21 votes
Accepted

What is the modal logic of outer multiverse?

I've noticed that recently you have asked a few questions about my work, and so let me thank you; you are kind to take an interest. This particular question can be seen as part of the subject of set-...
15 votes
Accepted

Are buttons really enough to bound validities by S4.2?

Unfortunately, the argument I had had in mind seems broken to me now, and I retract the claim. I wonder what of it can be salvaged? Let me explain what I had had in mind. The main idea was this: it ...
12 votes

A specific Model of ZFC

In a joint work with Mitchell, which is under preparation, we succeeded to give a full answer to Hamkins-Löwe question. Edit: The paper is ready. See On a Question of Hamkins and Löwe on the ...
10 votes

A specific Model of ZFC

$\newcommand\possible{\diamondsuit}\newcommand\necessary{\square}$The forcing modalities defined for a given definable class $\Gamma$ of forcing notions are $\varphi$ is $\Gamma$-possible, written $\...
10 votes
Accepted

Interpretations of modal logic where $\Box$ means "valid"

$\def\R{\mathrel R}$No, this is not possible. Recall that the depth of a point $x$ in a transitive frame $(W,R)$ is the maximal length $d$ of a strictly increasing chain starting at $x$, i.e., $x_1,\...
10 votes
Accepted

Difference between provability and the existence of a proof?

First note this isn't a constructive logic, so it's wrong to think of "$X$" as "there is a proof of $X$". (Even in constructive logic I find that dubious.) Second note that if $X$ ...
  • 122k
9 votes
Accepted

Modal vs First-Order Logic on finite models

On p. 30–31 of Van Benthem’s Notes on modal definability (Notre Dame Journal of Formal Logic 30 (1988), #1, pp. 20–35), you can find a (brief!) sketch of a proof that the modal formula $$(\Diamond\...
8 votes
Accepted

Substitutional modality

$\def\ml{\mathrm{ML}}\let\LOR\bigvee\let\ET\bigwedge$The question you asked is a variant of Problem 42 in Friedman [1]. It also has an intuitionistic analogue, Problem 41, which asks if there exists ...
8 votes

Non-trivial consequences of Lob's theorem

Löb's theorem can be used to show that there exist equilibria in games like prisoner's dilemma when the participants are computer programs that can read each other's source code. If player 1 can show ...
  • 7,566
6 votes

Is the axiom $\Diamond\Box\varphi\to\Box\Diamond\varphi$ in c.c.c. forcing potentialism equivalent to the productivity of c.c.c. forcing?

Failure of $\sf CH$ It seems I was a bit off on my initial go at showing $(.2)_{ccc}$ entails the failure of $\mathsf{CH}$. Here is the correct argument. Proposition 1: Assuming $(.2)_{ccc}$. If $...
  • 1,657
5 votes

In modal logic, is there a formula that could express the inverse of accessibility relation?

No there is not. To be a little more precise on the formalism involved, we consider the following language of propositional modal logic over a set $Var=\{p_0,p_1,\dots\}$ of propositional variables: ...
  • 163
5 votes
Accepted

Is there a good list of nomenclature for modal axioms?

One of the most complete online summaries of modal logic systems is still Halleck's list. Even though the axiom corresponding to density is not explicitly identified in the latter list, it is a ...
  • 418
5 votes

Are buttons really enough to bound validities by S4.2?

We now know that the existence of arbitrarily many independent buttons suffices to bound the validities by Grz.2. It is one of the results from my forthcoming work—Modal theory of the category of sets—...
4 votes
Accepted

How to get $\omega$-regular expression from buchi automaton

A Büchi automaton is a finite automaton that one runs on infinitely long strings (length $\omega$), with the proviso that the string is accepted if infinitely often the machine had visited an ...
4 votes

How can you formalize the metamathematics conventionally used to state Godel’s theorem?

I may not be understanding your question, but here goes: In most cases (aside from some metamathematics of set theory itself) the metalanguage is implicitly understood to be ZFC. So Gödel's first ...
  • 1,489
4 votes
Accepted

Translations between S4 and S5 modal logics

As written, there is a trivial such translation: just put $F'=\top$ for all formulas $F$. Assuming you actually wanted to formulate the condition as “if and only if” rather than just “if”, a simple ...
3 votes
Accepted

Superintuitionistic logics which are not hereditary/monotonic: impossible or possible?

Every propositional logic $L$ weaker than classical logic (i.e., any logic whose provable propositions are a subset of classically provable propositions) has a Kripke model. Just take $W = \{\star\}$, ...
  • 44.5k
3 votes

Literature on Kripke models

Brian F. Chellas: Modal Logic: An Introduction, 1980. Starts very basic but covers in detail the beautiful completeness theorem proofs for the basic systems like $S5$, $S4$, $K$.
3 votes
Accepted

Counterexample equivalent in relevant logic DL

From D3 and D4, 1- $A\wedge\neg B\rightarrow A$. 2- $A\wedge\neg B\rightarrow \neg B$. From 2, D8 and R1, 3- $B\rightarrow \neg(A\wedge\neg B)$. From 1, 3 and R3 (taking $B=A$, $C=B$, $A=A\wedge\neg B$...
3 votes

Initial reference on Gödel-Löb axiom in Kripke semantic of $GL$

The paper Provability: the emergence of a mathematical modality by Boolos and Sambin says (bottom of page $9$) that the first fact was independently gotten by Kripke and Segerberg, and gives the ...
2 votes

On a modal correspondence

Let $\mathrm{Z}$ be the formula in question, which can be rewritten as: $(\Diamond\alpha\land\Box(\alpha\rightarrow\Box\alpha))\rightarrow\Box\alpha$. If I understand correctly "No world sees two ...
  • 318
2 votes

Literature on Kripke models

I can't say that this is the 'best' introduction to Kripke models (as 'best' is always a relative term), but John P. Burgess's survey article "Kripke Models" presumes only knowledge of propositional ...
2 votes
Accepted

What kind of set theory is obtained from the canonical models of K?

There is a 'research report(?)' from the Institute for Logic, Language, and Computation (a rersearch institute of the University of Amsterdam) by Goivanna D'Agostino, Angelo Montanari, and Alberto ...
2 votes
Accepted

Axioms for modal logics based upon counterfactuals

In "Completeness and decidability...", Lewis shows the system C1 to be decidable and complete with respect to the semantics of corresponding canonical $\alpha$-models (81-4). It seems that K holds in ...
  • 146
2 votes
Accepted

Modal collapse upon addition of the law of the excluded middle to an Intuitionistic modal logic

I'll write $\to$ instead of $\supset$, and $\bot$ instead of false, below. Since Law of Excluded Middle is given, I'll argue using classical propositional logic. Since $M\to\bigcirc M$ is already ...
2 votes
Accepted

Can we avoid the modal collapse in a certain Intuitionistic modal logic by abandoning ¬◯⊥ but retaining the law of the excluded middle?

Notice that by the inference rule $$\frac{M\supset N}{\bigcirc M\supset\bigcirc N}\tag{@}$$ we have $$\frac{\bot\supset N}{\bigcirc \bot\supset\bigcirc N}$$ But $\bot\supset N$ always holds. So either ...
2 votes

How can you formalize the metamathematics conventionally used to state Godel’s theorem?

I think this question is clearly asking for provability logic. This is the basic modal logic K with the additional axiom $$\square(\square A \to A) \to \square A.$$ "$\square A$" is interpreted as "$...
  • 39.2k
2 votes

Modal logics which have an algebraic semantics but not a Kripke semantics

There is a semantics based on neighborhood frames. A logic of a class of neighborhood frames is not necessarily normal and even not necessarily monotone (i.e. closed under the ϕ→ψ / □ϕ→□ψ rule). On ...
  • 21
2 votes
Accepted

Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory

My suggestion would be to start from your intended semantics. First ask what kind of "predicative topos" you have in mind where the theory you're asking about would have a model, and ...

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