Tag: OpenLab 8
Logic on Math StackExchange
- Determining whether a given set of sentences is satisfiable August 17, 2022I recently ran into this seemingly simple, yet tricky problem, and I would like your help solving it. Determine whether the following set of sentences is satisfiable or not. If not, use natural deduction to prove contradiction. $$ S = \left\{ \exists x \exists y R(x, y), \exists x \forall y (x = y), \exists […]Scorate
- How are $∀y∃x\:P(x,y)$ and $∃x∀y\:P(x,y)$ different? August 17, 2022MIT Opencourseware Notes 6.042J says that $$\exists x \forall y \:P(x, y) \implies \forall y \exists x \: P(x, y)$$ is a valid assertion. I am confused because of a counter model that I thought of: if the domain is the integers and $P(x,y)$ means $x > y,$ then isn't the hypothesis wrong? Because in […]photon
- left-adjoint to join in a Heyting algebra August 17, 2022Define a Heyting algebra to be a bounded lattice $L$ with an operation $\to : L^{op}\times L \to L$ such that for any $x, a, b \in L$ we have $x\wedge a \leqslant b$ iff $x \leqslant a \to b$. Thinking of $L$ as a category, this says $-\wedge a$ is left-adjoint to $a\to -$. […]Matthew Towers
- Are (p → q) ∨ (q → r) and p → r equivalent statements? [closed] August 17, 2022Could anyone pls help me with this question, please? thank you :>tama
- A sound and complete proof system for $\forall$-first-order logic August 17, 2022There is, as is well-known, a sound and complete proof system for first-order logic. It is also known that equational logic, which is the fragment of first-order logic that concerns only universally quantified equations, has a sound and complete proof system. I am interested in a logic that is between equational logic and full first-order […]user107952
- Why is this true? $(\exists x, Px \to r) \iff (\forall x, Px) \to r$ August 16, 2022I can totally understand the forward direction: if there is an $x$ such that $Px$ implies $r$, then clearly having $Px$ true for all $x$ will imply $r$. But the other direction doesn't make any sense to me. If I give a more natural example, assume that "If it rains every day this week, then […]Robin
- Axioms for multiplicative number theory? August 16, 2022Multiplicative number theory is concerned with divisibility, modular arithmetic and the primes. Of course all of the relevant multiplicative properties are theorems of Peano arithmetic and this is usually the way things are done. But I'm looking for an alternative that has models that are not rings. In particular, I'm looking for a set of […]saintali
- Semi-computable sets are not closed under set subtraction August 16, 2022Showing that $A \setminus B$ is not semi-computable for semi-computable sets $A$, $B$ is not too difficult: $\mathbb{N} \setminus C$ is not a semi-computable set for a semi-computable, but not computable, set $C$. But what about the case where $A$, $B$ are semi-computable, but neither is computable? Is there any such pair of sets with […]sanguine
- Is there a formal definition of "Proving theorem X without using theorem Y"? August 16, 2022In math textbooks and math classes, the author or professor sometimes says to prove a certain theorem without using another theorem. I understand what that means intuitively. But is there a formal definition of this notion somewhere, in some book or paper? Or, is this another one of those things that is "you know it […]user107952
- is $(\phi^2 > 2) \Rightarrow (\phi > 1.4)$ true or false? August 16, 2022First, I need to evaluate left and right hand sides of '$\Rightarrow$' to use definition of implication (its truth table). And, I'm simply lost in a question: "how should I evaluate truth or falsity of both $(\phi^2 > 2)$ and $(\phi > 1.4)$ ? Both of left hand side and right hand side of implication […]Oleksandr Khryplyvenko
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