# Tag: vi hart

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- If $f$ is surjective, it has a right inverse February 28, 2024I've been struggling to understand how the surjection of a function $f : X \rightarrow Y$ implies that it has a right inverse. My questions basically reside on the application of the axiom of choice to the proof. First question: The axiom of choice states that, for any set $Y$, if $\varnothing \not\in Y$, then […]TylerD007
- How to show that if $\Gamma \vdash_{\frak{L}'} \varphi$ then $\Gamma \vdash_{\frak{L}} \varphi$ for languages $\frak{L}\subseteq \frak{L}'$? February 28, 2024I have a course in mathmatical logic, primarily about first-order logic. I'm stuck with trying to prove the following statement: Let $\frak{L}\subseteq \frak{L}'$ be languages (I have proved that the set of all well-formed formulas of the language $\frak{L}$ is a subset of the set of well-formed formulas of $\frak{L}'$). Now, let $\vdash_{\frak{L}}$ and $\vdash_{\frak{L}'}$ […]compaq.43
- What did Richard Dedekind mean exactly by his statement about generality? February 28, 2024But—and in this mathematics is distinguished from other sciences—these extensions of definitions no longer allow scope for arbitrariness; on the contrary, they follow with compelling necessity from the earlier restricted definitions, provided one applies the following principle: Laws which emerge from the initial definitions and which are characteristic for the concepts that they designate are […]Demon
- Tautologies in classical logic February 28, 2024According to its truth table, $P \lor \neg P$ is a tautology, i.e. it is true for all truth values of its constituent propositions. But how come that is true in classical logic? $\lor$ is the inclusive 'or', so $P \lor \neg P$ means 'either $P$, or $\neg P$, or both of them'. But in […]Riccardo Iorio
- 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

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