Tag: Open Lab #6
Logic on Math StackExchange
- English to First order Logic October 4, 2024I came across this question So part (i) makes sense as it is obvious. part (iv) also makes sense as it obvious again. HOWEVER, part (ii) and part (iii) does not make much sense or maybe they do. I mean, to me Part (ii) says "for every C, if that is a country, then, if […]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 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
- Issue with contrapositive October 1, 2024So contrapositives are logically equivalent statements, which can be useful when proving things. But I'm not convinced that this is true. Take this example: $x \in \mathbb{Z}$. Then $$x \neq 2 \Rightarrow x+1 \neq 0.5$$ is true, but the contrapositive $$x+1 = 0.5 \Rightarrow x=2$$ is false. If one statement is always false/true, then is […]thedanktouch
- How can we know that a model is a valid model for a given set of axioms? October 1, 2024For a while I have thought (naively) that maths is constructed in the following way: pick a set of axioms and some rules of logic, then deduce all the theorems you can about it. The goal being obviously to have the simplest set of axioms (and simplest logic rules?) such that you recover the mathematical […]Flavien Hirsch
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