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Logic on Math StackExchange
- Kripke structure (model checking) October 2, 2023how can i build a kripke structure that refutes this formula? (A → B) ∨ (B → C) i tried this but i am not sure W1: A = true, B = false, C = false W2: A = false, B = true, C = false And now we define the accessibility relation R(W1, W2): […]anna
- What is the real difference between a schema logical system and propositional variable system? [duplicate] October 2, 2023Note: This question is not a duplicate of the suggested link. That link explains the schema system is great detail -- and is a great reference, and better than I have seen elsewhere. However, it does not address propositional variable systems, or the differences between them. Further, the fact that they also require substitution rules […]Michael Lee Finney
- logic question about the bijectivity of an application [closed] October 1, 2023Let $\mathrm{f}: \mathrm{E} \rightarrow \mathrm{F}$ be an application, and $\mathcal{S}=\left\{\mathrm{X} \subset \mathrm{E}\right .$ such $\left.\mathrm{f}^{-1}(\mathrm{f}(\mathrm{X}))=\mathrm{X}\right\}$. 1 For $A \subset E$, show that $f^{-1}(f(A)) \in \mathcal{S}$. 2 Show that $\mathcal{S}$ is stable by intersection and union. 3 Let $X \in \mathcal{S}$ and $A \subset E$ be such that $X \cap A=\emptyset$. Show that $X \cap f^{-1}(f(A))=\emptyset$. […]Saad Sigma
- Difference between locations of "for all" quantifier in a formula October 1, 2023I've seen statements of the type $\forall x \in X, P(x)$ and also $P(x) \forall x \in X$. Is there any logical difference?user129393192
- Is this $\alpha$-conversion correct? October 1, 2023I found this $\alpha$-conversion below in a certain book: $(\lambda x. x(\lambda z. xy)) = \alpha (\lambda z. z(\lambda x. zy))$ Well... this seems wrong to me, since $z$ is a binding variable in $(\lambda x. x(\lambda z. xy))$. Am I just not understanding?Paulo Argolo
- How to verbally describe $\forall$ elimination? October 1, 2023I know that the statement $\forall x {\in} \mathbb{R} \; P(x)$ means that we are free to choose any real value for $x$ and the $P(x)$ is true. In other words we have $P(1), P(2), \ldots$ and so on. In English, once we have said "$P(x)$ holds for any real $x$", how do we then […]Nav Bhatthal
- Does the universal set exist without allowing the concept of a set belonging to itself? September 30, 2023I have studied some elementary set theory and encountered a proof that a universal set containing everything cannot exist, as follows: Suppose, on the contrary, that there exists a set $\mathbb U$ containing everything. Then, by the axiom of specification, there exists a set $$\mathbb M= \{ x\in \mathbb U |\ x\ \text{is not a […]Aria
- Please simply explain “extension by definitions” in logic September 30, 2023I am self learning logic and have come across the concept of “extension by definitions”. Unfortunately, Wikipedia has proved to be a bit unfriendly to my dysfunctional brain. I understand the concept and idea behind it, just not the notational aspect. If anyone could walk me through the notation it would be a massive blessing. […]Hasan Zaeem
- How can I create logical OR out of IfTrue, IfFalse, and AND? September 30, 2023I am trying to get Anki to only create a card if field A or field B is not empty. Delightfully, the only operations available to me to do this are testing if the field is present (#A[text]/), if the field is not present(^A[text]/), and nesting these conditions (#A^B[text]/). Interestingly, there is one sort of […]grepgrok
- A question about S4 modal logic September 30, 2023I’ve been doing some proofs in S4 recently and noticed that the following holds: If $\phi$ is a wff and all positive literals $A$ that occur in $\phi$ are prefixed by either $\Diamond$ or $\Box$, then $\vDash_{S4} \Box \Diamond \phi \to \Diamond \Box \phi$. I’m trying to prove this by induction on the complexity of […]PW_246
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