Friday, May 23, 2008

Intuitionistic logic English www.tool-tool.com

Bewise Inc. www.tool-tool.com Reference source from the internet.

Intuitionistic logic, or constructivist logic, is the symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. The system preserves justification, rather than truth, across transformations yielding derived propositions. From a practical point of view, there is also a strong motivation for using intuitionistic logic, since it has the existence property, making it also suitable for other forms of mathematical constructivism.

[edit] Syntax
Intuitionistic propositional formulas in one variable (aka Rieger–Nishimura lattice)

Intuitionistic propositional formulas in one variable (aka Rieger–Nishimura lattice)

The syntax of formulæ of intuitionistic logic is similar to propositional logic or first-order logic. However, intuitionistic connectives are not interdefinable in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic it is customary to use →, ∧, ∨, ⊥ as the basic connectives, treating ¬ as the abbreviation ¬A = (A → ⊥). In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.

Many tautologies of classical logic can no longer be proven within intuitionistic logic. Examples include not only the law of excluded middle p ∨ ¬p, but also Peirce's law ((p → q) → p) → p, and even double negation elimination. In classical logic, both p → ¬¬p and also ¬¬p → p are theorems. In intuitionistic logic, only the former is a theorem: double negation can be introduced, but it cannot be eliminated.

The observation that many classically valid tautologies are not theorems of intuitionistic logic leads to the idea of weakening the proof theory of classical logic.

[edit] Sequent calculus

Main article: sequent calculus

Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system which is sound and complete with respect to intuitionistic logic. He called this system LJ.

[edit] Hilbert-style calculus

Intuitionistic logic can be defined using the following Hilbert-style calculus. Compare with the deduction system at Propositional calculus#Alternative calculus.

In propositional logic, the inference rule is modus ponens

* MP: from φ and φ → ψ infer ψ

and the axioms are

* THEN-1: φ → (χ → φ)
* THEN-2: (φ → (χ → ψ)) → ((φ → χ) → (φ → ψ))
* AND-1: φ ∧ χ → φ
* AND-2: φ ∧ χ → χ
* AND-3: φ → (χ → (φ ∧ χ))
* OR-1: φ → φ ∨ χ
* OR-2: χ → φ ∨ χ
* OR-3: (φ → ψ) → ((χ → ψ) → (φ ∨ χ → ψ))
* FALSE: ⊥ → φ

To make this a system of first-order predicate logic, the generalization rules

* ∀-GEN: from ψ → φ infer ψ → (∀x φ), if x is not free in ψ
* ∃-GEN: from φ → ψ infer (∃x φ) → ψ, if x is not free in ψ

are added, along with the axioms

* PRED-1: (∀x φ(x)) → φ(t), if no free occurrence of x in φ is bound by a quantifier quantifying a variable occurring in the term t
* PRED-2: φ(t) → (∃x φ(x)), with the same restriction as for PRED-1

[edit] Optional connectives

[edit] Negation

If one wishes to include a connective ¬ for negation rather than consider it an abbreviation for φ → ⊥, it is enough to add:

* NOT-1′: (φ → ⊥) → ¬φ
* NOT-2′: ¬φ → (φ → ⊥)

There are a number of alternatives available if one wishes to omit the connective ⊥ (false). For example, one may replace the three axioms FALSE, NOT-1′, and NOT-2′ with the two axioms

* NOT-1: (φ → χ) → ((φ → ¬χ) → ¬φ)
* NOT-2: φ → (¬φ → χ)

as at Propositional calculus#Axioms. Alternatives to NOT-1 are (φ → ¬χ) → (χ → ¬φ) or (φ → ¬φ) → ¬φ.

[edit] Equivalence

The connective ↔ for equivalence may be treated as an abbreviation, with φ ↔ χ standing for (φ → χ) ∧ (χ → φ). Alternatively, one may add the axioms

* IFF-1: (φ ↔ χ) → (φ → χ)
* IFF-2: (φ ↔ χ) → (χ → φ)
* IFF-3: (φ → χ) → ((χ → φ) → (φ ↔ χ))

IFF-1 and IFF-2 can, if desired, be combined into a single axiom (φ ↔ χ) → ((φ → χ) ∧ (χ → φ)) using conjunction.

