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Logic on Math StackExchange
- Type theory vs "Theory On Top of Logic" mantra in Set Theories April 25, 2024I have a question about (especially second part of) following statement following statement from wikipedia emphasizing intrinsical feature in which type theory substantially differs from set theories: Unlike set theories, type theories are not built on top of a logic like Frege's. So, each feature of the type theory does double duty as a feature […]user267839
- Explicit geometric machinery to check the truth of FOL formula April 25, 2024It is standard to use the semantics of topological spaces for checking the truth value of propositional formula in the absence of excluded middle. An explicit description is written down here, for example. https://grossack.site/2022/04/17/geometry-and-logic I suppose, there might be possible to have something similar to determine the truth value of first order formulas. i.e. There […]Y.X.
- Why is Gödel's Incompleteness Theorem relevant to anything beyond self-referential statements? April 24, 2024I feel like I understand the general idea behind how Gödel used self-reference to prove that there will always be holes in logical systems, even if you add the self-referential statement to the axioms because you can always create another statement that is true but cannot be proven via Cantor's diagonalization proof. I am confused […]user34140
- Examples of geometric theory without classical models April 24, 2024I encountered a post on locales and geometric theory here https://grossack.site/2022/05/22/locale-basics.html In about the middle of this blog, the author gives a geometric theory defining a function from $\Bbb N$ to $\Bbb R$, and asked: Do you see why this is a nontrivial theory with no classical models? I am not confident that I understand […]Y.X.
- can somone help me finish this fitch proof? [closed] April 24, 2024Premise 1: $P \to Q$ Premise 2: $\lnot P \to R$ Premise 3: $\lnot Q$ Goal: $R.$ $\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \fitch{~~1.~~P\to Q\\~~2.~~\neg P\to R\\~~3.~~\neg Q}{\fitch{~~4.~~\neg R}{\fitch{~~5.~~P}{~~6.~~Q\hspace{10ex}{\to}\text{Elim }5,1\\~~7.~~\neg Q\hspace{9ex}\text{Reit }3\\~~8.~~\bot\hspace{10ex}\bot\text{Intro }6,7}\\~~9.~~\neg P\hspace{12ex}\neg\text{Intro }6{-}8\\10.~~R\hspace{13ex}{\to}\text{Elim }9,2\\~}\\~}$ the book is Language, Proof and Logic The Second Edition authors are Jon Barwise and John EtchemendyAvrgDude
- Question relating isomorphisms and structures in logic April 24, 2024A question in a sheet was as follows: Let $T_n,$ $n \in\mathbb N \setminus 0,$ be the structure $⟨N; f_n)$ where $f_n(a) = \lfloor\frac{a}{n}\rfloor.$ Draw a picture of Tn for n = 1, 2, 3. For T1 I drew a number-line from 1 to infinity For T2 I drew a number-line of 0, 1, 1, […]Alice
- Derivation of the "Chain syllogism" rule in first order logic April 24, 2024I'm currently reading these: https://www.mcmp.philosophie.uni-muenchen.de/students/math/math_logic_munich.pdf lecture notes and am kind of stuck on the proof on page 58 line 4 where $$\Gamma \lnot \varphi \lnot \varphi$$ is derived using (Ass.). I don't understand how, none of the rule thus far introduced allow me to manipulated the consequent except contradiction which I can't use in this […]Fabio
- 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
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