τ(x₁, ..., x_n) be an arbitrary term where x₁, ..., x_n are not bound. The term τ(x₁, ..., x_n) must not contain all x₁, ..., x_n as free subject variables. Furthermore it can also have other subject variables either free or bound. If the following conditions are fulfilled we can obtain a new true formula by replacing each occurrence of σ(t₁, ..., t_n) with appropriate terms t₁, ..., t_n in α by τ(t₁, ..., t_n). 1. the free variables of τ(x₁, ..., x_n) without x₁, ..., x_n do not occur as bound variables in α 2. each occurrence of σ(t₁, ..., t_n) in α contains no bound variable of τ(x₁, ..., x_n) 3. the result of the substitution is a well-formed formula ☉ Rule 6 (Universal Generalization) [rule:universalGeneralization] If α → β(x₁) is a true formula and α does not contain the subject variable x₁, then α → (∀ x₁ (β(x₁))) is a true formula too. ☉ Rule 7 (Existential Generalization) [rule:existentialGeneralization] If α(x₁) → β is already proved to be true and β does not contain the subject variable x₁, then (∃ x₁ α(x₁)) → β is also a true formula. The usage and elimination of abbreviations and constants is also an inference rule. In many texts about mathematical logic these rules are not explicitly stated and this text is no exception. But in the exact QEDEQ format corresponding rules exist. ___________________________________________________ Chapter 3 Derived Propositions ════════════════════ Now we derive elementary propositions with the axioms and rules of inference of chapter chapter4. 3.1 Propositional Calculus ―――――――――――――――――――――――――― At first we look at the propositional calculus. To define the predicate t r u e we just combine a predicate and its negation. ☉ Definition 1 (True) [definition:True] ⊤ ↔ (A ∨ ¬ A) For a precise definition we should have written something like p⁰₀ = ⊤ and ⊤ :↔ (A ∨ ¬ A). ┌ │ In the deeper laying (see │ http://www.qedeq.org/current/doc/project/qedeq_logic_language_en.pdf ) │ formal language this predicate has the name T R U E and zero │ arguments. So we just have to map names to natural numbers to │ fulfill the exact definition. └ In the future we only write the symbol itself. It’s arity should be evident from the formula. For the predicate f a l s e we just negate t r u e . ┌ │ Analogous to the preceeding definition we can set p⁰₁ = ⊥ └ ☉ Definition 2 (False) [definition:False] ⊥ ↔ ¬ ⊤ We have the following basic propositions. ☉ Proposition 1 (Basic Propositions) [theorem:propositionalCalculus] (aa) ⊤ (ab) ¬ ⊥ (ac) A → A (ad) A ↔ A (ae) (A ∨ B) ↔ (B ∨ A) (af) (A ∧ B) ↔ (B ∧ A) (ag) (A ∧ B) → A (ah) (A ↔ B) ↔ (B ↔ A) (ai) (A ∨ (B ∨ C)) ↔ ((A ∨ B) ∨ C) (aj) (A ∧ (B ∧ C)) ↔ ((A ∧ B) ∧ C) (ak) A ↔ (A ∨ A) (al) A ↔ (A ∧ A) (am) A ↔ ¬ ¬ A (an) (A → B) ↔ (¬ B → ¬ A) (ao) (A ↔ B) ↔ (¬ A ↔ ¬ B) (ap) (A → (B → C)) ↔ (B → (A → C)) (aq) ¬ (A ∨ B) ↔ (¬ A ∧ ¬ B) (ar) ¬ (A ∧ B) ↔ (¬ A ∨ ¬ B) (as) (A ∨ (B ∧ C)) ↔ ((A ∨ B) ∧ (A ∨ C)) (at) (A ∧ (B ∨ C)) ↔ ((A ∧ B) ∨ (A ∧ C)) (au) (A ∧ ⊤) ↔ A (av) (A ∧ ⊥) ↔ ⊥ (aw) (A ∨ ⊤) ↔ ⊤ (ax) (A ∨ ⊥) ↔ A (ay) (A ∨ ¬ A) ↔ ⊤ (az) (A ∧ ¬ A) ↔ ⊥ (ba) (⊤ → A) ↔ A (bb) (⊥ → A) ↔ ⊤ (bc) (A → ⊥) ↔ ¬ A (bd) (A → ⊤) ↔ ⊤ (be) (A ↔ ⊤) ↔ A (bf) ((A → B) ∧ (B → C)) → (A → C) (bg) ((A ↔ B) ∧ (C ↔ B)) → (A ↔ C) (bh) ((A ∧ B) ↔ (A ∧ C)) ↔ (A → (B ↔ C)) (bi) ((A ∧ B) ↔ (A ∧ ¬ B)) ↔ ¬ A (bj) (A ↔ (A ∧ B)) ↔ (A → B) (bk) (A → B) → ((A ∧ C) → (B ∧ C)) (bl) (A ↔ B) → ((A ∧ C) ↔ (B ∧ C)) (bm) (A ∧ (A → B)) → B (bn) (A ∧ (A → B)) ↔ (A ∧ B) 3.2 Predicate Calculus ―――――――――――――――――――――― For predicate calculus we achieve the following propositions. We have the following basic propositions. ☉ Proposition 2 (Basic Propositions) [theorem:predicateCalculus] (a) ∀ x (φ(x) → ψ(x)) → (∀ x φ(x) → ∀ x ψ(x)) (b) ∀ x (φ(x) → ψ(x)) → (∃ x φ(x) → ∃ x ψ(x)) (c) ∀ x (φ(x) ↔ ψ(x)) → (∀ x φ(x) ↔ ∀ x ψ(x)) (d) ∃ x (φ(x) ∧ ψ(x)) → (∃ x φ(x) ∧ ∃ x ψ(x)) (e) (∀ x ψ(x) ∨ ∀ x ψ(x)) → ∀ x (φ(x) ∨ ψ(x)) (f) ∃ x (φ(x) ∨ ψ(x)) ↔ (∃ x φ(x) ∨ ∃ x ψ(x)) (g) ∀ x (φ(x) ∧ ψ(x)) ↔ (∀ x φ(x) ∧ ∀ x ψ(x)) (h) ∀ x ∀ y φ(x, y) ↔ ∀ y ∀ x φ(x, y) (i) ∃ x ∃ y φ(x, y) ↔ ∃ y ∃ x φ(x, y) (j) ∀ x (φ(x) → A) → (∀ x φ(x) → A) (k) ∀ x (A → φ(x)) ↔ (A → ∀ x φ(x)) (l) ∀ x (φ(x) ∧ A) ↔ (∀ x φ(x) ∧ A) (m) ∀ x (φ(x) ∨ A) ↔ (∀ x φ(x) ∨ A) (n) ∀ x (φ(x) ↔ A) → (∀ x φ(x) ↔ A) (o) ∀ x (φ(x) ↔ ψ(x)) → (∀ x φ(x) ↔ ∀ x ψ(x)) 3.3 Derived Rules ――――――――――――――――― Beginning with the logical basis logical propositions and metarules can be derived an enable a convenient argumentation. Only with these metarules and additional definitions and abbreviations the mathematical world is unfolded. Every additional syntax is c o n s e r v a t i v e . That means: within extended system no formulas can be derived, that are written in the old syntax but can not be derived in the old system. In the following such conservative extensions are introduced. ☉ Rule 8 (Replace by Logical Equivalent Formula) [rule:replaceEquiFormula] Let the formula α ↔ β be true. If in a formula δ we replace an arbitrary occurence of α by β and the result γ is also a formula ┌ │ During that substitution it might be necessary to rename bound │ variables of β. └ and contains all the free subject variables of δ, then δ ↔ γ is a true formula. ☉ Rule 9 (Replacement of ⊤ by already derived formula) [rule:replaceTrueByTrueFormula] Let α be an already derived true formula and β a formula that contains ⊤. If we get a well formed formula γ by replacing an arbitray occurence of ⊤ in β with α then the following formula is also true: β ↔ γ ☉ Rule 10 (Replacement of already derived formula by ⊤) [rule:replaceTrueFormulaByTrue] Let α be an already derived true formula and β a formula that contains α. If we get a well formed formula γ by replacing an arbitray occurence of α in β by ⊤ then the following formula is also true: β ↔ γ ☉ Rule 11 (Derived Quantification) [rule:derivedQuantification] If we have already derived the true formula α(x) and x is not bound in α(x) then the formula ∀ x α(x) is also true. ☉ Rule 12 (General Associativity) [rule:generalAssociativity] If an operator of arity two fulfills the associative law it also fulfills the general associative law. The operator 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. ┌ │ The operator of arity n is defined with a certain bracketing, but │ every other bracketing gives the same result. └ ☉ Rule 13 (General Commutativity) [rule:generalCommutativity] If an operator fulfills the general associative law and is commutative then all permutations of parameters are equal or equivalent. ┌ │ That depends on the operator type: term or formula operator. └ 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 β is d e d u c i b l e f r o m t h e f o r m u l a α if the formula β from the totality of all true formulas of the predicate calculus and the formula α by means of application of all the rules of the predicate calculus, in which connection both rules for binding by a quantifier, the rules for substitution in place of predicate variables and in place of free subject variables must be applied only to predicate variables or subject variables which do not occur in the formula α and α → β is a formula. Notation: α ⊢ β. That a formula β is deducible from th formula α must be strictly distinguished from 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 added to the axioms then the formula B can be derived. But B is not deducible from A. ☉ Rule 15 (Deduction Theorem) [rule:deductionTheorem] If the formula β is deducible from the formula α, then the formula α → β can be derived from the predicate calculus. ___________________________________________________ Chapter 4 Identity ════════ Everything that exists has a specific nature. Each entity exists as something in particular and it has characteristics that are a part of what it is. Identity is whatever makes an entity definable and recognizable, in terms of possessing a set of qualities or characteristics that distinguish it from entities of a different type. An entity can have more than one characteristic, but any characteristic it has is a part of its identity. 