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I'm new to modal logic and I am trying to understand it more intuitively.

If something is true, is it necessarilly true? I.e. $$P\implies\square P$$

This seems intuitive but it is not an axiom. This is not a book problem but a question I am wondering myself.

Let's take S5, which contains within it the following as axiom or theorem:

B $P\implies\square\Diamond P$
D $\square P\implies\Diamond P$
K $\square (P\implies Q)\implies(\square P\implies \square Q)$
T $\square P\implies P$
4 $\square P\implies\square\square P$
5 $\Diamond P\implies\square\Diamond P$

Is it proveable that $$P\implies\square P$$ ?

Isaac Sechslingloff
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  • Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – Shaun Feb 04 '23 at 23:18
  • Please add some context to your question such as what book you are reading where this appears. Alternatively, please include your attempt to prove or disprove $P \implies \square P$ and where you got stuck. – Greg Nisbet Feb 04 '23 at 23:42
  • Isaac, do you know or have you heard of the Kripke semantics for modal logic? Have you tried using it to prove or disprove $P \implies \square P$. Have you tried producing a proof in some proof calculus? – Greg Nisbet Feb 05 '23 at 00:03
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    Alternatively, instead of a proof attempt, you could include links to other questions such as this one that refer to necessitation (that is the name for the property $P \implies \square P$) and describe what parts of the answers don't make sense. Or you could read through a resource like SEP and quote the part of it that does not make sense yet. – Greg Nisbet Feb 05 '23 at 00:10
  • @GregNisbet Your last answer helped. I misunderstood the necessitation rule previously. It completely went over my head. Thank you. – Isaac Sechslingloff Feb 05 '23 at 00:23
  • How do I close the post now? – Isaac Sechslingloff Feb 05 '23 at 00:24
  • The last thing I wrote was intended to be a comment directing you to sources you could cite to provide context for this question, not to be an answer per se. I am glad you found it helpful, though. I have voted to close this question as a duplicate of the question I linked to. You can vote to accept the duplicate assignment. You can also still edit this question and cite that question as context. In my judgment, this question is not an exact duplicate of the linked question, but is similar in spirit. – Greg Nisbet Feb 05 '23 at 01:07

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