Are mathematical theorems entailments, or merely implications?

Question

I have a background in Electrical Engineering and not Mathematics. Recently I've been self-studying "Charles Chapman Pugh - Real Mathematical Analysis", and I was reminded of the old question that I've always had in my mind, but to my surprise, I never find (or can't see) this simple question ever being explicitly asked or addressed. I understand that many (if not all) of the theorems I've seen are in the form of implication: $P \implies Q$

What if both $P$ and $Q$ are true, but irrelevant/independent? Consider the following cases:

$P_1$: There are infinitely many prime numbers $\implies$ $Q_1$: The square root of 2 is irrational

$P_2$: The number $\pi$ is transcendental $\implies$ $Q_2$: Pythagorean theorem

I can guess these apparently irrelevant antecedents and consequents could make up for theorems if they are stated in an axiomatic system where all the required axioms are included. But generally speaking, shouldn't a theorem in the form of $P \implies Q$ be something that means we can 'derive' the truth of $Q$ from the truth of $P$ (given the auxiliary facts)? If so, does it mean that theorems are always in the form of entailments, not merely implications? I haven't seen this issue being discussed in the math books I've encountered, although it seems to me that it's an essential component of being a theorem.

Please guide me through, or where I can find the minimum proper terminology to talk about this issue.

Thanks.

If $P$ and $Q$ are true, then the implication $(P\implies Q)$ is also true. Indeed, many theorems are phrased as implications, but what we're often interested in is the truth value of $Q$. Now, we have that [$P$ and ($P\implies Q$)] is logically equivalent to $Q$, I believe this is called Modus Ponens. But this bit of logic is considered 'trivial', so we're happy to phrase many theorems as implications (e.g if a function is differentiable at a point, then it is continuous at that point) because we can then check whether in our case the hypothesis is satisfied. — peek-a-boo, Sep 16 '22 at 08:14
Example theorem: If $x \in \mathbb R$ then $0 \cdot x = 0$. The meaning is clear, isn’t it? I read your question but I still don’t understand what the confusion is. — littleO, Sep 16 '22 at 08:15
Note that (with some exceptions) interesting implications tend to share variables in their antecedents and consequents. Take e.g. " if $f$ is a bounded entire function then $f$ is a constant function": the variable $f$ is shared, and the theorem has interesting content. In contrast, while "if $f$ is a bounded entire function then the square root of 2 is irrational" is certainly a theorem, and certainly an implication, it's not an interesting implication. — Z. A. K., Sep 16 '22 at 08:15
In "real" mathematics theorems are proven in a context of a theory, made of axioms. Thus, when we prove a theorem $Q$ in the context of the theory $T$, where $T$ is a specified set of axioms, we have $T \vdash Q$. If $P$ is an "irrelevant" statement whatever, due to the fact that $Q \to (P \to Q)$ is a law of logic, we have also $T \vdash Q \to (P \to Q)$ and thus, by Modus Ponens: $T \vdash P \to Q$. — Mauro ALLEGRANZA, Sep 16 '22 at 08:18
«$\pi$ is transcendental» is a theorem without implication. So is «There are infinitely many prime numbers». — Gribouillis, Sep 16 '22 at 08:21
Thus if our theory $T$ is Euclidean geometry and in it we have proved the Pythagorean theorem, we have that $T$ proves also "if the number $\pi$ is transcendental, then the Pythagorean theorem". — Mauro ALLEGRANZA, Sep 16 '22 at 08:21
@Gribouillis $\pi$ is transcendental is implied by the definitions of $\pi$ and transcendental. If you were asking whether a Thai musical instrument existed outside the material world then you might have a different conclusion. — Henry, Sep 16 '22 at 08:38
@Henry A definition is not a proposition, it is neither true nor false. How could it imply anything? Furthermore, $\pi$ is given to us by Nature. — Gribouillis, Sep 16 '22 at 08:45
@Gribouillis $\pi$ did not "fall from the sky". It was defined by humans. Definitions are not arbitary, they must be designed in consistency to what we already have. So, a definition is of course a true statement (once it is accepted by the math community) , and of course there are facts that are implications of this. I do not understand this comment. — Peter, Sep 16 '22 at 09:15
Just because $A$ and $B$ are both true (which implies that $A\implies B$ is true as well) does not mean that there is a (reasonable) way to derive $B$ directly from $A$. It is hard to imagine that we start with the four colour theorem and end with Fermat's last theorem. Maybe this is technically possible , but it would be extremely hard to follow this implication, if it would be possible. — Peter, Sep 16 '22 at 09:19
@Peter $\pi$ fell from the sky. It is the ratio between the perimeter of the circle and the diameter. Humans didn't choose 3.1415... — Gribouillis, Sep 16 '22 at 09:35
And that we defined this as $\pi$ was our choice , of course we did not choose the value. — Peter, Sep 16 '22 at 09:43
@Peter It is a purely dialectical point of vue. It is 3.141592654... that is transcendental, not the name $\pi$. If we call it Peter, we obtain the «theorem» that Peter is transcendental. — Gribouillis, Sep 16 '22 at 09:45
Not very clear for me the later discussion... Transcendental number is a definition. "$\pi$ is transcendental" is a theorem: see Lindemann–Weierstrass theorem. — Mauro ALLEGRANZA, Sep 16 '22 at 09:59
@MauroALLEGRANZA The discussion is about whether $\pi$ was defined or "fell from the sky" — Peter, Sep 16 '22 at 10:05
I pointed out here and here that mathematical theorems/implications are not logical entailments. — ryang, Jan 29 '23 at 15:25

