These are some examples in modal logic, I am not too sure why the first two are valid and why the last one specifically is not valid. Hope someone can explain!

  1. $\square(P \rightarrow P)$ (VALID)

  2. $\square P \rightarrow \square P$ (VALID)

  3. $\square P \rightarrow P$ (NOT VALID)

  • $\begingroup$ Your question shows up with square boxes at the beginning of each statement, so I don't know what it actually says. You may want to consider using LaTeX to write out the math. $\endgroup$ – Jonathan Christensen Dec 6 '12 at 19:39
  • $\begingroup$ @JonathanChristensen: I hope the edits have made everything clear! $\endgroup$ – user642796 Dec 6 '12 at 19:46

Something is perhaps amiss, as the third is certainly a modal validity (in normal alethic modal logics which aim to regiment the notion of necessary truth). Which modal logic are you working in?

(1) holds in any normal modal logic, as any tautology like $(P \to P)$ is a theorem, and necessitations of theorems are theorems.

(2) is an instance of a tautology so is a theorem.

As I say (3) is in most systems (an instance of) a modal axiom, reflecting the truism that if $P$ is necessarily true, then it is plain true. [If the box is read e.g. deontically, as meaning "it ought to be the case that", then the principle obviously fails, as things that ought to be the case aren't always the case. But in a modal logic with the box intended to be read alethically, as meaning "it is necessarily true that" the principle is evidently sound.]

[I'd recommend the opening chapters of Rod Girle, Modal Logics and Philosophy (Acumen 2000, 2009) to anyone starting out on modal logic.]

  • $\begingroup$ General modal logic is the one I am working in if that makes sense, I know there is a k system $\endgroup$ – Masterminder Dec 6 '12 at 19:18
  • 2
    $\begingroup$ There is no such thing as "general modal logic": there is rather a whole family of logics of varying strengths, of which the key ones you ought to know about are T, S4 and S5. The system K is weaker than all those, and in that system, $\Box P \to P$ fails (but that's why $K$ isn't an adequate modal logic if what you are aiming to regiment is the notion of necessary truth). $\Box P \to P$ is a theorem of T, S4 and S5. $\endgroup$ – Peter Smith Dec 6 '12 at 19:43

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