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Logic on Math StackExchange
- Greatest satisfying assignment to Horn clause theory March 23, 2023Consider a Horn clause theory $\mathcal H$ over some propositional atoms $\mathcal A$. Call a function $f : \mathcal A \to Bool$ an assignment. If $f$ and $g$ are assignments, write $f\leq g$ when $\forall{A\in\mathcal A}.f(A)\Rightarrow g(A)$. It is not hard to see that this is a partial order. Call $f$ a satisfying assignment when […]Jim
- What does $\bigwedge$ mean? March 23, 2023I came across the notation $\renewcommand{vec}[1]{\mathbf{#1}}$ $$\text{flow}(\vec{x},\vec{y},t) \equiv \bigwedge_{j=1}^{n} \exists \vec{z}\in I : y_j-x_j = t\cdot f_j(\vec{z}).$$ equation when reading the paper Light-weight hybrid model checking facilitating online prediction of temporal properties in preparation for my master's thesis, but I have never seen the triangle symbol (with the $n$ on top) in front of the […]Anton Reinecke
- Why are there 5 propositional connectives in propositional calculus? March 23, 2023I don't understand why there are 5 propositional connectives (conjuction, disjunction, implication, negation and equivalence) that are regularly given a priori in propositional calculus? If the goal was to minimize the number of them why was equivalence given a priori even though it can be defined as a conjucture of an implication and its reverse […]Inquisitor
- When should one use transfinite induction? March 22, 2023I've come across it multiple times now in proof theory papers that authors use (sometimes quite elaborate) inductions in order to prove easy results. The most striking example is the following, where O is the class of countable ordinals: From: Jean H. Gallier, "What's so special about Kruskal's theorem and the ordinal $Γ_0$?", Annals of […]10012511
- Gödel on the “True Reason” for Incompleteness March 21, 2023In footnote 48a of his famous paper on incompleteness, Gödel writes: [T]he true reason for the incompleteness inherent in all formal systems of mathematics is that the formation of ever higher types can be continued into the transfinite (see Hilbert 1926, page 184), while in any formal system at most denumerably many of them are […]neddo
- Intuitionistic Logic vs Constant Domains March 21, 2023Quantified modal logic is a controversial field, specifically since it forces one to consider what is meant by “world” in Kripke Semantics. For example, the formula $\Box \forall x \varphi \implies \forall x \Box \varphi$ requires that if $\Box \forall x \varphi$ is true at a world $w$, then every object in $w$’s domain must […]RW_123
- Determining a statement's truth value from given definitions [closed] March 21, 2023Let the predicates $P(x)$ and $Q(x)$ be defined on set $\{a, b, c\}$ as \begin{array}{|c|c|c|c|} \hline x& a & b & c \\ \hline P(x) & 1& 1&0\\ \hline \end{array} \begin{array}{|c|c|c|c|} \hline x& a & b & c \\ \hline Q(x) & 1& 0&0\\ \hline \end{array} What is the truth value of $$∃x∀y\;(P(x) \leftrightarrow Q(y))?$$Traa
- use the law of logic to show that ~(p->~q) ^(r->q)^q is equivalent to p^q [closed] March 21, 2023Use the law of logic to show that ~(p->~q)^(r->q)^q is equivalent to p ^ q. Explain each step fullyLuke
- Ambiguity about principle of excluded middle March 21, 2023According to the principle of excluded middle, every statement is either true or false. It might sound a little ridiculous, but consider the following statement: Mountains believe in God. Believing in God has no meaning for things and beings that do not have intelligence. So what truth value should be determined for the above statement? […]Amin
- How to proof that Classic (Propositional) Logic and Priest's (Propositional) Logic of Paradox have the same logical truths? March 21, 2023As far as I understand it, in Priest's "Logic of paradox" there is a proof to the effect that $\phi$ is classically valid IFF $\phi$ is valid in the Logic of Paradox (LP), that is: $\vDash_C \phi$ IFF $\vDash_{LP} \phi$. Priest says that the proof from right to left is immediate since every two valued […]Frank
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