Tag: “Math Improve”
Logic on Math StackExchange
- Does this Weird Correspondence of $\Box$ to $\top \to$ to the S4 Axioms Allow for a Companion to IS4 with No Modal Operators? April 26, 2025This just struck me as weird while I was reading a bunch of papers on modal decision procedures for intuitionistic propositional logic: N : If $A$ is a theorem, derive $\Box A$. ~ If $A$ is a theorem, derive $\top \to A$. K : $\Box (A \to B) \to (\Box A \to \Box B)$ ~ […]Joshua Harwood
- Help walking through difficulties understanding the difference between $\forall x$ and infinite conjunction April 26, 2025I (and possibly others) naively thought of $\forall$ as an infinite conjunction (only vaguely aware of others saying to avoid this interpretation of the $\forall$ sign). Years later I'm confronting this belief. I've read over Derek Elkins' blog post on the matter. I only partially probably have the logical prerequisites to approach the problem, but […]Pineapple Fish
- Does $p,q ∈ S$ require $p$ and $q$ to be distinct? April 25, 2025I have noticed that different authors use different rules for such simple statements such as "Let there be two numbers $p$ and $q$." Most allow the possibility that $p$ and $q$ are the same number. Others treat $p$ and $q$ as necessarily distinct; for example, Stephenson in An Introduction to Matrices, Sets and Groups specifically […]Steven Thomas Hatton
- Ordering the set of all filters on a partial order. April 25, 2025Fix two partial orders, $R$ and $R'$, on the same set $X$. Denote by $F(R)$ and $F(R')$ the set of all filters of $R$ and $R'$ respectively. Specifically, for $f \subseteq X$ we have $f \in F(R)$ if $f$ is non-empty $x \in f$ and $xRy$ implies $y \in f$ $x,y \in f$ implies there […]201p
- Explanation of a fallacy during Structural Induction [closed] April 25, 2025Here is the description of Structural Induction extracted from "Discrete Mathematics and Its Applications - 8th edition" by Kenneth Rosen: BASIS STEP: Show that the result holds for all elements specified in the basis step of the recursive definition to be in the set. RECURSIVE STEP: Show that if the statement is true for each […]Vlad Mikheenko
- Are there infinite conjunctions that have no uniform proof? April 25, 2025I've been discussing the difference between infinite conjunction and universal quantification with my friend. I believe he has set me the following problem, which may help resolve some of the discussions we were having, but I am not creative enough to know how to answer it: Define a family of constants $c_i$ for $i$ ranging […]Pineapple Fish
- Proving $(A\rightarrow (B\rightarrow C))\rightarrow ((A\rightarrow B)\rightarrow (A\rightarrow C))$ in Hilbert/Ackermann axiomatic system April 24, 2025I am working upon exercises from "Introduction to Mathematical Logic" by Mendelson chapter 1.6. Given a formal system L1 : $\vee$ and $\neg$ are the primitive connectives. We use $B\rightarrow C$ as an abbreviation for $\neg B\vee C$. We have four axiom schemas: (1) $B\vee B\rightarrow B$ (2) $B\rightarrow B\vee C$ (3) $B\vee C\rightarrow C\vee […]user4035
- Implication versus conjunction [duplicate] April 24, 2025So I was going through the Rosen's book on Discrete Mathematics and I stumbled upon an example which had me confused from hours. This was the question: Consider these statements, of which the first three are premises and the fourth is a valid conclusion. i)“All hummingbirds are richly colored.” ii)“No large birds live on honey.” […]Ajay
- Do we have a term to describe all the cases of a sentence? April 24, 2025Does formal logic have a word for all the possible permutations of the truth values of a sentence's atomic sentences? I'm looking for a word to describe the set $$\{(T,T),(T,F),(F,T),(F,F)\}$$ associated with the sentence $$A \land B.$$linear_combinatori_probabi
- If a language with at least one constant and ψ(x) has no quantifiers then exist a finite number of terms with no variables then ⊢ V_(i=1)^nψ_(ti/x) [closed] April 23, 2025The exercise is for an assigment of first order logic, concretly about formal proofs and satisfaction, I've working on it for a long time but I fail to realease what I'm missing.It's also very different from what we've seen during class and that's a big part of why I want to solve it. It says: […]Bañó
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