First-Order Theorem Proving and Vampire

... http://www.tptp.org contains a large collection of first-order problems. For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax this group theory problem can be written down as follows: %---- 1 * x = 1 fof(lef ...

... http://www.tptp.org contains a large collection of first-order problems. For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax this group theory problem can be written down as follows: %---- 1 * x = 1 fof(lef ...

Enumerations in computable structure theory

... Families of sets with special enumeration properties have been used to produce a number of interesting examples in computable structure theory. Selivanov [22] constructed a family of sets that Goncharov [13] used to produce a structure that is computably categorical but not relatively computably cat ...

... Families of sets with special enumeration properties have been used to produce a number of interesting examples in computable structure theory. Selivanov [22] constructed a family of sets that Goncharov [13] used to produce a structure that is computably categorical but not relatively computably cat ...

Enumerations in computable structure theory

... It would be pleasant if computable categoricity and relative computable categoricity were the same–then we could drop the eﬀectiveness conditions from Goncharov’s result. However, Goncharov [11] showed that this is not the case, using an enumeration result of Selivanov [17]. There are examples with ...

... It would be pleasant if computable categoricity and relative computable categoricity were the same–then we could drop the eﬀectiveness conditions from Goncharov’s result. However, Goncharov [11] showed that this is not the case, using an enumeration result of Selivanov [17]. There are examples with ...

Per Lindström FIRST

... The next task of logic is then to formulate suitable logical rules of inference by means of which theorems of T can be derived from the axioms of T. Again, without formalization, such rules could not be investigated or even precisely defined. What the logical rules are, their properties, and their r ...

... The next task of logic is then to formulate suitable logical rules of inference by means of which theorems of T can be derived from the axioms of T. Again, without formalization, such rules could not be investigated or even precisely defined. What the logical rules are, their properties, and their r ...

Proofs

... Case 2. We show that if x 2 is even then x must be even (the if part or sufficiency) . We use an indirect proof: Assume x is not even and show x2 is not even. If x is not even then it must be odd. So, x = 2k + 1 for some k. Then x2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 which is odd and hence ...

... Case 2. We show that if x 2 is even then x must be even (the if part or sufficiency) . We use an indirect proof: Assume x is not even and show x2 is not even. If x is not even then it must be odd. So, x = 2k + 1 for some k. Then x2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 which is odd and hence ...

Harmony, Normality and Stability

... and the eliminations are in the end only consequences thereof, which could be expressed thus: In the elimination of a symbol, the formula in question, whose outer symbol it concerns, may only “be used as that which it means on the basis of the introduction of this symbol”.’2 Gentzen’s Thesis invites ...

... and the eliminations are in the end only consequences thereof, which could be expressed thus: In the elimination of a symbol, the formula in question, whose outer symbol it concerns, may only “be used as that which it means on the basis of the introduction of this symbol”.’2 Gentzen’s Thesis invites ...

How to Go Nonmonotonic Contents David Makinson

... Both of the representations are useful. Sometimes one is more convenient than another. For example, it is often easier to visualize things in terms of the relation, but more concise to formulate and prove them using the operation. The same will be true when we come to nonclassical consequence. For t ...

... Both of the representations are useful. Sometimes one is more convenient than another. For example, it is often easier to visualize things in terms of the relation, but more concise to formulate and prove them using the operation. The same will be true when we come to nonclassical consequence. For t ...

The Computer Modelling of Mathematical Reasoning Alan Bundy

... theorem proving’ techniques could be readily brought into a Resolution framework, and how this helped us to relate the various techniques – creating coherence from confusion. In order to achieve this goal I have taken strong historical liberties in my descriptions of the work of Boyer and Moore, Gel ...

... theorem proving’ techniques could be readily brought into a Resolution framework, and how this helped us to relate the various techniques – creating coherence from confusion. In order to achieve this goal I have taken strong historical liberties in my descriptions of the work of Boyer and Moore, Gel ...

Intuitionistic completeness part I

... Definition 1. A first order language L is a symbol D and a finite set of relation symbols {Ri |i ∈ I} with given arities {ni |i ∈ I}. First order formulas, F(L), over L are defined as usual. The variables in a formula (which range over D) are taken from a fixed set Var = {di |i ∈ N}. Negation ¬ψ can ...

... Definition 1. A first order language L is a symbol D and a finite set of relation symbols {Ri |i ∈ I} with given arities {ni |i ∈ I}. First order formulas, F(L), over L are defined as usual. The variables in a formula (which range over D) are taken from a fixed set Var = {di |i ∈ N}. Negation ¬ψ can ...

Henkin`s Method and the Completeness Theorem

... (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “|= ϕ” for ϕ is valid (that is, ϕ is satisfied in every interpretation of L). The soundness theorem for L states that if ` ϕ, then |= ϕ; and the completeness theorem for L states that if |= ϕ, then ` ϕ. Put to ...

... (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “|= ϕ” for ϕ is valid (that is, ϕ is satisfied in every interpretation of L). The soundness theorem for L states that if ` ϕ, then |= ϕ; and the completeness theorem for L states that if |= ϕ, then ` ϕ. Put to ...

Modal Logic for Artificial Intelligence

... Some rules are very simple: if you can prove ϕ and you can prove ψ, then you can also prove their conjunction ϕ ∧ ψ. Other rules are more complicated. For example, the only way to ‘eliminate’ the disjunction ϕ ∨ ψ is by proving, first that ϕ ∨ ψ, and second, that some conclusion χ can be proven both ...

... Some rules are very simple: if you can prove ϕ and you can prove ψ, then you can also prove their conjunction ϕ ∧ ψ. Other rules are more complicated. For example, the only way to ‘eliminate’ the disjunction ϕ ∨ ψ is by proving, first that ϕ ∨ ψ, and second, that some conclusion χ can be proven both ...

Chapter 9: Initial Theorems about Axiom System AS1

... Ignoring the implicit universal quantifiers, the main functor of this formula is ‘→’, which is the metalinguistic ‘if…then’ connective. Similarly, ‘&’ is the metalinguistic ‘and’ connective. Translating this into English, we have: if „α, and „α→β, then „β. The remaining occurrence of ‘→’ is under th ...

... Ignoring the implicit universal quantifiers, the main functor of this formula is ‘→’, which is the metalinguistic ‘if…then’ connective. Similarly, ‘&’ is the metalinguistic ‘and’ connective. Translating this into English, we have: if „α, and „α→β, then „β. The remaining occurrence of ‘→’ is under th ...

Self-Referential Probability

... In Chapter 4, we will consider another Kripke-style semantics but now based on a supervaluational evaluation scheme. This variation is particularly interesting because it bears a close relationship to imprecise probabilities where agents’ credal states are taken to be sets of probability functions. ...

... In Chapter 4, we will consider another Kripke-style semantics but now based on a supervaluational evaluation scheme. This variation is particularly interesting because it bears a close relationship to imprecise probabilities where agents’ credal states are taken to be sets of probability functions. ...

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 ...

Using linear logic to reason about sequent systems

... described, in part, by where the structural rules of thinning and contraction can be applied. In classical logic, these structural rules are allowed on both sides of the sequent arrow; in intuitionistic logic, no structural rules are allowed on the right of the sequent arrow; and in linear logic, th ...

... described, in part, by where the structural rules of thinning and contraction can be applied. In classical logic, these structural rules are allowed on both sides of the sequent arrow; in intuitionistic logic, no structural rules are allowed on the right of the sequent arrow; and in linear logic, th ...

Chapter 6: The Deductive Characterization of Logic

... Formally, a finite sequence is just like a finite string, the difference being purely pragmatic. Generally, a sequence σ has a first element σ 1, a second element σ 2, etc. If σ is n-long, then σ n is the last element of σ. Also, to say that σ is a sequence of so-and-so’s is to say that each σ i is ...

... Formally, a finite sequence is just like a finite string, the difference being purely pragmatic. Generally, a sequence σ has a first element σ 1, a second element σ 2, etc. If σ is n-long, then σ n is the last element of σ. Also, to say that σ is a sequence of so-and-so’s is to say that each σ i is ...

Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as evidence is acquired. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called ""Bayesian probability"".