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My question is :

Can any one exhibit a mathematical formula in which a conditional has to be STRICTLY understood as a MATERIAL one, so that the formula would become false if the material conditional were (mistakenly) replaced by the logical implication symbol ( ==> ). (I would be satisfied even though the conditional were not the main operator in that formula.)

Explanation.

In logic, strictly speaking, material implication ( symbol : ' --> ') has to be carefully distinguished from logical implication ( symbol ' ==> ').

I have noticed that in mathematical books, the distinction is not emphasized, as if, in that field, all implication was logical implication. Is it actually the case?

Reference: On this distinction and on the symbols I use , Seymour Lipschutz, Schaum's Outline of Set Theory , ch. 14 " Algebra Of Propositions". ( at Archive.org).

Example to illustrate the difference between material and logical implication. ( The example is taken from (basic) set theory but I'm expecting a formula from a more classical/traditional part of mathematics).

Suppose I define a set A in the following way:

A = { x | x is a mathematician --> x is a musician }

Since the conditional is a material one, the set is simply the collection of people who ( contingently) happen not to be both (1) mathematician and (2) non-musician.

The formula ( and the set) would be very different if I substituted ' ==> ' for ' --> '.

In my mind ( with ' ==> ' for logical implication) the set :

  B = { x| x is a mathematician ==> x is a musician } 

is the set of persons such that it is or would have been logically impossible for them to be mathematician without being musician. Depending on one's opinion concerning the relationship between mathematics and fine arts, one will probably tend to answer either that B = U (necessarily, a mathematician is a musician) or that B = Ø.

Remark. - I think substituting ' --> ' for ' ==> 'cannot lead to important problems, since, if A logically implies B, A should also materially impliy B ("A==> B" meaning, that ( A--> B) is true in all possible cases, all possible "interpretations"). The question I'm asking is the reverse question: is it always correct to substitue ' ==> ' for ' --> ' in mathematics? Or, in other words, is it correct to use always " ==>" in mathematics?

Remark (2) - My question is not on symbols.

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  • $\begingroup$ Logical implication usually is material implication. $\endgroup$ – PyRulez Mar 21 at 17:24
  • $\begingroup$ Here is a helpful reference on typesetting mathematical symbols. $\endgroup$ – Théophile Mar 21 at 17:35
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    $\begingroup$ In mainstream mathematics, there is only one type of implication. I've never heard of the distinction you mention, but it sounds like the one type is material implication, because if it is true that Paris is the capital of France, then the implication is true. $\endgroup$ – Matt Samuel Mar 21 at 17:37
  • $\begingroup$ @MattSamuel If you have never heard of it, with what authority are you saying that "there is only one type of implication"? Actually, you are wrong that the concept of material implication doesn't exist in mathematics. $\endgroup$ – user647486 Mar 21 at 17:48
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    $\begingroup$ Could shed some light on the issue math.stackexchange.com/questions/68932/… $\endgroup$ – chhro Mar 21 at 18:12
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Consider the statement

"If x is even, then x is divisible by $2$"

In 'math world' we regard this statement as true.

But this is not a logical truth. That is, logically one is allowed to interpret 'even', 'divisible by' and '$2$' in a way that would make the statement false.

So, the statement is a mathematical truth, but not a logical truth. More to the point: the 'if' part does not logically imply the 'then part. Indeed, if we were to symbolize it, we should be using the material implication, and not the logical implication.

Of course, if we are given the (normal!) definitions of 'even', 'divisible by' and '$2$', then we can logically infer the truth of the statement above as a whole. That is, the statement as a whole is logically implied by the relevant definitions.

Also, if we fill in a specific value for $x$, say $4$, then the statement becomes:

"If $4$ is even, then $4$ is divisible by $2$"

And now, given the standard definitions/axioms (let's refer to that as a set of statements $A$), we have that $A$ together with "$4$ is even" logically implies that "$4$ is divisible by $2$" ... but we still don't have that "$4$ is even" by itself logically implies that "$4$ is divisible by $2$"

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  • $\begingroup$ Very clear and precise answer, thanks! Contrary to what I thought mathematicians hardly ever use ' ==> ' as a symbol for logical implication! $\endgroup$ – Ray LittleRock Mar 21 at 18:38
  • $\begingroup$ @RayLittleRock Right. Or to be exact: when a mathematician makes an 'if ... then .. ' claim, it is rarely a claim of logical implication. $\endgroup$ – Bram28 Mar 21 at 18:44
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The symbols you want are $\to$ (\to) for material implication and $\implies$ (\implies) for logical implication. Insofar as mainstream mathematics distinguishes them, $p\implies q$ means that $p\to q$ is (a) true in all models of a theory of interest (however, in that context we'd usually write $\models$ (\models) instead of $\implies$ to make it clear) or (b) a tautology. And in modal logic, we can rewrite $p\implies q$ as $\Box(p\to q)$ (note the use of \Box). But in practice, $\implies$ is often used in proofs to indicate an inference from what was already known.

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  • $\begingroup$ Thanks for the clarification concerning symbols and underlying concepts. How would you symbollically translate statement such that : " if 4² is even then 4 is even ". With the (\Box) or without it? $\endgroup$ – Ray LittleRock Mar 21 at 22:26
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    $\begingroup$ @RayLittleRock Always make your statement only as strong as you intend. The statement $2|4^2\to 2|4$ will be implied whatever symbols you use, as all alternatives to $\to$ are at least as strong. If you want to make the further statement that all models yield the above, change $\to$ to $\models$; if you deem it "necessary" (whatever you take that to mean metaphysically), by all means wrap the statement in $\Box()$; if you dare claim definitions have sufficed to make it a tautology, use $\implies$. $\endgroup$ – J.G. Mar 21 at 22:29
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In mathematics based on classical logic, there appears to be no difference between material and logical implication.

Consider, for example, the implication: If it is raining ($R$), then it is cloudy ($C$).

$$R \implies C$$

This implication does not mean that rain causes cloudiness, or that cloudiness causes rain. It means only that, at the moment, it is not both raining and not cloudy.

$$\neg [R \land \neg C]$$

This is often used in one form or another as The Definition of $\implies$ in introductory textbooks, but it can also be derived from other widely accepted properties of implication, conjunction and negation in classical logic:

  • Introducing and Eliminating $\land$
  • Eliminating $\neg\neg$
  • Conditional proof
  • Proof by contradiction
  • Detachment (Modus Ponens)

See my formal proof.

So, in classical logic anyway, the above "definition" would seem to apply to every implication.

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  • $\begingroup$ Neat proof. Thanks! How do you understand this equivalence of ( A-->B) and ~ (A& ~B) $\endgroup$ – Ray LittleRock Mar 22 at 17:37
  • $\begingroup$ @RayLittleRock I see it as applying in mathematics and in natural language as well, provided you are talking about a pair of unambiguous true-or-false propositions at a given moment in time. In mathematics, of course, there is no notion of the passage of time, no future or past, essentially only the present -- that which IS true. $\endgroup$ – Dan Christensen Mar 22 at 17:58

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