Tag: OpenLab 8
Logic on Math StackExchange
- How to prove $\phi \models \psi$ iff $\phi \cup \{\neg \psi\}$? September 22, 2023I want to prove $\phi \models \psi$ iff $\phi \cup \{\neg \psi\}$ unsatisfiable. What's the meaning of the union operator w.r.t. satisfiability in FOL? Does it mean that all such interpretations $\mathcal{I}$ of the LHS also have to model $\phi$ and $\{\neg \psi\}$? So $\phi \cup \{\neg \psi\} \Leftrightarrow \phi \wedge \neg \psi$ ? I […]Martin Stuwe
- Statement with conjunctions and disjunctions [closed] September 22, 2023Is it correct to say $$ (~a \&\& ~ b) \mid\mid ( ~a \&\& b \&\& c) = ( ~a \&\& ~b) \mid\mid ( ~a \&\& c), $$ where $\&\&$ means "conjunction" and $\mid\mid$ means "disjuncton". I am also not sure whether I can simplify it further, so please explain if that is possible and […]Nadia
- How can one claim they proved $φ$ or $¬φ$ without knowing which? September 22, 2023I've been told True Arithmetic is syntactically complete(either $φ$ or $¬φ$ is provable for every $φ$), yet one cannot decide which of the two. This makes no sense to me. How can one say they proved either $φ $ or $¬φ$, without knowing which?setblack7
- Cardinality in an instance where evolution describes everything? [closed] September 22, 2023The simplest algorithm to represent evolution I can think of (as a physics student) is this: Given a system with probability outcomes $p_1$ and $1-p_1$. Life will search if $p_1 > 1- p_1$ or $1-p_1 > p_1$. There are 2 values over here required such as certainty of accuracy of $p_1$ and judgement of $p_1 […]More Anonymous
- Finding the relationship (equivalence or implication) between two expressions September 22, 2023I am trying to find the relationship between $$\exists X \; (p(X) ∧ q(X))$$ and $$\exists X \; p(X) ∧ \forall X \; q(X).$$ I believe that quantifiers cannot be used in forming truth tables, after all both expressions become equal if we take out the quantifiers. Without a truth table I do not know […]Govt_employee
- Does $\omega$-consistency depend on the encoding? September 21, 2023A given theory $T$ can interpret the language of arithmetic in different ways, and therefore it seems to me like it is possible that in one of these ways $T$ would be $\omega$-consistent but not in the other. Assuming PA is $\omega$-consistent, there seems to be an easy way to construct such a theory: Let […]Carla_
- Does semantic consistency imply syntactic consistency? September 21, 2023Semantic consistency of a theory T is defined by there being a model M in which all theorems of T are true. Syntactic consistency of a theory T is defined by there being no formulas A such that both A and it's negation ~A are provable. So let's assume semantic consistency and then assume syntactic […]Anon
- How to turn a $p \Rightarrow q$-type statement into a $\neg q \Rightarrow \neg p$-type one? September 21, 2023I am working with the supposition that $\lim\limits_{x\to \infty}{f(x)}=\infty$. I have rewritten it as $$\forall R>0\;\exists r>0\;\text{ such that }\;|x|>r\Rightarrow |f(x)|>R.$$ I want to turn this "$p\Rightarrow q$"-type supposition into a $\neg q\Rightarrow \neg p$ one, but I am not sure how to negate it. Is $$\forall R>0\;\exists r>0\;\text{ such that }\;|f(x)|\leq R\Rightarrow |x|\leq r$$ […]khalelbm
- Encoding lexicographic order into CNF September 21, 2023Suppose we have two Boolean vectors, $x$ and $y$, of length $n$. How can I encode $xJova
- How consider Emergence in Combinatory Logic September 21, 2023Introduction & Presentation of emergence concept My perspective about emergence is not new... since this concept of emergence dates from at least the time of Aristotle. Emergent structures are patterns that emerge via the collective actions of many individual entities. To explain such patterns, according to Aristotle, emergent structures are more than the sum of […]alamata
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