[edit] Relation to classical logic

The system of classical logic is obtained by adding any one of the following axioms:

* φ ∨ ¬φ (Law of the excluded middle. May also be formulated as (φ → χ) → ((¬φ → χ) → χ).)
* ¬¬φ → φ (Double negation elimination)
* ((φ → χ) → φ) → φ (Peirce's law)

In general, one may take as the extra axiom any classical tautology that is not valid in the two-element Kripke frame \circ{\longrightarrow}\circ (in other words, that is not included in Smetanich's logic).

[edit] Non-interdefinability of operators

In classical propositional logic, it is possible to take one of conjunction, disjunction, or implication as primitive, and define the other two in terms of it together with negation, such as in Łukasiewicz's three axioms of propositional logic. It is even possible to define all four in terms of a sole sufficient operator such as the Peirce arrow (NOR) or Sheffer stroke (NAND). Similarly, in classical first-order logic, one of the quantifiers can be defined in terms of the other and negation.

These are fundamentally consequences of the law of bivalence, which makes all such connectives merely boolean functions. The law of bivalence does not hold in intuitionistic logic, only the law of non-contradiction. As a result none of the basic connectives can be dispensed with, and the above axioms are all necessary. Most of the classical identities are only theorems of intuitionistic logic in one direction, although some are theorems in both directions. They are as follows:

Conjunction versus disjunction:

* (\phi \wedge \psi) \to \neg (\neg \phi \vee \neg \psi)
* (\phi \vee \psi) \to \neg (\neg \phi \wedge \neg \psi)
* (\neg \phi \vee \neg \psi) \to \neg (\phi \wedge \psi)
* (\neg \phi \wedge \neg \psi) \leftrightarrow \neg (\phi \vee \psi)

Conjunction versus implication:

* (\phi \wedge \psi) \to \neg (\phi \to \neg \psi)
* (\phi \to \psi) \to \neg (\phi \wedge \neg \psi)
* (\phi \wedge \neg \psi) \to \neg (\phi \to \psi)
* (\phi \to \neg \psi) \leftrightarrow \neg (\phi \wedge \psi)

Disjunction versus implication:

* (\phi \vee \psi) \to (\neg \phi \to \psi)
* (\neg \phi \vee \psi) \to (\phi \to \psi)
* \neg (\phi \to \psi) \to \neg (\neg \phi \vee \psi)
* \neg (\phi \vee \psi) \leftrightarrow \neg (\neg \phi \to \psi)

Universal versus existential quantification:

* (\forall x \ \phi(x)) \to \neg (\exist x \ \neg \phi(x))
* (\exist x \ \phi(x)) \to \neg (\forall x \ \neg \phi(x))
* (\exist x \ \neg \phi(x)) \to \neg (\forall x \ \phi(x))
* (\forall x \ \neg \phi(x)) \leftrightarrow \neg (\exist x \ \phi(x))

So, for example, "a or b" is a stronger statement than "if not a, then b", whereas these are classically interchangeable. On the other hand, "neither a nor b" is equivalent to "not a, and also not b".

If we include equivalence in the list of connectives, some of the connectives become definable from others:

* (\phi\leftrightarrow \psi) \leftrightarrow ((\phi \to \psi)\land(\psi\to\phi))
* (\phi\to\psi) \leftrightarrow ((\phi\lor\psi) \leftrightarrow \psi)
* (\phi\to\psi) \leftrightarrow ((\phi\land\psi) \leftrightarrow \phi)
* (\phi\land\psi) \leftrightarrow ((\phi\to\psi)\leftrightarrow\phi)
* (\phi\land\psi) \leftrightarrow (((\phi\lor\psi)\leftrightarrow\psi)\leftrightarrow\phi)

In particular, {∨, ↔, ⊥} and {∨, ↔, ¬} are complete bases of intuitionistic connectives.

As shown by Kuznetsov, either of the following defined connectives can serve the role of a sole sufficient operator for intuitionistic logic:[1]

* ((p\lor q)\land\neg r)\lor(\neg p\land(q\leftrightarrow r)),
* p\to(q\land\neg r\land(s\lor t)).

[edit] Semantics

The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics.

[edit] Heyting algebra semantics

In classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra. The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra. Then we have the useful theorem that a formula is a valid sentence of classical logic if and only if its value is 1 for every valuation—that is, for any assignment of values to its variables.

A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case. A formula is valid in intuitionistic logic if and only if it receives the value of the top element for any valuation on any Heyting algebra.

