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Can I use gaifman locality theorem to ground an first order logic formula to a DNF?

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#first-order-logic
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Is the claim $(f\in P_1\cap P_2)\Rightarrow[(f\in P_3)\Leftrightarrow(f\in P_4)]$ identical to $(f\in P_1\cap P_2\cap P_3)\Leftrightarrow(f\in P_4)$?

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#logic #first-order-logic
29
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Dependence of Skolem functions on existentially quantified variables

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#logic #first-order-logic
81
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Proof of the deduction theorem in first-order logic

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#logic #first-order-logic
40
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Understanding quantifier rules in sequent calculus

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#logic #first-order-logic #sequent-calculus
56
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complement of a property in the lambda calculus

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#first-order-logic #lambda-calculus
88
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If the exponential is definable in an expansion of $\mathbb{\overline{R}},$ then it is definable without parameters

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#first-order-logic #model-theory
67
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Understand a proof on Craig's interpolation Theorem

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#logic #first-order-logic
32
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Composing Substitution Sets in Predicate Calculus

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#first-order-logic
122
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How to say in FOL that P(f) is true and everything else with property P is false?

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#logic #propositional-calculus #first-order-logic
62
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Subtleties on comprehension sets

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#set-theory #first-order-logic
673
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Finding a property that is true for every left ideal but not for right ideals

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#ring-theory #first-order-logic #model-theory
235
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Precise axiomatic definition for the equality "=" as a binary relation

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#logic #set-theory #first-order-logic #relations #axioms
81
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Proof of Hahn-Banach theorem from Compactness theorem of FOL

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#functional-analysis #logic #first-order-logic #alternative-proof #hahn-banach-theorem
46
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Why in First order logic, Variables mapping is not included in Structure definition

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#logic #first-order-logic

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