Tag: Wau
Logic on Math StackExchange
- Is proving a biconditional statement circular reasoning? [duplicate] November 25, 2024Suppose that you want to prove the biconditional proposition $p\iff q.$ How do we do that? Well, the most common way to prove it has two steps: first, assume $p$ and use that assumption to prove $q$ then, assume $q$ and use that assumption to prove $p.$ Hmm, wait. Isn't this circular reasoning? After all, […]Just_A_User
- Can a contradiction exist in the antecedent of a sequent? November 24, 2024Consider the following sequent. Please note that starting from the bottom, only the relevant branches were developed (the other branches did not pose a problem). $\dfrac{ \dfrac{ \dfrac{ \dfrac{...}{A,B,A→¬C⊢ B,C} ~~~\dfrac{A,C⊢A,B,C~~~~\textbf{A,C,¬C⊢ B,C}} {~~A,C,A→¬C⊢B,C~~}~~~→L } {A,B∨C,A→¬C⊢ B,C}∨L } {A∧(B∨C) , A→¬C ⊢ B,C}∧L } {A∧(B∨C) , A→¬C ⊢ B∨C}∨R $ If the antecedent corresponds to a […]A Koscianski
- Proving soundness of sequent calculus November 24, 2024I am currently working on a presentation about the sequent calculus for propositional logic. In this video, the interpretation for sequents is given as $v(\Gamma \vdash \Delta) = v\left(\bigwedge\Gamma \to \bigvee\Delta\right)$ Now, if we want to prove soundness, wouldn't it be enough to show that the interpretations of the sequents in a rule are equivalent? […]Snek
- Prenex-normal form theorem for many sorted logic November 24, 2024I am looking for a reference that contains an analogous of the first-order prenex normal form theorem for many sorts.user1868607
- Are these SOL formulas correct? November 24, 2024I need to know if these formulas are correct in the SOL "language" and if my "translations" of them are correct, more examples of easy to understand SOL formulas are also appreciated. (1) $X$ is a property, $\sigma$ is the successor function and $c$ is the number $0$: $\forall X(Xc \ \land \ \forall x(Xx […]EscandinavoEducado
- how to populate the sets in this domain(logic) [closed] November 23, 2024I'm struggling to understand how to populate these sets in this domain, but not just in this exercise, in general, what's the general way to do it easily? Here's an example: "All cats that are not black are liked by children" basically, we need to create domain variables and show how they're distributed: M(Cat) = […]Andrew 0.0
- Transfinite induction in non-standard logics November 23, 2024I will describe the issue through a concrete example. We have a class of sets indexed by ordinals. We want to prove a certain property about those sets and an argument by induction works, but this is not a first-order property. For example, suppose that the sets are posets and that the property is the […]TiskarnaBrother
- Condition on a signature for union of substructures to be a substructure November 23, 2024Let $A_1,A_2 \subseteq \mathcal M$ be two substructures of a structure $\mathcal M$ with signature $S$. What is a sufficient condition on the signature $S$ that ensures the union $ A_1 \cup A_2 $ is also a substructure of $\mathcal M$? My guess is, that if $A_1 \cap A_2$ is a substructure and the closures […]anonymous
- If x belongs to A and A is not a proper subset of B, then x belongs to B. Is this statement is true? [closed] November 23, 2024If x belongs to A and A is not a proper subset of B, then x belongs to B. Is this statement is true ? So the solution of this question is as. Let A = {3,6,12} and B = {2,5,10}. Now 6 belongs to A and A is not a proper subset of B […]Arjun Sharma
- General frame for GL satisfying a consistent set November 23, 2024It is a theorem of modal logic that every normal modal logic is sound and strongly complete with respect to some class of general frames. As such, the provability logic GL is sound and strongly complete with respect to some class of general frames. It is easy to say that for all consistent sets of […]tses
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