It can be shown that to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elements are the open subsets of the real line R.[2] In this algebra, the ∧ and ∨ operations correspond to set intersection and union, and the value assigned to a formula A → B is int(AC ∪ B), the interior of the union of the value of B and the complement of the value of A. The bottom element is the empty set ∅, and the top element is the entire line R. Negation is as usual defined as ¬A = A → ∅, so the value of ¬A reduces to int(AC), the interior of the complement of the value of A, also known as the exterior of A. With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line.[2]

For example, the formula ¬(A ∧ ¬A) is valid, because no matter what set X is chosen as the value of the formula A, the value of ¬(A ∧ ¬A) can be shown to be the entire line:

Value(¬(A ∧ ¬A)) =
int((Value(A ∧ ¬A))C) =
int((Value(A) ∩ Value(¬A))C) =
int((X ∩ int((Value(A))C))C) =
int((X ∩ int(XC))C)

A theorem of topology tells us that int(XC) is a subset of XC, so the intersection is empty, leaving:

int(∅C) = int(R) = R

So the valuation of this formula is true, and indeed the formula is valid.

But the law of the excluded middle, A ∨ ¬A, can be shown to be invalid by letting the value of A be {y : y > 0 }. Then the value of ¬A is the interior of {y : y ≤ 0 }, which is {y : y <>y : y > 0 } and {y : y <>y : y ≠ 0 }, not the entire line.

The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in the top element, representing true, as the valuation of the formula, regardless of what values from the algebra are assigned to the variables of the formula.[2] Conversely, for every invalid formula, there is an assignment of values to the variables that yields a valuation that differs from the top element.[3][4] No finite Heyting algebra has both these properties.[2]

[edit] Kripke semantics

Main article: Kripke semantics

Building upon his work on semantics of modal logic, Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics

歡迎來到Bewise Inc.的世界,首先恭喜您來到這接受新的資訊讓產業更有競爭力,我們是提供專業刀具製造商,應對客戶高品質的刀具需求,我們可以協助客戶滿足您對產業的不同要求,我們有能力達到非常卓越的客戶需求品質,這是現有相關技術無法比擬的,我們成功的滿足了各行各業的要求,包括:精密HSS DIN切削刀具、協助客戶設計刀具流程、DIN or JIS 鎢鋼切削刀具設計、NAS986 NAS965 NAS897 NAS937orNAS907 航太切削刀具,NAS航太刀具設計、超高硬度的切削刀具、BW捨棄式鑽石V卡刀’BW捨棄式金屬圓鋸片、木工捨棄式金屬圓鋸片、PCD木工圓鋸片、醫療配件刀具設計、汽車業刀具設計、電子產業鑽石刀具、全鎢鋼V卡刀-電路版專用’全鎢鋼鋸片、焊刃式側銑刀、焊刃式千鳥側銑刀、焊刃式T型銑刀、焊刃式千鳥T型銑刀、焊刃式螺旋機械鉸刀、全鎢鋼斜邊刀電路版專用、鎢鋼焊刃式高速鉸刀、超微粒鎢鋼機械鉸刀、超微粒鎢鋼定點鑽、焊刃式帶柄角度銑刀、焊刃式螺旋立銑刀、焊刃式帶柄倒角銑刀、焊刃式角度銑刀、焊刃式筒型平面銑刀、木工產業鑽石刀具等等。我們的產品涵蓋了從民生刀具到工業級的刀具設計;從微細刀具到大型刀具;從小型生產到大型量產;全自動整合;我們的技術可提供您連續生產的效能,我們整體的服務及卓越的技術,恭迎您親自體驗!!

BW Bewise Inc. Willy Chen willy@tool-tool.com bw@tool-tool.com www.tool-tool.com skype:willy_chen_bw mobile:0937-618-190 Head &Administration Office No.13,Shiang Shang 2nd St., West Chiu Taichung,Taiwan 40356 http://www.tool-tool..com / FAX:+886 4 2471 4839 N.Branch 5F,No.460,Fu Shin North Rd.,Taipei,Taiwan S.Branch No.24,Sec.1,Chia Pu East Rd.,Taipao City,Chiayi Hsien,Taiwan

