Euclidian Roles in Description Logics
... The last years much research effort has been spend towards increasing the expressiveness of Description Logics with respect to what can be said about roles. For example, in [2] the Description Logic RIQ is extended with several role axioms, like reflexive and irreflexive role axioms, disjoint role a ...
... The last years much research effort has been spend towards increasing the expressiveness of Description Logics with respect to what can be said about roles. For example, in [2] the Description Logic RIQ is extended with several role axioms, like reflexive and irreflexive role axioms, disjoint role a ...
Introduction to Mathematical Logic lecture notes
... Then S = n∈N Sn is the set of all formulae. We call Sn the set of formulae constructed in n steps. For example, (P → Q) and (R ∨ (P ∧ (¬Q))) are formulae, but (∧P ) and ¬Q) are not. With time we will allow ourselves to omit some parentheses if the meaning remains clear: for example, instead of (¬((¬ ...
... Then S = n∈N Sn is the set of all formulae. We call Sn the set of formulae constructed in n steps. For example, (P → Q) and (R ∨ (P ∧ (¬Q))) are formulae, but (∧P ) and ¬Q) are not. With time we will allow ourselves to omit some parentheses if the meaning remains clear: for example, instead of (¬((¬ ...
QUASI-MINIMAL DEGREES FOR DEGREE SPECTRA 1
... Let τ be a partial finite part and e, x ∈ N. The forcing relation is defined as follows: (i) τ Fe (x) if and only if there exists a v such that hv, xi ∈ We and for all u ∈ Dv , there is 1 ≤ i ≤ k such that (u =h0, i, xu1 , . . . , xuri i, & xu1 , . . . , xuri ∈ dom(τ ) & (τ (xu1 ), . . . , τ (xuri ...
... Let τ be a partial finite part and e, x ∈ N. The forcing relation is defined as follows: (i) τ Fe (x) if and only if there exists a v such that hv, xi ∈ We and for all u ∈ Dv , there is 1 ≤ i ≤ k such that (u =h0, i, xu1 , . . . , xuri i, & xu1 , . . . , xuri ∈ dom(τ ) & (τ (xu1 ), . . . , τ (xuri ...
Chapter 6: The Deductive Characterization of Logic
... To say that α follows from Γ by R is to say that the pair 〈Γ,α〉 stand in relation R. To say that a relation R is computable means that a computer can, in principle, decide whether a given pair 〈Γ,α〉 stand in the relation. We will ignore this formal issue at the moment; suffice it to say that the rul ...
... To say that α follows from Γ by R is to say that the pair 〈Γ,α〉 stand in relation R. To say that a relation R is computable means that a computer can, in principle, decide whether a given pair 〈Γ,α〉 stand in the relation. We will ignore this formal issue at the moment; suffice it to say that the rul ...
On the Question of Absolute Undecidability
... significantly alter the nature of the case for the failure of CH. More importantly, it opens up the way for an important new inner model, something we discuss in items 3 and 4 of the postscript.] ...
... significantly alter the nature of the case for the failure of CH. More importantly, it opens up the way for an important new inner model, something we discuss in items 3 and 4 of the postscript.] ...
Back to Basics: Revisiting the Incompleteness
... associating expressions with code numbers for them will be taken as well understood. Assume some such system as fixed throughout. If e is an expression, or a sequence of expressions, then in informal contexts peq is the Gödel number for e, and in formal contexts peq, i.e. the same expression1 , sta ...
... associating expressions with code numbers for them will be taken as well understood. Assume some such system as fixed throughout. If e is an expression, or a sequence of expressions, then in informal contexts peq is the Gödel number for e, and in formal contexts peq, i.e. the same expression1 , sta ...
Provability as a Modal Operator with the models of PA as the Worlds
... Theorem 3: MB , A (ψ → ψ) → ψ for all worlds A and arbitrary modal formulae ψ. Proof: B is Löbian so all three of Löb’s conditions hold. Furthermore, the diagonalization lemma applies to any formula B of first order arithmetic, so we can construct γ such that PA ⊢ γ ↔ [Bγ → ϕ]. These are all th ...
... Theorem 3: MB , A (ψ → ψ) → ψ for all worlds A and arbitrary modal formulae ψ. Proof: B is Löbian so all three of Löb’s conditions hold. Furthermore, the diagonalization lemma applies to any formula B of first order arithmetic, so we can construct γ such that PA ⊢ γ ↔ [Bγ → ϕ]. These are all th ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based ...
... uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based ...
Completeness theorems and lambda
... For example, the formula (∧nX(n)) ∨ ∨n¬X(n) because for each n0 the formula X(n0) ∨ ∨n¬X(n) is valid The fact that such a statement is true iff it is provable is known as Henkin-Orey completeness theorem for ω-logic In this logic, a proof tree is a well-founded countably branching tree ...
... For example, the formula (∧nX(n)) ∨ ∨n¬X(n) because for each n0 the formula X(n0) ∨ ∨n¬X(n) is valid The fact that such a statement is true iff it is provable is known as Henkin-Orey completeness theorem for ω-logic In this logic, a proof tree is a well-founded countably branching tree ...
page 135 ADAPTIVE LOGICS FOR QUESTION EVOCATION
... is followed, the resulting reasoning processes not only exhibit an external form of dynamics, but also an internal one — the withdrawal of previously derived conclusions may be caused by merely analysing the premises. There are several arguments in favour of this last option. The first is that unwan ...
... is followed, the resulting reasoning processes not only exhibit an external form of dynamics, but also an internal one — the withdrawal of previously derived conclusions may be caused by merely analysing the premises. There are several arguments in favour of this last option. The first is that unwan ...
On the structure of honest elementary degrees - FAU Math
... the same material as we do in Part I of [9], but we give more detailed proofs and more elaborated explanations. This should be the most thorough and readable introduction to the honest elementary degrees available so far. But be aware that we are talking about a technical introduction, and it is bey ...
... the same material as we do in Part I of [9], but we give more detailed proofs and more elaborated explanations. This should be the most thorough and readable introduction to the honest elementary degrees available so far. But be aware that we are talking about a technical introduction, and it is bey ...
Introduction to Logic
... 12.3. Turing Machines . . . . . . . . . . . . . . . . 12.4. Formal Systems in general . . . . . . . . . . . 12.4.1. Axiomatic System – the syntactic part 12.4.2. Semantics . . . . . . . . . . . . . . . . 12.4.3. Syntax vs. Semantics . . . . . . . . . 12.5. Statement Logic . . . . . . . . . . . . . . ...
... 12.3. Turing Machines . . . . . . . . . . . . . . . . 12.4. Formal Systems in general . . . . . . . . . . . 12.4.1. Axiomatic System – the syntactic part 12.4.2. Semantics . . . . . . . . . . . . . . . . 12.4.3. Syntax vs. Semantics . . . . . . . . . 12.5. Statement Logic . . . . . . . . . . . . . . ...
Introduction to Logic
... 12.3. Turing Machines . . . . . . . . . . . . . . . . 12.4. Formal Systems in general . . . . . . . . . . . 12.4.1. Axiomatic System – the syntactic part 12.4.2. Semantics . . . . . . . . . . . . . . . . 12.4.3. Syntax vs. Semantics . . . . . . . . . 12.5. Statement Logic . . . . . . . . . . . . . . ...
... 12.3. Turing Machines . . . . . . . . . . . . . . . . 12.4. Formal Systems in general . . . . . . . . . . . 12.4.1. Axiomatic System – the syntactic part 12.4.2. Semantics . . . . . . . . . . . . . . . . 12.4.3. Syntax vs. Semantics . . . . . . . . . 12.5. Statement Logic . . . . . . . . . . . . . . ...
Notes on First Order Logic
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
Gödel Without (Too Many) Tears
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
you can this version here
... 10.4.1 Truth-predicates and truth-definitions . . . . . 10.4.2 The undefinability of truth . . . . . . . . . . . 10.4.3 The inexpressibility of truth . . . . . . . . . . . Box: The Master Argument for incompleteness . . . . . . . 10.5 Rosser’s Theorem . . . . . . . . . . . . . . . . . . . . . 10.5.1 ...
... 10.4.1 Truth-predicates and truth-definitions . . . . . 10.4.2 The undefinability of truth . . . . . . . . . . . 10.4.3 The inexpressibility of truth . . . . . . . . . . . Box: The Master Argument for incompleteness . . . . . . . 10.5 Rosser’s Theorem . . . . . . . . . . . . . . . . . . . . . 10.5.1 ...
Introduction to Logic
... 12.3. Turing Machines . . . . . . . . . . . . . . . . 12.4. Formal Systems in general . . . . . . . . . . . 12.4.1. Axiomatic System – the syntactic part 12.4.2. Semantics . . . . . . . . . . . . . . . . 12.4.3. Syntax vs. Semantics . . . . . . . . . 12.5. Statement Logic . . . . . . . . . . . . . . ...
... 12.3. Turing Machines . . . . . . . . . . . . . . . . 12.4. Formal Systems in general . . . . . . . . . . . 12.4.1. Axiomatic System – the syntactic part 12.4.2. Semantics . . . . . . . . . . . . . . . . 12.4.3. Syntax vs. Semantics . . . . . . . . . 12.5. Statement Logic . . . . . . . . . . . . . . ...
Safety Metric Temporal Logic is Fully Decidable
... In [16] we already gave a procedure for model checking Alur-Dill timed automata against Safety MTL formulas. As a special case we obtained the decidability of the validity problem for Safety MTL (‘Is a given formula satisfied by every timed word?’). The two main contributions of the present paper co ...
... In [16] we already gave a procedure for model checking Alur-Dill timed automata against Safety MTL formulas. As a special case we obtained the decidability of the validity problem for Safety MTL (‘Is a given formula satisfied by every timed word?’). The two main contributions of the present paper co ...
Notes on Classical Propositional Logic
... might talk about the leftmost occurrence of the symbol + in the equatiion (x + y) = (y + x). Here we are using syntax. Or we might say 3 + 5 is the number 8. Here we are applying an operation. While these are quite different, there is usually no problem keeping the differences sorted out. Now, here ...
... might talk about the leftmost occurrence of the symbol + in the equatiion (x + y) = (y + x). Here we are using syntax. Or we might say 3 + 5 is the number 8. Here we are applying an operation. While these are quite different, there is usually no problem keeping the differences sorted out. Now, here ...
Logic and Discrete Mathematics for Computer Scientists
... one, students are left in doubt as to what a proof actually is and what might count as one. We are convinced that this process, of learning by example only works for students who have some innate ability to understand the distinctions being made by repeated exposure to examples. But these distinctio ...
... one, students are left in doubt as to what a proof actually is and what might count as one. We are convinced that this process, of learning by example only works for students who have some innate ability to understand the distinctions being made by repeated exposure to examples. But these distinctio ...
Axiomatic Set Teory P.D.Welch.
... B. Jensen : Modelle der Mengenlehre (Springer Lecture Notes in Maths, vol 37,1967), and his subsequent lecture notes. ...
... B. Jensen : Modelle der Mengenlehre (Springer Lecture Notes in Maths, vol 37,1967), and his subsequent lecture notes. ...
The Dedekind Reals in Abstract Stone Duality
... to “turn the symbols upside down” (>/⊥, ∧/∨, =/6=, ∀/∃), often giving a new theorem. In this context, we shall see what the foundational roles of Dedekind completeness and the Heine–Borel theorem actually are. The former is the way in which the logical manipulation of topology has an impact on numer ...
... to “turn the symbols upside down” (>/⊥, ∧/∨, =/6=, ∀/∃), often giving a new theorem. In this context, we shall see what the foundational roles of Dedekind completeness and the Heine–Borel theorem actually are. The former is the way in which the logical manipulation of topology has an impact on numer ...
First-Order Proof Theory of Arithmetic
... can prove the arithmetized version of the cut-elimination theorem and those which cannot; in practice, this is equivalent to whether the theory can prove that the superexponential function i 7→ 21i is total. The very weak theories are theories which do not admit any induction axioms. Non-logical sym ...
... can prove the arithmetized version of the cut-elimination theorem and those which cannot; in practice, this is equivalent to whether the theory can prove that the superexponential function i 7→ 21i is total. The very weak theories are theories which do not admit any induction axioms. Non-logical sym ...
210ch2 - Dr. Djamel Bouchaffra
... • Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. • Hence, g(a) g(b) It follows that g must be an injection. However, f need not be an injection (you show). CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, ...
... • Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. • Hence, g(a) g(b) It follows that g must be an injection. However, f need not be an injection (you show). CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, ...
Lecture 09
... – Suppose that P(1) holds and P(k) → P(k + 1) is true for all positive integers k. – Assume there is at least one positive integer n for which P(n) is false. Then the set S of positive integers for which P(n) is false is nonempty. – By the well-ordering property, S has a least element, say m. – We k ...
... – Suppose that P(1) holds and P(k) → P(k + 1) is true for all positive integers k. – Assume there is at least one positive integer n for which P(n) is false. Then the set S of positive integers for which P(n) is false is nonempty. – By the well-ordering property, S has a least element, say m. – We k ...