Tag: OpenLab8
Logic on Math StackExchange
- 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
- How many kilometers of highway are there in (any country)? [closed] September 30, 2023Do you know how to approach this question reasonably? It's some sort of a brain teaser. If we take any country, let’s say France or Germany, how to count highways in total without using any additional sources? Thanks in advance for your ideas! P.S. I recently got this question in a job interview and it’s […]Lila
- 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
- Negation of a statement validation September 29, 2023Let the statement be : "For any two real numbers $x,y,$ if $xAlex03
- In Presburger arithmetic: express "all numbers are odd or even" without using ∃ September 29, 2023I read in many places that in Presburger arithmetic the formula $∀x ∃y ((y + y = x) ∨ (y + y + 1 = x))$ expresses that every number is either even or odd. But as I understand, the restriction in Presburger arithmetic is on the use of existential quantification ($∃$), meaning one cannot […]Alfa Beta
- Logic equivalence question September 29, 2023I have a question about discrete mathematics question that I have been struggling to solve. Here is the question: Each of the two rooms (room I and room II) contains either a lady or a tiger. If a room contains a lady, the sign on its door is true. If it contains a tiger, the […]Rozhan Mosaferfard
- Vacuous falsehood - does it exist, and are there examples? September 28, 2023I've ben struggling with the concept of vacuous truth, as used (1) in proving implications, (2) as base cases for induction proofs. To help me understand, it would be useful to understand if the concept of vacuous falsehood exists, and if so, what simple examples might be. Discussion This question comes from the notion that […]Penelope
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