Tag: OpenLab 8
Logic on Math StackExchange
- Truth-tabling $\left[\,(p\lor q)\land(p\to r)\land(q\to r)\,\right]\implies r $ October 5, 2024Our professor assigned this homework. By constructing a truth table, prove this logical implications: $$\left[\,(p\lor q)\land(p\to r)\land(q\to r)\,\right]\implies r $$ How to handle two AND's in one bracket?Alix
- can self-referential statements be regarded as statements(propositions)? October 5, 2024By definition, a proposition (statement) is a declarative sentence that is either True or False, but not both. Hence the question, is a referential statement, such as "This sentence is false" can be regarded as a statement? BestBurakhan Aksoy
- "Every Cat loves its mother or father" October 5, 2024"Every Cat loves its mother or father" $\forall x ( \operatorname {Cat}(x) \land (\operatorname {Loves}(x,\operatorname{Mother}) \lor \operatorname {Loves}(x,\operatorname {Father}) )$ Although I know that a universal quantifier cannot be used with conjunction (not because of syntax error), I have issues understanding why the above translation is incorrect. I mean, it does sound correct: for every […]dikshank
- The logic subtlety behind solving differential equations. October 5, 2024Let me first explain what has led me to ask this question. When solving functional equations, it is often the case that through a link of implications (that is, uni-directional implications), we get several possible solutions for the functional equation. Then, we have to plug these functions into the original equation to see whether each […]The_Eureka
- Possible Error in Poizat's A Course in Model Theory (Chapter 7, Arithmetic) October 4, 2024I was reading Chapter 7 on Arithmetic in Bruno Poizat's A Course in Model Theory and noticed a potential error in Section 7.1 regarding the axioms defining the successor function. The axiom is given as: ($\forall x) (x \neq 0)$ This seems incorrect since it should express that 0 is not the successor of any […]Jackson Willoughby
- "All countries that border Ecuador are in South America" October 4, 2024Parts (i) and (iv) make sense as they are obvious. Part (ii) says "for every C, if that is a country, then, if it borders Ecuador then it is in SouthAmerica." I want to know whether there is a method to convert "and ($\land$)" to implication; this way, parts (i) and (ii) are interchangable. I […]dikshank
- Prove that a translation is (or is not) essentially surjective October 4, 2024Let $L_1$ be a first-order language with only one extralogical symbol, i.e. a unary predicate $Px$. Let $L_2$ be a first-order language with only one extralogical symbol, i.e. a binary predicate $Rxy$. Let $T_1$ be the empty theory, i.e. $T_1= \emptyset$. Let $T_2$ be a theory that just says that $R$ is symmetric: $T_2 = […]Soennecken
- “Richard’s brothers are John and Geoffrey” October 4, 2024From Chapter 8 of the book Artificial intelligence: The modern approach: “Richard’s brothers are John and Geoffrey”: Brother (John, Richard) $∧$ Brother (Geoffrey, Richard) $∧$ John $\neq$ Geoffrey $∧$ ∀x Brother (x, Richard) ⇒ (x=John $∨$ x=Geoffrey) Isn't translation this simpler, with the exact same meaning? If not, what is wrong with it? ∀x Brother […]dikshank
- Formal proof of a statement involving the interchange of quantifiers [duplicate] October 3, 2024Let $P$ be a formula with two variables. Given the premises: $\forall x \exists y P(x,y)$ $\forall x \forall x' \forall y \forall y' (P(x, y) \land P(x',y') \to y = y')$ I would like to derive $\exists y \forall x P(x,y)$. Informally, the idea is to introduce a function $(f(x) = y) \leftrightarrow P(x,y)$ […]FR09
- Can a Program Certainly Distinguish Normal Distribution from a Discrete Finite One October 3, 2024Suppose we have two distributions $\mathcal{A}$ and $\mathcal{B}$. One of the distributions is normal $\mathcal{N}(0,1)$ and one is a discrete distribution with finite support, i.e. can be represented as a finite sum of weighted delta measures $\sum_{k = 1}^n c_k\cdot\delta_{b_k}$, but we don't know which one is which. Now a natural question to ask if […]Sergey Novozhilov
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