# Tag: OpenLab7

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### Logic on Math StackExchange

- 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
- First order logic "Artificial intelligence: The modern approach" October 4, 2024In the book Artificial intelligence: The modern approach" chapter 8, it says “Richard’s brothers are John and Geoffrey” is as follows: Brother (John, Richard) $∧$ Brother (Geoffrey, Richard) $∧$ John $\neq$ Geoffrey $∧$ ∀x Brother (x, Richard) ⇒ (x=John $∨$ x=Geoffrey) WHY IS IT NOT ∀x Brother (x, Richard) ⇒ ((x=John $∨$ x=Geoffrey) $∧$ John […]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
- Is $A $ implying $B$ really captured by $A \implies B$? [closed] October 2, 2024Consider for example a common statement for a function f from basic calculus : $$ \text{differentiability of (f)} \implies \text{continuity of (f)} -(1)$$ Now, if we have a function is discontinuous, then it would be an acceptable deduction (... for most math students) to deduce from the above that it can not be differentiable.(*) But, […]Cantor Dust Drachen
- two issues on first order logic's GEN rule [duplicate] October 2, 2024The most popular axiom system for first order logic contains 5 axioms and 2 rules,the rules of inference of any first-order theory are: 1 Modus ponens(MP rule): C follows from B and B → C 2 Generalization(GEN rule): (∀x)B follows from B In the book Introduction to Mathematical Logic (MENDELSON 6th Ed - CRC Press) […]showkey
- Strength of Axiom of Choice vs. Law of Excluded Middle vs. Dependent Choice October 1, 2024tl;dr Are there results comparing how many results in ZF can be proven with axiom of dependent choice (DC) vs. law of excluded middle (LEM) vs. axiom of choice (AC)? To hopefully nip in the bud any issues of ambiguity, let me clarify from the outset that I am speaking in terms of constructive set […]DiracComb16796

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