Tag: Stars
Logic on Math StackExchange
- How to prove that: if Σ ⊨ α and Σ ∪ {α} ⊨ β , then Σ ⊨ β? June 3, 2023In the PDF that I was reading, it says that this property is called "Transitivity of logical consequence". I don't know how to prove it. My attempt: if Σ ⊨ α , so for any truth assignment v that satisfies Σ ,v(α) = T. if Σ ∪ {α} ⊨ β, so for any truth assignment […]Eduardo Alves
- Proving a logic axiom involving counterfactuals June 3, 2023I just read David Lewis' (1973) "Counterfactuals and Comparative Possibility" where he states an axiom (p. 441) I'd like to prove. $$ ((a \vee b) \Box\rightarrow a) \vee ((a \vee b) \Box\rightarrow b) \vee (((a \vee b) \Box\rightarrow c) \leftrightarrow (a \Box\rightarrow c) \wedge (b \Box\rightarrow c)) $$ Suppose for reductio that the negated axiom […]JGD
- Second Godel's incompleteness theorem June 3, 2023Godel's 2nd theorem says "For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself on other word F can't porve Cons(F). My question is there any correct proof in math leads to a false result please give me example since this is logically refused […]Mahmoud Mrowi
- How do I show that some rule is derivable from another rule? June 3, 2023I know that $P\rightarrow Q$ means $Q$ can be derived from $P$. For example $\forall x(x=2\rightarrow x\neq 3)$, and you would prove this by assuming $x=2$ and showing that $x$ is not $3$. But I dont really understand what it means to prove $P\rightarrow Q$, when $P$ and $Q$ are some kind of rules. For […]lightyourassonfire
- How to show that the introduction and elimination rules for $\exists$ can be derived from the rules for $\forall$? June 3, 2023An instructive exercise, given the negation rules for classical logic, is to show that the rules for ∃ (for ∀) are derivable from the rules for ∀ (for ∃) and the definition of ∃x(A(x)) as ¬∀x¬(A(x)) (the definition of ∀x(A(x)) as ¬∃x¬(A(x))) What am I supposed to do here exactly? Assume that the rules for […]lightyourassonfire
- How to formalize "if and only" in propositional logic June 3, 2023I want to formalize and proof the validity of "If I train or if I don't train if and only my friend competes, I'll go to the meeting" p = I train q = my friend competes r = go to the meeting My guess is (p ∨ (¬ p ↔ q)) → r, but […]Lita
- Prove A $\times$ A equinumerous with A, for well ordered A without AC? [duplicate] June 2, 2023For everything below we assume A is infinite For sets A and B, let $A \equiv B$ denote there is a bijection between A and B. I know $\forall A, A \times A \equiv A \iff \text{Axiom of Choice is true}$ However, for particular A, such as the natural numbers, there is a way to […]wsz_fantasy
- Formalising the sentence "someone is plotting against me" June 2, 2023Someone is plotting against me. Can the above sentence be translated into predicate logic without using the existential quantifier? If not, is it because the sentence is self-referential?Richard Gasquet
- Cardinal number of the iterated set $A^{*}$, where $A=\{ a,b,c\}$. Why can't I use Cantor's diagonal argument? [duplicate] June 2, 2023I have a question which may sound silly to you, but I'm confident that I don't understand Cantor's diagonal argument very well to use it. Any provided insight would be appreciated. I was tasked with finding the cardinal number of the set $$A^{*} = \{\epsilon, a, b, c, aa, ab, ac ... \}$$ Which is […]THE_CRANIUM
- A question on existence of ultrafilters and cardinality. June 2, 2023Let $S$ be an infinite set with cardinality $k$. Let $U$ be an ultrafilter containing the filter of cofinite sets. Then, for any set $P \subset U$ such that the cardinal of $P$ is $Nulhomologous
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