# Tag: calculus

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- How to interpret functional symbols in a many-sorted language in the corresponding single-sorted language? February 27, 2024I was reading about Many-sorted logic and I kept seeing a lot of authors claiming that "When there are only finitely many sorts in a theory, many-sorted first-order logic can be reduced to single-sorted first-order logic". I get that this is done by introducing, for every sort $a$, a unary predicate symbol $P_a$ and by […]Eduardo Magalhães
- Forcing as a Quotient February 27, 2024I'm reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here. Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^B$ as follows: $M^B_0 = \emptyset$ $M^B_{\alpha+1} = \{ \text{partial functions} \ f : M^B_\alpha \to B \}$ $M^B_\Lambda = \bigcup_\lambda M^B_\lambda$ for limits […]ASheard
- Existence and uniqueness of "minimal p-object"? February 27, 2024I'm currently working on an analytical philosophy problem, and I think formal logic tools could help me. First of all, I introduce some definitions. Let us give ourselves a language $\mathcal{L}$ and a theory $\mathcal{T}$. Unless otherwise stated, we place ourselves within this theory, and do not deviate from it. Let us take a property […]K. Kapa
- Defining the Y combinator in terms of S, K and I February 27, 2024We know that the Y-combinator is defined as: $$\text{Y}:=\lambda f.(\lambda x.f(xx))(\lambda x.f(xx))$$ Wikipedia says :$$\text{Y}:=\text{S(K(SII))(S(S(KS)K)(K(SII)))}$$ Now the question is: What logical steps can we take to convert the first definition to the second? While it is easy to show the equivalence between the two definitions, finding how the first definition can motivate and lead to […]Soham Saha
- Logic behind contrapositive proofs that involves De Morgan's Laws February 27, 2024Suppose $a,b\in\mathbb{Z}$. If both $ab$ and $a+b$ are even, then both $a$ and $b$ are even Proof by contrapositive. Propositions: $P$: $ab$ is even $Q$: $a+b$ is even $R$: $a$ is even $S$: $b$ is even Then logically we have $(P\land Q)\implies (R\land S)$. We have to negate $R\land S$, so $\neg(R\land S)$, by De […]Alexis SM
- Why we have to proof both $Q$ and $R$ in $P\implies (Q\lor R)$ February 27, 2024I'm studying proofs trying to use logic before starting with the proof. A direct proof can be written as $P\implies Q$, by forcing $P$ to be true, we have to force $Q$ to be true so the statement stays true. But in a case of the form $P\implies (Q \lor R)$, why do we have […]Alexis SM
- Can a CL-term have multiple fixed points? February 27, 2024Given a CL term $E$, can there exist multiple non-equivalent fixed points for $E$? I think: any fixed point of $E$ can be expressed as $Y(E)$, this expression cannot reduce to multiple non-equivalent forms, due to confluence. So, I think that any CL term can have only 1 fixed point. Is my logic correct? All […]Soham Saha
- Complexity of entailment between equivalences of dual formulas February 27, 2024Consider a propositional language over the set of propositional variables $\{p^+,p^-,q^+,q^-,\ldots\}$ and connectives $\{\wedge,\vee,\rightarrow,\equiv\}$ (conjunction, disjunction, implication, equivalence). We call a formula $\phi$ monotone if it only contains $\wedge$ and $\vee$. Additionally, we call $\phi^\partial$ the dual of $\phi$ if all $\wedge$'s are swapped for $\vee$'s (and vice versa) and all $p^+$'s with $p^-$'s (and […]Daniil Kozhemiachenko
- Why is there no conditional inference rule in Sequent Calculus of these forms? February 27, 2024I'm wondering why sequent calculus doesn't have rules like these (at least in the ones I've come across): $$ \Gamma \vdash A \rightarrow B, \Pi \qquad \Delta \vdash A, \Sigma \over \Gamma, \Delta \vdash B, \Pi, \Sigma $$ And $$ \Gamma, A \rightarrow B \vdash \Pi \qquad \Delta, A \vdash \Sigma \over \Gamma, \Delta, A […]Allen Liao
- Propper way to express a Set Equivalence in logical form February 27, 2024This is my first post on the Math Stack Exchange so i'm sorry if questions of this nature are supposed to be tagged in a specific way, if they are please let me know and I will try to update my post to reflect it! I'm currently trying to self study my way through Vellemans […]Cibo

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