# Tag Info

1

This is simply part of the definition of the word "know". In this context, we are declaring that if a statement is not true, then you do not know it, no matter how strongly you believe it. This may or may not coincide with your understanding of the English word "know" in ordinary language, but it is the meaning that is being modeled ...

1

You are right in that agents may believe some fact that is not actually true. In this case, as the authors say, the truth axiom does not hold. Moreover, the epistemic relation in models ceases to be the equivalence. If we are talking about knowledge in the context of epistemic logic, then the notion of knowledge is pretty strong. First, if an agent knows ...

2

Assuming my clarification in the comments is what you intended: Consider a Kripke frame with a single world $w$ and no accessibility ($w$ is not acessible from $w$). Then at $w$, $\square \varphi$ holds vacuously for all $\varphi$. It follows that all instances of 4 and B are valid on this frame. But T is not valid on this frame. For example, suppose $P$ ...

1

There are several motivations behind taking possibility relations to be equivalence relations. Some are logical, some have to do with applications of epistemic logic. From a logical point of view, using equivalence models (i.e. modal models with equivalence relations) allows reducing modal models to a very simple form, at least in the monomodal case (just ...

1

Your proof sketch aims to show that the 1.1-formula defines the class of frames whose accessibility relations are partial functions. But that is a frame correspondence result and not a frame completeness result, which you are asked to show in the exercise. But there is a connection here between frame correspondence and frame completeness. To show the ...

4

This looks to me like the distinction between Semantics and Syntax, as applied to modal logic. What we want to do with formal logic is allow the manipulation of strings of symbols according to mechanical rules - the Syntax - to allow us to learn about logical facts about certain mathematical structures - the Semantics. From the point of view of the Semantics,...

6

You are conflating the proof theory of modal logic with the semantics of modal logic. We can approach verifying statements in two ways: 1) formal proofs from the axioms. 2) informal arguments for their validity in terms of Kripke frames. (Which is not to say these are the only two approaches, just the two relevant to this question.) The theorems of system K (...

2

So, you want to prove the completeness of a modal logic ($\mathit{K}$, $S5$, etc. ), which is formally can be stated as follows: Completeness: For all $\varphi \in \mathcal{K}$, if $\varphi$ is valid, then $\varphi$ is derivable in the proof system $\mathbb{K}$. In the above, $\mathcal{K}$ is the language of $K$, and $\mathbb{K}$ is the axiomatisation of $K$....

1

Since you mention that you are OK with the first four chapters of Blackburn's et al. Modal Logic, then, technically, you already know Epistemic Logic (EL), which is essentially S5. A classic textbook on EL, which includes various notions of knowledge, completeness and complexity results, etc. is Reasoning about knowledge by Fagin, Halpern, Moses, and Vardi. ...

1

For the first part of (c). If $\diamond \diamond \diamond \phi$ holds at $w$ then we have this picture for $R$ $$w \to x \to y \to z$$ where $\phi$ holds at $z$, and $\diamond \phi$ holds at $y$, and $\diamond \diamond \phi$ at $x$ From the Euclidean property we have $$R(w,x) \wedge R(w,x) \to R(x,x),$$ $$R(x,y) \wedge R(x,x) \to R(y,x),$$ R(y,x) \wedge R(...

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