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- Prove that formulas in $\{\land,\lor,0,1\}$ represents exactly the monotonic boolean functions March 18, 2025The Problem from Rautenberg's A Concise Introduction to Mathematical Logic: Show that the formulas in $\land,\lor,0,1$ represents exactly the monotonic Boolean functions. These are the constants from boolean function of order $0$, $B_0$, and for $n>0$ the $f\in B_n$ such that for all $i$ with $1\le i\le n$, $$f(x_1,\cdots,x_{i-1},0,\cdots,x_n)\leq f(x_1,\cdots,x_{i-1},1,x_{i+1},\cdots,x_n)$$ The hint given is here: […]Liyang Meng
- Logic: is LHS of equality distributive across RHS OR? [duplicate] March 18, 2025Is LHS of equality distributive across RHS OR? For example, is the following a valid change of form to solve for $y$? $T = T \land y$ Negate both sides, DeMorgans Law: $F = F \lor \lnot y$ Distribute across equality? $\lnot y = F, F = F$ Negate both sides: $y = T, T […]atod
- Generalization in propositional calculus (Mendelson's book on Mathematical Logic) March 17, 2025In Section 2.4 of Mendelson's Introduction to Mathematical Logic (6th Edition), below the proof of Corollary 2.3, the author cautions against an unqualified application of the deduction theorem à la propositional calculus in the predicate one. To clarify this distinction, a small counterexample is presented — one that I can't fully understand... Counterexample: "...for any […]WhatAbouty
- Prove that each formula $\alpha$ in signature $\{\neg, +\}$ represents a linear Boolean function March 17, 2025$f\in\boldsymbol{B}_n$ is called linear if $f(x_1,\ldots,x_n)=a_0+a_1x_1+\cdots+a_nx_n$ for suitable coefficients $a_0, \ldots , a_n\in \{ 0, 1\}.$ Here + denotes exclusive disjunction (addition modulo 2) and the not written multiplication is conjunction (i.e., $a_ix_i=x_i$ for $a_i=1$ and $a_ix_i=0$ for $a_i=0).$ (a) Show that the above representation of a linear function $f$ is unique. (b) Determine the […]Liyang Meng
- Does ZF-FA $\vdash$ $\exists x (x = \{x\})?$ March 17, 2025A set is called reflexive iff x = {x}. An atom or unrelement is not a set but it can be an element of a set. Given ZF.We get ZFA if we allow atom and don't allow reflexive sets. If we want get reflexive sets, we must remove FA. It seems that $\vdash Con ($ZF-FA […]whiston
- Theorem on the Equivalence of True and False Propositions [closed] March 16, 2025Title: Theorem on the Universality of Truth and Its Epistemological Implications Body: I propose the following theorem, which challenges the traditional way we distinguish between true and false in classical axiomatic systems: Theorem (Universality of Truth – Saad Sigma): In a system where truth is defined absolutely by a “theory of everything,” all true propositions […]Saad Sigma
- Laver's theorem, reference March 16, 2025Is there a recent paper or book's chapter that proves Laver's theorem on Fraïssé's conjecture, apart from Laver's original paper On Fraissé’s order type conjecture? ThanksLorenzo
- Can $\mathsf{PRA}$ prove the consistency of the axioms of group theory? March 16, 2025In usual, consistency proofs is carried out by model theoretic methods or syntactic methods. And what consistency proof can be carried out depends on the meta-theory in which the consistency proof takes place. Then $\mathsf{PRA}$ can prove the consistency of the axioms of group theory by model theoretic methods or syntactic methods? Of course $\mathsf{PRA}$ […]Matcha
- Can Self-Verifying Theories define their own truth predicate? March 15, 2025All of the following information is from Dan Williard's Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles Self-verifying theories are theories can prove their own consistency while still retaining their consistency. This is because of the following: A class of formulas can be defined where we consider the mapping of all $\Pi^0_1$-sentences $\phi$ […]SJe967
- Is every intermediate logic determined by its implicational fragment? March 14, 2025An intermediate logic is a propositional logic whose axioms are a subset of classical logic, but which at least includes all the axioms of intuitionistic logic. The implicational fragment of such a logic is the set of all implicational formulae which it proves, where an implicational formula is one where the only logical connective is […]Jade Vanadium
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