Written work – None.
WeBWorK – Assignment #5, due Tuesday, October 10th, at midnight.
OpenLab – OpenLab #5, due Thursday, October 12th, before class.
Handy Links
Logic on Math StackExchange
- Negating quantifiers or statements September 23, 2024How do I symbolize the statement "there does not exist a $x$ for all $y$, $B(x,y)$". Using $\neg(\exists x) (\forall y) B(x,y)$ would mean it is not the case that there exist a $x$ for all $y$, $B(x,y)$ or that the negation would be for both the quantifiers. If we write $(\neg (\exists x))(\forall y) […]user221985
- Issue with proving relativization of $\Sigma_2$ sentences September 23, 2024Let $\Lambda$ be the class of all cardinals less than any inaccessible cardinal and let $\mathsf{IC}$ be the first-order sentence stating the existence of one such inaccessible cardinal. As I understand it, "is an inaccessible cardinal" is a $\Pi_1$ sentence, which would make $\mathsf{IC}$ (resp. $\neg\mathsf{IC}$) $\Sigma_2$ (resp. $\Pi_2$). I want to prove the following: […]Sho
- There exist only one dream that all Americans share" vs "All Americans have only one and the same dream" September 22, 2024For the following examples, let $A$ be the set of all Americans, $D$ the set of all dreams, and $H(a,d)$ be the proposition $\text{American } a \in A \text{ has dream } d \in D.$ There exist only one dream that all Americans have (they may have more than one dream, but there is only […]Tripola
- Russell's-Paradox. Why can't something be true and false at the same time? September 22, 2024Since the real world allows something that is false to be true at the same time, why doesn't math support this fact? In Quantum mechanics a cat in Schrodinger's box is dead and alive at the same time. You could also have the box constructed so that Russell's set of all sets that don't contain […]Jeff Guarino
- Is this rule of inference sound? September 22, 2024Suppose the inference rule Modus Ponens is true/sound. I want to prove the following. From $\vdash \varphi(x)\to \psi$, infer $\vdash \exists x \varphi(x)\to \psi$ given that $x$ doesn't occur free in $\psi$. Here's my reasoning. Proof. Let $\mathfrak{A}\vDash \varphi(x)\to\psi$. And assume $\mathfrak{A}\vDash \exists x \varphi(x)$. Then $\mathfrak{A}\vDash \varphi(a)$ for some $a\in A$. Since $x$ doesn't […]caligulasremorse
- How many constant symbols can a set of intuitionistic formulas have for completeness to hold? September 22, 2024Fitting's proves a version of the completeness theorem for intuitionistic FOL in his book on intuitionistic model theory and forcing. Let $U$ be any set of formulas without parameters (i.e. constant symbols). Then $U \vdash X$ (in the intuitionistic sense) iff in any model $\mathcal{G}$, for any $\Gamma \in \mathcal{G}$, if $\Gamma \vDash U$, $\Gamma […]zaq
- Logical Equivalence in Universal Quantification September 22, 2024If I know that $$\forall x\, \big(A(x) \Leftrightarrow B(x)\big)$$ is true, how can I prove that $$\forall x\, A(x) \Leftrightarrow \forall y\, B(y)$$ From the perspective of the meaning of quantifiers, this seems obvious, but is there a standard method to prove it?user1361001
- to proof are these two formula equal? ∀a[∃bP (a, b) → F] ≡ ∀a∃b[P (a, b) → F] [closed] September 21, 2024i am trying to make it =∀a[¬∃bP (a, b)∨ F] =∀a[∀b¬P (a, b)∨ F] and use double negation but still cannot proofDasiy
- A logical system in Grundlagen der Mathematik(vol.1) September 21, 2024In [1] a logical system (let us temporarily name it HB1) close to first-order logic(FOL) was introduced. The main difference between HB1 and FOL is that HB1 has formula variables with arguments. Is there an accepted name for the logical system HB1? More generally, is there a comprehensive list of formal logical systems used in […]Victor M
- Verify if formula is tautology in Intuitionistic logic [closed] September 21, 2024I have following formula: $\left( A \rightarrow (B \vee \neg B) \right) \vee \left( \neg A \rightarrow (B \vee \neg B) \right)$ I know a method to veify that formula is not a tautology in intuitionistic logic that uses topological spaces but not sure what else I can use... I also know that formula $(B […]leeeeeeeeess
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