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Quantifiers
... some UD is truth-functionally invalid, then the original argument is FO invalid, but if it is truth-functionally valid, then that does not mean that the original argument is FO valid. • For example, with UD = {a}, the expansion of the argument would be truth-functionally valid. In general, it is alw ...
... some UD is truth-functionally invalid, then the original argument is FO invalid, but if it is truth-functionally valid, then that does not mean that the original argument is FO valid. • For example, with UD = {a}, the expansion of the argument would be truth-functionally valid. In general, it is alw ...
1 Deductive Reasoning and Logical Connectives
... and make statements about these objects. For example, consider the statement: No matter what number might be chosen, if it is greater than 3, then its square is greater 9. We introduce a variable to replace the words referring to the number: No matter what number n might be chosen, if n is greater t ...
... and make statements about these objects. For example, consider the statement: No matter what number might be chosen, if it is greater than 3, then its square is greater 9. We introduce a variable to replace the words referring to the number: No matter what number n might be chosen, if n is greater t ...
Lecture 2 - inst.eecs.berkeley.edu
... either an axiom or its truth follows easily from the fact that the previous statements are true. For example, in high school geometry you may have written two-column proofs where one column lists the statements and the other column lists the justifications for each statement. The justifications invo ...
... either an axiom or its truth follows easily from the fact that the previous statements are true. For example, in high school geometry you may have written two-column proofs where one column lists the statements and the other column lists the justifications for each statement. The justifications invo ...
Model Theory of Second Order Logic
... Let P be Cohen forcing for adding a generic set G ⊆ N. Let F G be the set of A ⊆ N with A∆G finite. We show that M = (ω ∪ F G , ω, <, E), where nEx ⇐⇒ n ∈ X , is not characterizable by a second order theory. Let M0 = (ω ∪ F −G , ω, <, E). Since M M0 , it suffices to prove that M and M0 are second ...
... Let P be Cohen forcing for adding a generic set G ⊆ N. Let F G be the set of A ⊆ N with A∆G finite. We show that M = (ω ∪ F G , ω, <, E), where nEx ⇐⇒ n ∈ X , is not characterizable by a second order theory. Let M0 = (ω ∪ F −G , ω, <, E). Since M M0 , it suffices to prove that M and M0 are second ...
Propositional/First
... • A valid sentence is true in all worlds under all interpretations • If an implication sentence can be shown to be valid, then—given its premise—its consequent can be derived • Different logics make different commitments about what the world is made of and what kind of beliefs we can have regarding ...
... • A valid sentence is true in all worlds under all interpretations • If an implication sentence can be shown to be valid, then—given its premise—its consequent can be derived • Different logics make different commitments about what the world is made of and what kind of beliefs we can have regarding ...
REVERSE MATHEMATICS AND RECURSIVE GRAPH THEORY
... of the problem of determining which graphs in a sequence have Euler paths. Sharper bounds can be found in the work of Beigel and Gasarch (see [6]). Remark. A two-way or endless Euler path is a bijection between the integers (both positive and negative) and the set of edges of G such that each edge ...
... of the problem of determining which graphs in a sequence have Euler paths. Sharper bounds can be found in the work of Beigel and Gasarch (see [6]). Remark. A two-way or endless Euler path is a bijection between the integers (both positive and negative) and the set of edges of G such that each edge ...
Proofs in Propositional Logic
... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...
... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...
The Dedekind Reals in Abstract Stone Duality
... i.e. a choice principle. However, this is weaker than that found in other constructive foundational systems, because for us the free variables of the predicate must also be of overt discrete Hausdorff type. That is, they must be either natural numbers or something very similar, such as rationals. Th ...
... i.e. a choice principle. However, this is weaker than that found in other constructive foundational systems, because for us the free variables of the predicate must also be of overt discrete Hausdorff type. That is, they must be either natural numbers or something very similar, such as rationals. Th ...
PROOFS BY INDUCTION AND CONTRADICTION, AND WELL
... Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 0 ∈ S, and (ii) whenever {0, . . . , k} ⊂ S, then k + 1 ∈ S, then S = N. Remark. Note the difference from the principle of induction above. In the second property we require the stronger assumption ...
... Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 0 ∈ S, and (ii) whenever {0, . . . , k} ⊂ S, then k + 1 ∈ S, then S = N. Remark. Note the difference from the principle of induction above. In the second property we require the stronger assumption ...
Topological aspects of real-valued logic
... compactness theorem. A version of Lω1 ,ω for metric structures, which extends continuous first-order logic, was introduced by Ben Yaacov and Iovino [14]. In their logic formulas of the form supi<ω ϕi and inf i<ω ϕi are permitted, provided that the total number of free variables remains finite, and t ...
... compactness theorem. A version of Lω1 ,ω for metric structures, which extends continuous first-order logic, was introduced by Ben Yaacov and Iovino [14]. In their logic formulas of the form supi<ω ϕi and inf i<ω ϕi are permitted, provided that the total number of free variables remains finite, and t ...
An Introduction to Elementary Set Theory
... The beginning of Dedekind’s friendship with Cantor dates back to 1874, when they first met each other while on holidays at Interlaken, Switzerland. Their friendship and mutual respect lasted until the end of their lives. Dedekind was one of the first who recognized the importance of Cantor’s ideas, ...
... The beginning of Dedekind’s friendship with Cantor dates back to 1874, when they first met each other while on holidays at Interlaken, Switzerland. Their friendship and mutual respect lasted until the end of their lives. Dedekind was one of the first who recognized the importance of Cantor’s ideas, ...
chapter1p3 - WordPress.com
... In math, CS, and other disciplines, informal proofs which are generally shorter, are generally used. ...
... In math, CS, and other disciplines, informal proofs which are generally shorter, are generally used. ...
MODAL LANGUAGES AND BOUNDED FRAGMENTS OF
... Here, a bisimulation is a binary relation between the domains of two first-order models linking points with the same unary predicates P , corresponding to modal proposition letters p , and satisfying two 'back-and-forth' or 'zigzag clauses' with respect to relational R-successors. (More precisely, ...
... Here, a bisimulation is a binary relation between the domains of two first-order models linking points with the same unary predicates P , corresponding to modal proposition letters p , and satisfying two 'back-and-forth' or 'zigzag clauses' with respect to relational R-successors. (More precisely, ...
Logic programming slides
... = {Pa, x Px} has a model but no minimal Herbrand model. The Herbrand universe of is {a}, but no model on this domain satisfies . ' = {Pa Qa} has two minimal Herbrand models: one wherein Pa is true and Qa is false, and one wherein Qa is true and Pa is false. Properties of the minimal i ...
... = {Pa, x Px} has a model but no minimal Herbrand model. The Herbrand universe of is {a}, but no model on this domain satisfies . ' = {Pa Qa} has two minimal Herbrand models: one wherein Pa is true and Qa is false, and one wherein Qa is true and Pa is false. Properties of the minimal i ...
16 - Institute for Logic, Language and Computation
... = {Pa, x Px} has a model but no minimal Herbrand model. The Herbrand universe of is {a}, but no model on this domain satisfies . ' = {Pa Qa} has two minimal Herbrand models: one wherein Pa is true and Qa is false, and one wherein Qa is true and Pa is false. Properties of the minimal i ...
... = {Pa, x Px} has a model but no minimal Herbrand model. The Herbrand universe of is {a}, but no model on this domain satisfies . ' = {Pa Qa} has two minimal Herbrand models: one wherein Pa is true and Qa is false, and one wherein Qa is true and Pa is false. Properties of the minimal i ...
Document
... An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. ...
... An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. ...