If a proposition is necessarily true, does it follow that it's a tautology?

Question

If □P, does it follow that P is a tautology?

I know in K modal logic, the law of NEC states

⊢ P; therefore □P.

The corresponding conditional of the previous argument is

If ⊢ P then □P.

Now ⊢P iff P is a tautology, where I, as a monist, have concluded the propositional calculus is the only logic that makes sense.

So if that conclusion of mine is correct, then by the rule of replacement

If P is a tautology, then □P.

Can anyone out there can prove the converse of the preceding statement?

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EDIT -

Let Rx be the statement x is red all over, and let Bx be the statement x is blue all over. If Rx then not Bx. From the definition of IF, not Rx or not Bx. Now by DeMorgan's Law not(Rx & Bx). The previous statement is not a tautology unless Bx iff not Rx. Since there's more than two colors, it's not the case that Bx iff not Rx. So not(Bx and Rx) is not a theorem of any formal logistic system that is analytic. It is a statement about reality, that denotes a proposition that is true at all moments in time. The proposition it denotes has a truth value that is constant in time, so only one row of its truth table is possible, for any specific object x. Consider the following truth table, for an arbitrary object x.

Rx	Bx	not(Rx and Bx)
0	0	1
0	1	1
1	0	1
1	1	0

The fourth state row is impossible. As you can see, the statement in column three isn't a tautology, unless we carefully define tautology so that it is. As only three rows are possible we may say

(not Rx and not Bx) or (not Rx and Bx) or (Rx and not Bx), for any object x.

Thus if we define a tautology to be a compound statement that is true in every row of its truth table that is possible, then the statement not(Rx and Bx) is a tautology.

Can we redefine tautology in this manner?

Answer 1

In effect, your question is asking whether logical necessity is the only kind of necessity. It is fairly standard to hold that there are many kinds, of which logical necessity is only one. There is epistemic necessity, physical necessity, practical necessity, perhaps moral necessity, and others.

Many theorists also consider that there is a kind of metaphysical necessity that is stronger than physical necessity but less strong than logical necessity. Kripke in particular defended the thesis that there are metaphysical necessities arising from the necessity of identity and the causal theory of reference. According to Kripke, statements such as, "Water is H2O" or "Hesperus is Phosphorus" are necessarily true. They are not a priori knowable, and not logical truths, but necessary in a broader sense.

Some theorists also like to speak of a kind of necessity that attaches to statements such as, "If A north of B then B is south of A", "no human being is a number", "anything that is red is coloured", "no person is taller than themselves". These propositions are not true under all interpretations, so they are not logical truths or tautologies. But they are more than merely contingently true statements. Some writers refer to this class of statements under the heading of 'conceptual necessity' or 'broad logical necessity'.

Mathematical theorems are often held to be necessarily true, but they are not tautologous in the logical sense. For example, "2+2=4" is a theorem of arithmetic, but it is not true under all interpretations.

In modal logic, the box operator is typically used to indicate some kind of necessity, and the various different modal systems correspond to different kinds.

Answer 2

But what is a necessary truth? "A proposition that is true in all possible worlds." Well, is it possible for there to be something that exists in all possible worlds? Given the variety of conditions in an infinite multiverse, one would need an entity that could exist in worlds that did unfold from an initial expansion, as well as in worlds whose histories go back forever, in worlds that popped into place "last Thursday," and so on and on. One wonders whether it is only beings like God which could be necessary (compatible with arbitrary other building-block propositions for object-talk), but then one admits that if it is never analytic ("tautologous") that something exists, then for there to be necessary-existence claims that go through, one will have to forego equating "being a necessary truth" with "being a tautology."

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Are tautologies themselves true in all possible worlds? For a flippant example: consider a world where intuitionistic logic is somehow ontologically robust enough to outweigh its main competitor (FOL, or SOL, or whatever; we're roaming the worlds, so feel free to choose which random opponent, if any, that you please). In a sense, this will be a world where at least some tautologies of F/SOL "aren't true" (though, being an intuitionistic world, that won't of course be enough to say that there are actively false tautologies in that world). Then again ("of course"), worlds whose ambient logic is so variable are often classed among impossible worlds, so perhaps we have made no headway on your question.

But so are tautologies true in possible worlds or at them, so to speak? We could well have a system where all the first-order worlds W₁ are contained by some metaworld W₂. We could replace, "X is necessarily true if and only if true in all possible worlds," with, "X is necessarily true if and only if X is contained by W₂, and one way to get contained by that is to be contained by all the W₁'s." So there would be abstract things subsisting in the transworld space between the W₁'s and which "is" W₂; we might say that abstract logical truths subsist there, and not in the W₁'s themselves; tautologies are "merely necessary," so to speak (not actual or possible, here; but not "inactual" or "impossible," quite, either).

