# Tag: OpenLab7

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### Logic on Math StackExchange

- How to prove that these formulas are uniquely readable. June 4, 2023We have the inductively defined set, M, as follows(this is a language over the alpabet $(e,a,\bar{a},b,\bar{b})$: We let $M$ be the smallest set so that $e\in M$ If $x,y\in M$ then $xy\in M$ If $x \in M$ then $ax\bar{a}\in M$ If $x \in M$ then $bx\bar{b}\in M$ How to I prove that this set is […]user394334
- Prove $\exists$-$E$ can be derived from $\forall$-$E$ and $\forall-I$ June 4, 2023Here are the definitions of the rules: This answer helped me to prove the $\exists$-$I$ rule, but im struggling to prove $\exists$-$E$ How to show that the introduction and elimination rules for $\exists$ can be derived from the rules for $\forall$? So basically, im supposed to show that given $\neg\forall x\neg A(x)$ and a subproof […]lightyourassonfire
- Mendelson, Logic, Lemma 2.16: a possible typo or flawed understanding? June 4, 2023Here is the statement of Lemma 2.16: Let $J$ be a consistent, complete scapegoat theory. Then $J$ has a model $M$ whose domain is the set $D$ of closed terms of $J$. Definition of scapegoat theory in his book is: A theory $K$ is a scapegoat theory if, for any wf $\mathscr B(x)$ that has […]noballpointpen
- About some patterns I've noticed in truth functions June 4, 2023https://en.wikipedia.org/wiki/Truth_function I was looking at the 16 truth functions to see if I notice any patterns. I've found: There are 1 set with 4 ordered pairs, 4 sets with 3 ordered pairs, 6 sets with 2 ordered pairs, 4 sets with 1 ordered pair, and 1 set with 0 ordered pair. If looked at as […]csp
- 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
- How do I show that some rule is derivable from another rule? [duplicate] 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

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