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I have this problem statement:

Let $P$ a proposition, now if we suppose $P$ is $\texttt{true}$, and the proof gives $P$. What demonstrates this?

Solution:

$\ \ P\Rightarrow P$

$\equiv \langle \texttt{Definition }\Rightarrow\rangle$

$\ \ \neg P \vee P$

$\equiv \langle \texttt{Law of excluded middle} \rangle$

$\ \ true$

Therefore,

$P\Rightarrow P \equiv true$

But, I'm wandering if this is proving something? Or it just only shows how to apply the law of excluded middle?

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    $\begingroup$ “$P\Rightarrow P$” is a tautology, so it proves nothing. For instance ”$0\ne1\Rightarrow 0\ne1$” is surely true, but says nothing about the truth of “$0\ne1$”. $\endgroup$ – egreg Feb 20 '16 at 20:38
  • $\begingroup$ @egreg, what name can receive this type of proof? $\endgroup$ – InfZero Feb 20 '16 at 20:41
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    $\begingroup$ @JohnOrtizOrdoñez You have proved that “$P\Rightarrow P$” is a tautology. $\endgroup$ – egreg Feb 20 '16 at 20:43
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We have proved the tautology:

$P \to P$,

i.e. a formula (or schema) of truth-functional sentential logic that is always true, i.e. true in every possible interpretation.

In other terms, whatever is the natural language sentence that we can use to interpret the sentential letter (or variable) $P$, the "complex" sentence we will get is true.

Interpreting $P$ with the sentence:

"The snow is white",

the "complex" sentence:

"if the snow is white, then the snow is white"

is true.

And the same with "The sky is green", due to the truth-functional condition for $\to$ (i.e. $FALSE \to FALSE$ is $TRUE$).

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Here are a couple of ways to prove $P \Rightarrow P$.

  1. Use a truth table:

enter image description here

The proof that this statement is a tautology comes from the column in red: all rows in that column are true.

  1. Use a derivation system with inference rules:

enter image description here

There are two inference rules used in this Fitch-style proof checker after an assumption is made on line 1. One is called reiteration (R) where a statement that one already accepts may be used again. The second rule is conditional introduction (→I). This rule summarizes the subproof on lines 2 and 3: if I assume $P$ is true then any line correctly derived after that can be used as a conclusion. It is also called the deduction theorem.


Here is the OP's question:

But, I'm wandering if this is proving something? Or it just only shows how to apply the law of excluded middle?

In the truth table one can see that any statement $P$ can have one of two truth values, true or false. That would be the principle of bivalence. The law of excluded middle claims that any statement $P$ is either true or $\lnot P$ is true which would be useful in a derivation.

The derivation doesn't use the law of excluded middle as an inference rule to derive $P \Rightarrow P$. Admittedly, this derived conditional doesn't show much. However, one can look at $P$ as the common ground for two people engaging in an argument. After arguing they may be no further than $P$. Then all they can claim is $P \Rightarrow P$.


Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

Michael Rieppel Truth Table Generator https://mrieppel.net/prog/truthtable.html

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