Here are the final drafts of the “Group Process Papers.” Assessment details will be distributed in class. Great work, everyone!
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- Are there any statements about natural numbers P(m, n) that are such that P(m, n) for all m does not necessarily imply P(m, n) for all n? [closed] April 23, 2024Just out of interest. I'm trying to understand double induction, but there's not much more context.jacob2222
- Does “false” imply “true”? [duplicate] April 23, 2024I think calling a proposition an "implicature" means something like this: If there is a proposition "p->q" and this proposition is true, then we think that p "implies" q. So if p is false and q is true, does “false” imply “true”? My own thoughts are, Premise: ((I am a cup) and (the cup is […]Display name
- Show that Proof by Contradiction rule is derivable from all instances of it with atomic conclusion April 23, 2024The problem shows up in Troelstra's Basic Proof Theory, exercise 2.3.6A. Show that in [classical logic's Natural Deduction], for the languages without ∨, ∃, all instances of ⊥c derivable from instances ⊥c with atomic conclusions. I have a solution to the problem, but I'm not sure if it's correct as it seems trivials, perhaps because […]confusedcius
- Define something based on the property of another set. April 22, 2024I have always been confused about a logic problem, which I can not form it into a general statement. Here is an example that may express my ideas. We know given a set $A$ and an equivalent relation $\sim$ on it. We can define equivalence class by $\forall x\in A, [x]=\{y,|y \sim x\}$. It seems […]Andrew_Ren
- Prove $\forall x \forall y(xEy \rightarrow \neg x=y)$ in the vocabulary of graphs April 22, 2024Problem: Prove the sentence $\forall x \forall y(xEy \rightarrow \neg x=y)$ in the vocabulary of graphs using the axioms of graph theory. So far: The axioms of graph theory given are antireflexivity and symmetry. Here $xEy$ means there's an edge between the vertices $x$ and $y$. We were told we'd need identity axioms for this […]categoricallystupid
- Proving $\exists x P(x) \rightarrow \forall x P(x)$ from $\forall x\forall y(x=y)$ April 22, 2024Problem: Using identity axioms, prove $\exists x P(x) \rightarrow \forall x P(x)$ from $\forall x\forall y \, x=y$. So far: I'm quite stuck on where to even begin. Working backward, I know we want $P(x)$ and $\exists xP(x)$, but I can't quite see how to write it down. I'm used to proof trees over Fitch-style […]categoricallystupid
- Show that $\exists x \forall y R(x,y) \rightarrow \forall x \exists y R(x,y)$ is not provable by natural deduction April 22, 2024Problem: Show that the formula $$\exists x \forall y R(x,y) \rightarrow \forall x \exists y R(x,y)$$ is not provable by natural deduction. So far: First thought that came to mind was simply considering the natural numbers with an order relation to give a counter-example (considering an assignment such that $s(x)=0$). However now $R(x,x)$ is not […]categoricallystupid
- How is the universal generalization rule used? April 22, 2024The following deduction seems correct: $(x+1)^2=x^2+2x+1$ $\forall x (x+1)^2=x^2+2x+1$ But the following deduction is obviously incorrect: $x+1=2$ $\forall x (x+1)=2$ Did I use the universal generalization rule in a wrong way?William
- Automated theorem proving and set theory April 22, 2024I already know a few things about theorem proving however I was wondering how to use this when we are dealing with a non finite amount of axioms. In particular I was hoping to understand how to use automated theorem proving for set theory like ZFC for example. The method I know is using sequent […]Le Grand Spectacle
- How to construct a type $S$ such that for any $f:S, a:A, b:B$, $f(a,b):T(a)[b/y]$? April 22, 2024Convention: when a type depends on a variable from the context, this variable will be omitted from the type. For example, I will write $\Gamma, x:A \vdash B \text{ type}$ as opposed to $\Gamma, x:A \vdash B(x) \text{ type}$. The notation $B(a)$ will be reserved for the cases when $a$ appears in the premise; e.g. […]user837242
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