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Does using constant symbols in a first-order theory implicitely induce an existence axiom?

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#logic #first-order-logic #formal-languages #axioms
757
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Does weak completeness ("If $\vDash\phi$, then $\vdash\phi$") imply strong completeness ("If $\Gamma\vDash\phi$, then $\Gamma\vdash\phi$")?

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#logic #predicate-logic #first-order-logic #metalogic
442
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Where to read on Lindström's theorem?

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#logic #reference-request #first-order-logic
262
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How to deduce using rules of inference

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#discrete-mathematics #first-order-logic
72
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How to extend metatheorems regarding finite sets of formulas to infinite sets when compactness theorem is available?

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#compactness #propositional-calculus #first-order-logic #infinity
286
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Question about Existential Generalization

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#logic #predicate-logic #first-order-logic
2.5k
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Why can we use implication with the universal quantifier but not with the existential quantifier?

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#logic #first-order-logic #quantifiers
473
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Do we really need constant symbols in first-order theories?

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#logic #first-order-logic #formal-languages
73
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why this formulas are equivalent? $\exists x \phi(x) \equiv \neg \forall x \neg \phi(x)$

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#first-order-logic
58
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$\forall x P(x) \land \exists x Q(x) \equiv \forall x \exists y (P(x) \land Q(y))$

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#first-order-logic
223
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Proof using contradiction and a contrary assumption

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#calculus #multivariable-calculus #proof-verification #first-order-logic #quantifiers
191
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Theory of infinitely many constants vs. Theory of an infinite set

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#logic #model-theory #first-order-logic #formal-languages
727
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How to remove constant symbols from a first-order theory?

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#logic #model-theory #first-order-logic #formal-languages
174
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Does there exist a first-order theory $T$ and a proof system $\vdash$ such that $T\vDash\phi$ iff $T\vdash\phi$, but $\vdash$ is not complete?

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#logic #predicate-logic #first-order-logic #metalogic
1.3k
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Analyzing the logical forms of the following statements (quantifiers and conditionals)

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#logic #propositional-calculus #predicate-logic #first-order-logic

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