Tag: ViHart
Logic on Math StackExchange
- Induction and background theory March 24, 2023I've been trying to understand philosophical versus mathematical induction and I'm using the Genesereth/Nilsson text Logical Foundations of Artificial Intelligence which has a chapter on induction. They start by giving the example of selecting cards from a regular deck and saying "certain cards are rewarded, others not." Then they give examples of rewarded cards chosen […]147pm
- Negation of a statement with an inequality. March 24, 2023Let $A\subset\mathbb{R}$ be an upper bounded set. Then $$\forall\varepsilon>0~\exists x\in A\text{ such that }\sup{A}-\varepsilon< x \leq \sup A$$ I want to negate that statement. Would it be: $$ \exists \varepsilon>0~\forall x\in A\text{ such that } \sup A-\varepsilon\geq x\text{ or }x>\sup A~~?$$Fabrizio Gambelín
- Does this english-language sentence translate to the drinker paradox? March 24, 2023There are (one or) two students such that if they pass the exam, then every student passes the exam. I'm tasked to formalize the above sentence and give either a proof or counterexample. If we take the task "formalize the sentence" as "construct a formula that best captures the real world meaning", then I think […]Michael Langowski
- logic question- PLE logic March 24, 2023Using the given symbolization key, translate the following English sentences into PLE. UD The set {people in Brandon} Cxy x is a child of y Fxy x is a friend of y Exy x is an enemy of y Yxy x is younger than y Mxy x is married to y Px x is a […]xyz
- Can ALL necessary conditions, except sufficient ones, for a proposition constitute its sufficient condition? March 24, 2023This seems obviouly. A proposition P may have many many necessary conditions, even infinit. Some of these conditions are also sufficient, and we don't consider them. For ALL rest of necessary conditions, do they universaly identifies the P? I mean, If there is another proposition Q, which has exactly the same necessary conditions as P, […]tocrafty
- Existential elimination rule in natural deduction March 24, 2023This is the rule of Existential Elimination: ∃xAx ⊢ B Can Ac be inferred directly from ∃xAx, given c is assumed to have the property A? For qS-tree proofs, it is a valid rule to infer Ac from ∃xAx and for fS trees, Ac can be inferred from ∃xAx given the assumption Ec. In both […]kwalia
- Terence Tao's definition of function equality March 23, 2023In his Analysis I book, Terence Tao defines two functions $f,g:X\to Y$ to be equal if $f(x)=g(x)$, for all $x\in X$. After giving some examples of this concept, he then says: This notion of equality obeys the usual axioms (Exercise 3.3.1). It is quite easy to show that this relation defined on the class of […]Gleison Stanlley
- Is "exist A such that if A, then not B" equivelant to the logic “A and Not B”? [closed] March 23, 2023Is my statement correct in logics? The statement “exist A such that if A, then not B” is equivalent to “A and Not B”. This is because “exist A such that if A, then not B” means that there exists an A such that if A is true, then B is false. This can be […]Eyal Cohen
- Counterexample to most general unifier example March 23, 2023I am reading these slides about unifiers and most general unifiers (MGUs): https://artint.info/2e/slides/ch13/lect3.pdf In particular, an MGU is defined as follows. A substitution $\sigma$ is an MGU of $e_1$ and $e_2$ if $\sigma$ is a unifier of $e_1$ and $e_2$ and If a substitution $\sigma'$ also unifies $e_1$ and $e_2$, then $e\sigma'$ is an instance […]OvettoNocciolato
- Greatest satisfying assignment to Horn clause theory March 23, 2023Consider a Horn clause theory $\mathcal H$ over some propositional atoms $\mathcal A$. Call a function $f : \mathcal A \to Bool$ an assignment. If $f$ and $g$ are assignments, write $f\leq g$ when $\forall{A\in\mathcal A}.f(A)\Rightarrow g(A)$. It is not hard to see that this is a partial order. Call $f$ a satisfying assignment when […]Jim
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