Jean-Armand Moroni · Accepted Answer · 2022-09-16T20:12:26.757

Your question is indeed interesting.

A theorem may or may not be in the form of $P \Rightarrow Q$. When it is $P \Rightarrow Q$, $P$ is usually not a tautology (i.e $P$ is not always true), otherwise that would be unuseful to mention $P$. When a $P$ is mentionned, it can be because:

$P$ includes one or more variables which are also used in $Q$, and $P$ is true for some values of the variable, false for others. Such as "a prime number greater than $3$ is odd": here the variable is an integer; when it is a prime number greater than $3$, $Q$ (the number is odd) is true; and in some cases where the number is not a prime number greater than $3$, $Q$ is false.
$P$ may be a tautology, but then the variability is the set of axioms you choose, and the theorem links some axioms to $Q$. Example, if one writes "the axiom of choice implies that every vector space has a basis", this is to compare situations where $P$ (the axiom is choice) is an axiom (with other axioms needed for the demonstration, that however are not mentioned), with situations where $P$ is not an axiom (but the other axioms are still there).

Obviously, when writing $P \Rightarrow Q$, mathematicians generally omit all the axioms that are used to prove $P \Rightarrow Q$, as well as the set of logical rules. This, depending upon the context, may be more or less sloppy. In some cases it may be mentioned that the theorem is proved in ZF, or ZFC, etc. The fact is, most usual math theorems can be proven in any usual axiom set - just because the usual axiom sets have been chosen so that they enable to prove all the basics.

Similarly, while there was a time when logic would be considered common sense and everybody more or less agreed upon a set of logic rules, logic itself can now be weakened. Cf. substructural logic. These different logic frameworks have notably found interest in computer science, because some CS concepts are in direct correspondance with them. Apart from logic specialists, these different logics are seldom mentioned; however some logic principles are more frequently mentioned, e.g. the law of excluded middle (which some logic theories do not accept) - this is the reason why a proof by contradiction is often considered less desirable than a proof that does not use contradiction.

Quite recently (beginning in the 70's), a systematic exploration of "what axioms are required to prove what" has been launched, called reverse mathematics. The linked Wikipedia page is interesting to read: it shows that some theorems require more axioms than others. For example in the weakest axiomatisation of arithmetics, "a continuous real function on $[0,1]$ is bounded" cannot be proven; nor the Jordan curve theorem (a closed, non-intersecting curve on the plane separates the plane in two regions, its interior and its exterior, that cannot be connected with a continuous path without intersecting the curve).

Your title uses entailment as opposed to implication. I would be embarrassed translating this into french, because both entailment and implication translate into implication. We could use causality in place of entailment, but that would be too strong. The nearest you can approach this idea of entailment, in maths, is to show that $P,Q \Rightarrow R$ (where $P$ and $Q$ represent at least all the axioms and logic rules used to prove $R$), and that $P$ alone does not $\Rightarrow R$. In this situation we could say that $Q$, in the context of $P$, entails $R$.

Dan Christensen · Answer 2 · 2022-09-17T15:23:21.633

I understand that many (if not all) of the theorems I've seen are in the form of implication: $P \implies Q$

What if both $P$ and $Q$ are true, but irrelevant/independent?

In classical logic anyway, it then seems unavoidable that the implication is true, regardless of the presence or absence of any other logical relationships between propositions $P$ and $Q$.

Using a truth table, we can verify that $(A\land B)\to (A\to B)$ is a tautology, i.e. it is always true: