First-Order Theorem Proving and Vampire

... 125. mult(X2,X3) = mult(X3,X2) [superposition 21,90] 90. mult(X4,mult(X3,X4)) = X3 [forward demodulation 75,27] 75. mult(inverse(X3),e) = mult(X4,mult(X3,X4)) [superposition 22,19] 27. mult(inverse(X2),e) = X2 [superposition 21,11] 22. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 17,10] ...

... 125. mult(X2,X3) = mult(X3,X2) [superposition 21,90] 90. mult(X4,mult(X3,X4)) = X3 [forward demodulation 75,27] 75. mult(inverse(X3),e) = mult(X4,mult(X3,X4)) [superposition 22,19] 27. mult(inverse(X2),e) = X2 [superposition 21,11] 22. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 17,10] ...

Assumption-Based Argumentation with Preferences

... particularly over assumptions, similarly to the well known structured argumentation formalism ASPIC+ [47, 21, 49, 46] (which, however, accommodates preferences over rules too). Most existing approaches assume (e.g. [6, 14]) or perform (e.g. [47, 60]) an aggregation of objectlevel preferences to give ...

... particularly over assumptions, similarly to the well known structured argumentation formalism ASPIC+ [47, 21, 49, 46] (which, however, accommodates preferences over rules too). Most existing approaches assume (e.g. [6, 14]) or perform (e.g. [47, 60]) an aggregation of objectlevel preferences to give ...

Independence logic and tuple existence atoms

... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...

... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...

Predicate Logic

... False. 3 is a counterexample. the set of positive integers not exceeding 4: {1, 2, 3, 4} False. 3 is a counterexample. Also note that here ∀P (x) is P (1) ∧ P (2) ∧ P (3) ∧ P (4), so its enough to observe that P (3) is false. the set of real numbers in the interval [10, 39.5] True. It takes a bit lo ...

... False. 3 is a counterexample. the set of positive integers not exceeding 4: {1, 2, 3, 4} False. 3 is a counterexample. Also note that here ∀P (x) is P (1) ∧ P (2) ∧ P (3) ∧ P (4), so its enough to observe that P (3) is false. the set of real numbers in the interval [10, 39.5] True. It takes a bit lo ...

Discrete Mathematics

... A propositional variable (lowercase letters p, q, r) is a proposition. These variables model true/false statements. The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, wr ...

... A propositional variable (lowercase letters p, q, r) is a proposition. These variables model true/false statements. The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, wr ...

Chapter 9

... First, the notion of logical implication will have to be refined because the behavior of these dependencies taken together is different depending on whether infinite instances are permitted. Second, both notions of logical implication are nonrecursive. And third, it can be proven in a formal sense t ...

... First, the notion of logical implication will have to be refined because the behavior of these dependencies taken together is different depending on whether infinite instances are permitted. Second, both notions of logical implication are nonrecursive. And third, it can be proven in a formal sense t ...

Duplication of directed graphs and exponential blow up of

... levels. The upper bound in Theorem 31 re ects well this idea and shows how patterns lying in cut-free proofs might be recoverable from the graph of the original proof with cuts. In Sections 5 and 13 we analyze how patterns in proofs evolve through cut elimination and which are the combinatorial stru ...

... levels. The upper bound in Theorem 31 re ects well this idea and shows how patterns lying in cut-free proofs might be recoverable from the graph of the original proof with cuts. In Sections 5 and 13 we analyze how patterns in proofs evolve through cut elimination and which are the combinatorial stru ...

Logic and Proof - Numeracy Workshop

... The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . its converse x2 = 4 ⇒ x = 2 is false, because x could be equal to −2. ...

... The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . its converse x2 = 4 ⇒ x = 2 is false, because x could be equal to −2. ...

Termination of Higher-order Rewrite Systems

... Rewriting and Termination The word rewriting suggests a process of computation. Typically, the objects of computation are syntactic expressions in some formal language. A rewrite system consists of a collection of rules (the program). A computation step is performed by replacing a part of an express ...

... Rewriting and Termination The word rewriting suggests a process of computation. Typically, the objects of computation are syntactic expressions in some formal language. A rewrite system consists of a collection of rules (the program). A computation step is performed by replacing a part of an express ...

Dialectica Interpretations A Categorical Analysis

... The work presented in this thesis is a contribution to the area of type theory and semantics for programming languages in that we develop and study new models for type theories and programming logics. It is also a contribution to the area of logic in computer science, in that our categorical analys ...

... The work presented in this thesis is a contribution to the area of type theory and semantics for programming languages in that we develop and study new models for type theories and programming logics. It is also a contribution to the area of logic in computer science, in that our categorical analys ...

Intuitionistic and Modal Logic

... Platonism and formalism. View that mathematics and mathematical truths are creations of the human mind: true = provable. N.B! provable in the informal, not formal sense. • Platonism. Most famous modern representatives: Frege, Gödel. View that mathematical objects have independent existence outside ...

... Platonism and formalism. View that mathematics and mathematical truths are creations of the human mind: true = provable. N.B! provable in the informal, not formal sense. • Platonism. Most famous modern representatives: Frege, Gödel. View that mathematical objects have independent existence outside ...

An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.