Is "It is raining or it is not raining." a tautology?

Question

Is the following proposition a tautology:

"It is raining or it is not raining."

I is obviously always true, so one thinks that it should be a tautology. However, i thought a tautology has free variables, which can be replaced by propositions. If then, no matter what proposition we take, the composed composition turns into a true statement, we speak of a tautology.

E.G. $T(A)$ is a tautology, where $T(A):= (A\Rightarrow A)$. So the tautology $T$ depends on A.

However, the sentence "It is raining or it is not raining." doesn't contain a free variable, right?

Answer 1

Of course we can "speculate" on philosophy of language issues ad infinitum.

BUT ... if we agree that propositional calculus can provide a very very simplified "model" of natural language, suitable for some limited applications, than we have to consider [see Dirk van Dalen, Logic and Structure (5th ed - 2013), page 5] :

The linguistic entities occurring in this kind of reasoning are taken to be sentences, i.e. entities that express a complete thought, or state of affairs. We call those sentences declarative. This means that, from the point of view of natural language, our class of acceptable linguistic objects is rather restricted.

Fortunately this class is wide enough when viewed from the mathematician’s point of view.

The sentences we have in mind are of the kind “27 is a square number”, “every positive integer is the sum of four squares”, “there is only one empty set”.

The propositional calculus is based on [see page 7] :

Definition 2.1.1 The language of propositional logic has an alphabet consisting of

(i) proposition symbols: $p_0,p_1,p_2,\ldots$,

(ii) connectives: $∧,∨,→,¬,↔,⊥$,

(iii) auxiliary symbols: $( , )$.

[...]

The proposition symbols and $\bot$ stand for the indecomposable propositions, which we call atoms, or atomic propositions.

[page 15 :] The task of interpreting propositional logic is simplified by the fact that the entities considered have a simple structure. The propositions are built up from rough blocks by adding connectives.

The simplest parts (atoms) are of the form “grass is green”, “Mary likes Goethe”, “$6−3 = 2$”, which are simply true or false. We extend this assignment of truth values to composite propositions, by reflection on the meaning of the logical connectives.

We define valuation a mapping $v$ from the set $PROP$ of atoms to the set $\{ 0, 1 \}$ of truth values :

$v : PROP \to \{ 0, 1 \}$.

Thus we have [page 18] :

Definition 2.2.4

(i) $\varphi$ is a tautology if $v(\varphi) = 1$ for all valuations $v$.

(ii) $\vDash \varphi$ stands for “$\varphi$ is a tautology”.

Thus, according to the truth-functional definition of connectives of classical logic, we have that :

$\vDash p \lor \lnot p$, i.e. : $p \lor \lnot p$ is a tautology.

And so :

“the grass is green or the grass is not green”

is a natural language's instance of the above tautology.

Answer 2

There is an implied variable in this proposition, namely the day (or hour, etc.) being referred to. The question whether or not this is a tautology therefore does involve quantification over a domain which is potentially infinite, at least in some obvious formalisations. Therefore the question whether the proposition is a tautology is a bit more complicated.

This is of course closely related to the law of excluded middle (LEM). If one uses formalisations in classical logic, then of course this is a tautology. However, if one's background logic is intuitionistic, then it is not a tautology. In certain constructive frameworks one cannot even assume trichotomy, asserting that for every real number $x$ one always has $(x<0)\vee(x=0)\vee(x>0)$.

For a recent discussion see http://dx.doi.org/10.1007/s11787-014-0102-8 To make this a bit more plausible, note that LEM is the basic ingredient in a proof by contradiction. Even classically trained mathematicians recognize that a proof by contradiction can be disappointing because it provides no way of exhibiting the object whose putative "existence" is being proved. This was of course the basic concern of Kronecker, Brouwer, Errett Bishop, and others.