Is the logic implication a tautology?

Question

I had a discussion with a person who says that logic implication is a tautology.

Reviewing this book, specifically on page $12$ ($19$ of the document), I see that there is a difference between what is a conditional, $⇒$, and what is an implication. https://www.karlin.mff.cuni.cz/~krajicek/mendelson.pdf

The conditional has the next truth table:

http://www.metafysica.nl/nature/truth_table_impl.gif

And the book says on page $16$ that $p$ implies $q$ if and only if $p ⇒ q$ is a tautology. But, does this mean that an implication is a tautology? Because if it does, then we cannot say something does not imply another thing, I think.

There is a difference between implication in the object language and implication in the metalanguage. In the object language of propositional logic, implication is a sentential connective: the material conditional. In the metalanguage we can speak of one sentence of the object language implying or entailing another. These are different notions. They are related by the deduction theorem. See http://math.stackexchange.com/questions/68932/whats-the-difference-between-material-implication-and-logical-implication — symplectomorphic, Oct 27 '16 at 21:59
Many post available on this subject; see e.g. implies-vs-entails-vs-provable as well as whats-the-difference-between-material-implication-and-logical-implication. — Mauro ALLEGRANZA, Oct 28 '16 at 06:10

score 0 · Accepted Answer · answered Oct 27 '16 at 22:00

A conditional has a condition: if A is true, then B will also be true.

An implication is a direct statement of fact: A is true, therefore B is true.

That becomes tautological because it is trivially true in all cases. A is true so B is true, so A is true so B is true.

score -1 · Answer 2 · answered Oct 28 '16 at 07:14

No, it doesn't.

A tautology consists of a certain type of statement form which satisfies the given formation rules and always holds true under all evaluations. Using bold letters, instead of capital letters, some tautologies are:

(B=>B)

and

(A=>(B=>A))

and

(($\lnot$A=>$\lnot$B)=>(($\lnot$A=>B)=>A))

On the other hand, though an implication will hold true for all valuations of the letters, it need not satisfy the formation rules for statement forms. As an example, consider the following:

A logically implies ((A=>B)=>B)

Or we might write |= as a short-hand for logically implies and write:

A |= ((A=>B)=>B)

That doesn't satisfy the formation rules for statement forms. To indicate that the premisses having the form "A" and the form "(A=>B)" logically imply B we might write

A, (A=>B) |= B

or

{A, (A=>B)} |= B

But, the component parts are different from those of tautologies. With the idea of having a set of premisses in mind we could rewrite the above as:

{(A=>B), A} |= B

or

{A, A, A, (A=>B), A, A, (A=>B), (A=>B), A} |= B

Those immediately above all represent a single logical implication, at least as a single concept.

But, a tautology like

((A=>(A=>B))=>(A=>B))

is distinct from the tautology:

(((A=>(A=>B))=>(A=>(A=>B)))=>(A=>B)).

Those consist of two different statement forms.

Is the logic implication a tautology?

2 Answers2