Tag: Wau
Logic on Math StackExchange
- Satisfiability in an Heyting algebra implies satisfiability in a Boolean algebra for propositional logic? May 28, 2023Let $\mathcal{L}$ be a propositional language and let $\text{Prop}(\mathcal{L})$ be the set of all the propositions of the language $\mathcal{L}$. Let $(H,\wedge,\vee,\rightarrow,1,0)$ be an Heyting algebra. An evaluation of the propositions of $\mathcal{L}$ in $(H,\wedge,\vee,\rightarrow,1,0)$ is a map $V:\text{Prop}(\mathcal{L}) \to H$ such that $V(\top)=1$, $V(\bot)=0$, $V(P \wedge Q)=V(P) \wedge V(Q)$, $V(P \vee Q)=V(P) \vee V(Q)$ […]effezeta
- Are there "Godel encodings" for non-arithmetic theories? May 28, 2023This is probably a very naive question, but is there something about Godel encoding that is essentially arithmetical, or is it possible to construct analogous mappings between the objects studied in a theory and statements in the theory itself for non-arithmetical theories (i.e, theories whose objects of study are not numbers). Are there any notable […]Rando McRandom
- How well should I know understand the theorem before proving it? May 28, 2023Im reading "How to prove it" by velleman, and in a lot of the exercises you need to prove some theorem is true. How well should I understand why the theorem is actually true before proving it? For me it's pretty easy to prove something just by following the rules, but idk if that's smart. […]lightyourassonfire
- Iff propositions where both directions require choice? May 28, 2023Recently, I have been revising a basic course on noncommutative rings and modules over them. One proposition proven early on is all left modules over $R$ are free iff $R$ is a division ring and an interesting point is that, at least in the standard proofs we were given (and I have seen repeated in […]Isky Mathews
- Satisfiability in greater languages May 28, 2023Let $\mathcal{L}, \mathcal{L}'$ two languages in first order logic such that $\mathcal{L}\subset \mathcal{L}'$, $\Gamma \subseteq Form_{\mathcal{L}}$ and $\varphi \in Form_{\mathcal{L}}$. Prove that if $\Gamma \vdash_{\mathcal{L}'} \varphi$ then $\Gamma \vdash_{\mathcal{L}}\varphi$. In my book, there is a proof when $\mathcal{L}'$ is an extension of $\mathcal{L}$ with only constant symbols, but this is the general case. I am […]Superdivinidad
- How to solve $(x-1)(x-2)=0$ constructively? May 28, 2023I want to prove that $$(x-1)(x-2)=0\Leftrightarrow x=1, 2$$ $\Leftarrow$ is easy. The problem is $\Rightarrow$. Assuming $x\neq 1, 2$, we can derive $1=0$ by dividing both sides of $(x-1)(x-2)=0$ by $x-1$ and $x-2$. Thus we get $\lnot \lnot (x=1, 2)$. However, intuitionistic logic cannot eliminate double negation.BonBon
- Can a contradiction prove a contradiction like this? May 28, 2023If I prove the implication ¬P ⇒ ¬R∧R, and then I prove the implications ¬R ⇒ Q, R ⇒ ¬Q, is it valid to say ¬P ⇒ Q∧¬Q? I am unsure because while it is the case that Q and ¬Q both follow from P, it seems that we may have assumed that R is […]God
- About inference in mathematical induction of first-order predicate logic. May 28, 2023In induction of first order logic, reasoning with n∈ℕ : P(n)➞P(n+1). {n=1,P(1)} ⊨ P(1) k∈ℕ, {∃k: P(k)} ⊢ P(k+1) K:={ k | P(k), {P(k)} ⊢ P(k+1)} (then K⊂ℕ), {P(1) ∧ (P(1) → P(2))} ⊨ (P(2) ∧ 1∈K), {P(2) ∧ (P(2) → P(3))} ⊨ (P(3) ∧ 2∈K), ...... ∴∀n∈K, ⊨ (∀n: P(n) ∧ (P(n) → P(n+1))) […]DoP
- Does $(x\in A\land y\in B)$ follow from $xRy$? May 27, 2023I have defined “being in $R$-relation to” as: Given a binary relation $R\subseteq A\times B$ and the elements $x\in A$ and $y\in B,$ $$xRy\iff (x,y)\in R.$$ For $R$ to be injective, is $$(xRz\land yRz)\implies x=y$$ acceptable or is $$\forall x, y, z\;\big((x,y\in A\land z\in B \land xRz\land yRz)\implies x=y\big)$$ necessary? But doesn't $(x\in A\land y\in […]Leonardo Orietti Del Duca
- Existence of some syntactic deduction May 27, 2023Given an $\mathcal{L}-$language, prove if it exists or not a deduction for: $$\exists x_1 \exists x_2 \neg \varphi \vdash \neg \exists x_1 \exists x_2 \varphi $$ My idea is that if it exists a syntactic deduction, by the Soundness Theorem it follows the semantic implication, so a set composed of the first formula and the […]Superdivinidad
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