Less extravagantly: consider the difference between "any" and "all." Illustrate this difference with reference to the definitions of disjunction and conjunction: A disjunction is true when any of its disjuncts is true; a conjunction is true when all of its disjuncts are [note the plural verb!] true. Thus we might distinguish between a sense of necessity extracted from "any" and one extracted from "all," and we might say, "A tautology is true in any world that it dwells within" (if an indweller of worlds it be) without meaning to say that they are actually true in every possible world directly and simpliciter (if there is no completed totality of possible worlds, say, on account of issues with unrestricted quantification and comprehension in set theory).^w

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^wOr consider going from possible-worlds talk to impossible-worlds talk, and then to modal-worlds talk generically. So to say, we would relativize having the modal predicates to being in the "right kind of world." So W₂ could be a second-order possible world itself, i.e. such that membership in it makes something into a (second-order) possibility. Then we regain the possibility of the W₁'s themselves, which was lost to the idea that worlds do not have to contain themselves (and we might try out an exotic reduction whereby actuality itself is conceived of in terms of some W_n being contained by itself, i.e. by being the source of its own possibility as such).

So then we would not say, "X is merely possible," or, "X is merely necessary," but, "X is first-order merely possible/necessary, but X is also second-order necessary/possible." And so on and on. Wherefore neither would the multiverse of modal logic really be "the" such thing, would it not? (A second-order actual world contains an ensemble of first-order possible worlds, just like a second-order possible world does; but so are all the worlds contained by a second-order actual world, also first-order actual? And what happens once we venture to W₃ or even something like W_ω (among others)?)

Answer 3

If □P, does it follow that P is a tautology?

There are two kinds of statements that P can be, temporal and atemporal. A temporal statement denotes a proposition whose truth value varies in time, whereas an atemporal statement denotes a proposition whose truth value is constant in time. Those are the only two choices for P, in your question.

Any P is composed of n simple statements a₁,a₂,...,a_n, also called atomic sentences. A simple statement is a statement with exactly one subject and exactly one predicate. Symbolically

P=P(a₁,a₂,...,a_n).

Now there is a rule of inference in SQML called NEC, which is

⊢ P; therefore □P.

It's corresponding conditional is

If ⊢P then □P.

P could be a contingency, that's why we can't have

If P then □P.

But, suppose P is an atemporal statement that denotes a true proposition. Then, for that statement I will argue we do have

If P then □P.

Take your example of metaphysical necessity.

No object can be red all over and blue all over.

Symbolize that statement by

P=∀x[~(Rx ∧ Bx)]

It's an atemporal statement, that denotes a true proposition. Thus P is metaphysically necessary, symbolized □P.

Now take an epistemic necessity like

Fred must have stolen the book.

Let P=Fred stole the book.

P is an atemporal statement. Suppose it denotes a true proposition. Not just any reasoning agent can draw the conclusion that □P. But an omniscient one always can. Thus for any reasoning agent who knows "Fred stole the book," that reasoning agent can know "Fred must have stolen the book.". Therefore □P.

Now let's analyze Saul Kripke's example from Naming and Necessity.

P=Water is H₂O.

It's an atemporal statement, that denotes a true proposition, so it's necessary. Thus, again, □P.

Now let's take physical necessity.

P=Nothing can travel faster than light.

Suppose it denotes a true proposition. Then since it's an atemporal statement, □P.

Now let's take mathematical necessity.

Let P be the statement 2+2=4.

It's an atemporal statement that denotes a true proposition, so □P.

There is a commonality in all these examples.

Let P be an atemporal statement. Then we have the valid rule of inference

P; therefore □P.

In all these examples, the necessity operator was undefined, and in all these examples P is necessary, but not a tautology.

Now, suppose we define it.

Instead of using Kripke semantics to define the necessity operator in terms of "possible worlds" define it in terms of "possible rows" of the truth table for P. Thus

□P iff ∀A₁∀A₂...∀A_n[P(A₁A₂...A_n)], where the domain of the variables is the set of simple statements.

Since the variables can be any simple statements, they can all be instantiated by temporal statements, and P denotes a true proposition regardless. Thus, all of the 2ⁿ rows of the truth table for P are possible. And in each row, P denotes a true proposition. Thus P is a tautology.

Therefore, "If □P then P is a tautology," only if P is defined using "possible rows" semantics.