Tag: OpenLab8
Logic on Math StackExchange
- Mathematical Deduction Theorem Induction proof, case of minimal two-line deductions April 16, 2024The mathematical deduction theorem from mathematical logic appears extensively in the mathematical logic literature (in my experience) and has been extensively discussed in Stack Exchange. One can see the wikipedia for the standard induction-based proof of the non trivial part. The theorem states: for a first order language with standard types of 3 axioms and […]grell6954
- Considering $\Gamma \vdash \varphi$, is $\Gamma$ a set or a list? April 16, 2024As far as I know, most of mathematical logic textbooks state the Weakening Lemma: Let $L$ be a first-order language. Then, for any sets $\Gamma_1$ and $\Gamma_2$ of $L$-formulas and any $L$-formula $\varphi$, if $\Gamma_1 \subseteq \Gamma_2$ and $\Gamma_1 \vdash \varphi$ then $\Gamma_2 \vdash \varphi$. I tried to prove it by induction on $\Gamma_1 \vdash […]Kijeong Lim
- Proof of the principle of induction [duplicate] April 15, 2024I will be referencing the proof I provided here. I don't understand the remark of a person, who states the incorrectness of such proof. From what I could understand, the proof is also valid not assuming well-order on $\mathbb R$. However, this person says that: $\mathbb N$ needs to be well-ordered, otherwise you would have […]Elvis
- What is the accurate definition of inference rule? April 15, 2024Take the inference rule "$P\land Q \Rightarrow P$" for example, what is its accurate definition? "For any string $P\land Q$, if P and Q are w.f.f, then we can derive P" "For any string $P\land Q$, we can derive P" I prefer the second definition because it is simpler and not much meta-knowledge is involved. […]William
- How is the Łoś–Vaught test applied applied in infinitary logics? April 15, 2024I've been researching model theoretic logics and all the related stuff (categoricity, Hanf numbers) lately, and have been confused on how the Łoś–Vaught test applied to infinitary logics. The Łoś–Vaught test specifically states: If a satisfiable theory is k-categorical for some k ≥ |σ| for σ is some signature and has no finite models, it […]SJe967
- What is the Domain of Discourse when one isn’t specified? April 14, 2024The null set is a set that contains no elements. I think I can express this statement like this: $\forall x (x \not\in \emptyset)$ Here x could be a number, a word, a function, really anything. What is the domain of discourse then? In general if a domain of discourse is specified what is it […]Dr. J
- Why is this sentence of predicate logic false on this model? April 14, 2024The sentence: $\forall x \forall y \forall z ((Rxy \land Rxz) \rightarrow Ryz)$ Model. Domain = 0,1,2. R: , This sentence is apparently false on that model, but I'm struggling to see why. Since there's no x,y,z in the extension of R that satisfy the antecedent of the conditional, shouldn't the conditional be true, and […]songdivision
- Proof within Łukasiewicz's infinite-valued logic $Ł_{א}$ April 13, 2024Using the axiom system for Łukasiewicz's infinite-valued logic $Ł_{א}$, I need to construct a proof of the following: ⊢ (A → B) ∨ (B → A) ⊢ (A → (B → C)) → (B → (A → C)) A → B ⊢ (A ∧ C) → B A → B ⊢ ¬B → ¬A But […]Amilio
- Relating the Semantics of Church's Simple Type Theory to Dependent Type Theory April 13, 2024I'm looking at the 'standard' semantics of Church's simple type theory/simply typed lambda calculus (see for example section 2 here or pp. 194-196 in the January 2024 version of this), and I'd like to understand the relationship of that semantics with the standard set-theoretic semantics of depndent type theories (see this answer). Let's focus on […]user837242
- Mathematical logic -informal statement calculus- Adequate sets of connectives [closed] April 12, 2024Prove that {∼,↔} is not an adequate set of connectives. Please help me with this exercise, I cannot prove it.thanksAlireza Alehoseini
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