4.1 Identity Axioms ――――――――――――――――――― We start with the identy axioms. We define a predicate constant of arity two that shall stand for the identity of subjects. ☉ initial Definition 3 (Identity) [definition:identity] x = y For convenience we also define the negation of the identity a predicate constant. ☉ Definition 4 (Not Identical) [definition:notEqual] x ≠ y ↔ ¬ x = y ☉ Axiom 7 (Reflexivity of Identity) [axiom:identityIsReflexive] x = x ☉ Axiom 8 (Leibniz’ replacement) [axiom:leibnizReplacement] x = y → (φ(x) → φ(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. ☉ Proposition 3 [theorem:leibnizEquivalence] x = y → (φ(x) ↔ φ(y)) ☉ Proposition 4 [theorem:identyImpliesFunctionalEquality] x = y → f(x) = f(y) 4.2 Restricted Quantifiers ―――――――――――――――――――――――――― Every quantification involves one specific subject variable and a domain of discourse or range of quantification of that variable. Until now we assumed a fixed domain of discourse for every quantification. Specification of the range of quantification allows us to express that a predicate holds only for a restricted domain. At the following definition the replacement formula for α(x) must „reveal” its quantification subject variable. This is usually the first following subject variable. ┌ │ For example: in the following formula we identify the subject │ variable m for the second quantification: ∀ n ∈ ℕ ∀ m ∈ n m < n. └ In the exact syntax of the QEDEQ format ┌ │ Again see http://www.qedeq.org/current/xml/qedeq/ . └ the quantification subject variable is always given. ☉ Axiom 11 (Restricted Universal Quantifier) [axiom:restrictedUniversalQuantifier] ∀ α(x) β(x) ↔ ∀ x (α(x) → β(x)) A matching definiton for the restricted existential quantifier is the following. ┌ │ Matching because of ¬ ∀ ψ(x) (φ(x)) ↔ ∃ x ¬ (ψ(x) → φ(x)) ↔ ∃ x │ (ψ(x) ∧ ¬φ(x)) ↔ ∃ ψ(x) (¬φ(x)). └ ☉ Axiom 12 (Restricted Existential Quantifier) [axiom:restrictedExistentialQuantifier] ∃ α(x) β(x) ↔ ∃ x (α(x) ∧ β(x)) For restricted quantifiers we find formulas according to Proposition proposition 2. +++ To express the existence of only one individuum with a certain property we introduce a new quantifier. ☉ Axiom 13 (Restricted Uniqueness Quantifier) [axiom:restrictedUniquenessQuantifier] ∃! α(x) β(x) ↔ ∃ α(x) (β(x) ∧ ∀ α(y) (β(y) → x = y)) [Term Definition by Formula]rule:termdef If the formula ∃! x α(x) holds, we can expand the term syntax by D(x, α(x)). May the formula alpha(x) doesn’t contain the variable y and let β(y) be a formula that doesn’t contain the variable x. Then we define a new formula β(D(x, α(x))) by β(y) ∧ ∃! x (α(x) ∧ x = y). Also in this abbreviate notation the subject variable x counts as bound, the subject variable y is arbitrary (if it fulfills the given conditions) and will be ignored in the abbreviation. Changes in α that lead to another formula α’ because of variable collision with β must also be done in the abbreviation. All term building rules are extended accordingly. The expression is also replaceble by ∃! y (β(y) ∧ α(y) or by β(y) ∧ α(y). ___________________________________________________ Bibliography ════════════ [witheruss] A . N . W h i t e h e a d , B . R u s s e l l , Principia Mathematica, Cambridge University Press, London 1910 [bernays] P . B e r n a y s , Axiomatische Untersuchung des Aussagen-Kalkuls der „Principia Mathematica”, Math. Zeitschr. 25 (1926), 305-320 [hilback] D . H i l b e r t , W . A c k e r m a n n , Grundzüge 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 [novikov] P . S . N o v i k o v , Elements of Mathematical Logic, Edinburgh: Oliver and Boyd, 1964. [mendelson] E . M e n d e l s o n , Introduction to Mathematical Logic, 3rd. ed., Belmont, CA: Wadsworth, 1987. [guenter] V .  G ü n t h e r , Lecture „Mathematik und Logik”, given at the University of Hamburg, 1994/1995. [meyling] M . M e y l i n g , Hilbert II, Presentation of Formal Correct Mathematical Knowledge, Basic Concept, http://www.qedeq.org/current/doc/project/qedeq_basic_concept_en.pdf . QEDEQ modules that use this one: qedeq_set_theory_v1 http://www.qedeq.org/0_04_02/doc/math/qedeq_set_theory_v1.xml