Welcome to BW tool world! We are an experienced tool maker specialized in cutting tools. We focus on what you need and endeavor to research the best cutter to satisfy users’ demand. Our customers involve wide range of industries, like mold & die, aerospace, electronic, machinery, etc. We are professional expert in cutting field. We would like to solve every problem from you. Please feel free to contact us, its our pleasure to serve for you. BW product including: cutting tool、aerospace tool .HSS DIN Cutting tool、Carbide end mills、Carbide cutting tool、NAS Cutting tool、NAS986 NAS965 NAS897 NAS937orNAS907 Cutting Tools,Carbide end mill、disc milling cutter,Aerospace cutting tool、hss drill’Фрезеры’Carbide drill、High speed steel、Milling cutter、CVDD(Chemical Vapor Deposition Diamond )’PCBN (Polycrystalline Cubic Boron Nitride) ’Core drill、Tapered end mills、CVD Diamond Tools Inserts’PCD Edge-Beveling Cutter(Golden Finger’Edge modifying knife’Solid carbide saw blade-V type’V-type locking-special use for PC board’Metal Slitting Sawa’Carbide Side milling Cutters’Carbide Side Milling Cutters With Staggered Teeth’Carbide T-Slot Milling Cutters’Carbide T-Slot Milling Cutters With Staggered Teeth’Carbide Machine Reamers’High speed reamer-standard type’High speed reamer-long type’’PCD V-Cutter’PCD Wood tools’PCD Cutting tools’PCD Circular Saw Blade’PVDD End Mills’diamond tool ‘V-type locking-special use for PC board ‘Single Crystal Diamond ‘Metric end mills、Miniature end mills、Специальные режущие инструменты ‘Пустотелое сверло ‘Pilot reamer、Fraises’Fresas con mango’ PCD (Polycrystalline diamond) ‘Frese’Electronics cutter、Step drill、Metal cutting saw、Double margin drill、Gun barrel、Angle milling cutter、Carbide burrs、Carbide tipped cutter、Chamfering tool、IC card engraving cutter、Side cutter、NAS tool、DIN or JIS tool、Special tool、Metal slitting saws、Shell end mills、Side and face milling cutters、Side chip clearance saws、Long end mills、Stub roughing end mills、Dovetail milling cutters、Carbide slot drills、Carbide torus cutters、Angel carbide end mills、Carbide torus cutters、Carbide ball-nosed slot drills、Mould cutter、Tool manufacturer.

Bewise Inc. www.tool-tool.com

ようこそBewise Inc.の世界へお越し下さいませ、先ず御目出度たいのは新たな

情報を受け取って頂き、もっと各産業に競争力プラス展開。

弊社は専門なエンド・ミルの製造メーカーで、客先に色んな分野のニーズ、

豊富なパリエーションを満足させ、特にハイテク品質要求にサポート致します。

弊社は各領域に供給できる内容は:

(1)精密HSSエンド・ミルのR&D

(2)Carbide Cutting tools設計

(3)鎢鋼エンド・ミル設計

(4)航空エンド・ミル設計

(5)超高硬度エンド・ミル

(6)ダイヤモンド・エンド・ミル

(7)医療用品エンド・ミル設計

(8)自動車部品&材料加工向けエンド・ミル設計

弊社の製品の供給調達機能は:

(1)生活産業~ハイテク工業までのエンド・ミル設計

(2)ミクロ・エンド・ミル~大型エンド・ミル供給

(3)小Lot生産~大量発注対応供給

(4)オートメーション整備調達

(5)スポット対応~流れ生産対応

弊社の全般供給体制及び技術自慢の総合専門製造メーカーに貴方のご体験を御待ちしております。

BW специализируется в научных исследованиях и разработках, и снабжаем самым высокотехнологичным карбидовым материалом для поставки режущих / фрезеровочных инструментов для почвы, воздушного пространства и электронной индустрии. В нашу основную продукцию входит твердый карбид / быстрорежущая сталь, а также двигатели, микроэлектрические дрели, IC картонорезальные машины, фрезы для гравирования, режущие пилы, фрезеры-расширители, фрезеры-расширители с резцом, дрели, резаки форм для шлицевого вала / звездочки роликовой цепи, и специальные нано инструменты. Пожалуйста, посетите сайт www.tool-tool.com для получения большей информации.

BW is specialized in R&D and sourcing the most advanced carbide material with high-tech coating to supply cutting / milling tool for mould & die, aero space and electronic industry. Our main products include solid carbide / HSS end mills, micro electronic drill, IC card cutter, engraving cutter, shell end mills, cutting saw, reamer, thread reamer, leading drill, involute gear cutter for spur wheel, rack and worm milling cutter, thread milling cutter, form cutters for spline shaft/roller chain sprocket, and special tool, with nano grade. Please visit our web www.tool-tool.com for more info.

No comments: