I have this problem with Modal Logic that I am not sure I see which propositions in this logic are necessary besides tautological propositions.

I mean metaphysically, something is necessary if it's true in every world; besides tautologies and if P is a tautology then L^n P where L is the necessary operator and n is a natural number, is also a necessary statements.

But besides these two categories, what can be included in this category of "necessary" truths?

  • What kind of necessity are you talking about, logical, metaphysical, physical? Each kind produces its own selection of "all" possible worlds, and hence its own list of necessary truths. – Conifold May 3 '17 at 17:56
  • I am discussing metaphysical. As far as I can tell there's nothing necessary in physical truths; I mean the equations and rules are only approximations to reality; with different assumptions on reality you get different truths. Are we made of strings, points or loops? is it even meaningful to ask such a question? I mean strings of what? is it just the word "string" that makes us what we are made of? – MathematicalPhysicist May 4 '17 at 17:08
  • Within the colloquial use of modality we are assuming that well-established laws of nature (conservation of energy, etc.) hold, therefore in all physically possible worlds they are necessary truths. On Kripke's conception of "metaphysical" modality we are also assuming that water is H2O, you are a child of your parents, human, etc., essentially (meaning in each possible world either true or vacuous because the term fails to refer). – Conifold May 4 '17 at 17:49
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    You are thinking of "one true necessity", this is not how it works. The set up is conventional, we stipulate what we wish to keep fixed, laws of logic, laws of physics, background knowledge, plausible common sense assumptions, etc., all, some or none. These choices determine which worlds are possible, and hence what is necessary, see SEP's Varieties of Modality. If your possible worlds are stipulated to allow everything one can think of (coherently) they will produce just logical necessity, which is not very interesting. – Conifold May 4 '17 at 21:05
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    As always, everything in philosophy boils down to the terminology being used. – MathematicalPhysicist May 5 '17 at 14:08

metaphysically, something is necessary if it's true in every world

Sounds like you may be looking for the system of modal logic called S5. This has been studied extensively; it is decidable but it includes other theorems than L^n P where P is a tautology, for instance LP->LLP.

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