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In top voted answer by Bijesh K.S. Does author means material implication or logical implication?

That is, does

Here, $B \implies A$ is material implication or logical implication?

Here, are we talking about A→B where '→" is a connective? Or, we are talking about A⟹B where '⟹' is a logical implication?

Ubi.B
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  • If you want to delve into details, I think this should perhaps be $\vDash$ – Hagen von Eitzen Jul 01 '21 at 20:37
  • @HagenvonEitzen many authors treats material implication with symbol $A \rightarrow B$ and other treats as $A \implies B$. I am still in learning phase and I got confused about what it means in the answer. So, requires clarification. From your comment it looks that author in the answer intends to suggest it is material implication. – Ubi.B Jul 01 '21 at 20:44
  • I didn't realize there was a distinction between "material implication" and "logical implication", but apparently in some circles there is. Also, that older answer wasn't clear to me until I looked at the question (parentheses would have helped). Anyway, my take on the answer you cited is that $\iff$ is being used in a metalanguage sense as the ordinary mathematical usage of the phrase "if and only if" in ordinary math proofs, and $\implies$ is being used as the formal logic language connective for implication. – Dave L. Renfro Jul 01 '21 at 21:11
  • Actually, in looking more at the question and thinking of the context, it appears that neither symbol is being used in a formal logic sense, but rather simply as shorthand for "if and only if" and "implies" as understood in ordinary mathematical discourse (which assumes a material implication interpretation). But honestly, the issues being raised here are way below the radar for what is being discussed in that earlier question/answer. It's like trying to diagram a twitter sentence . . . – Dave L. Renfro Jul 01 '21 at 21:15
  • Maybe both. When the author says: "if a number is divisible by 2, then it is divisible by 6" we can formalize it in propositional calculus with material implication: "(a number is divisible by 2) $\to$ (it is divisible by 6)" or we may express a logical consequence relation: "$x$ is divisible by 2 $\vDash$ $x$ is divisible by 6". In the second case, sometimes we use $\Rightarrow$ as a shortand for implication. – Mauro ALLEGRANZA Jul 02 '21 at 06:03
  • @MauroALLEGRANZA how can it be both? If it is material implication then there are total of four cases to be considered. But in formal logic we don't pay much attention to proposition (or proposition variable which carry those propositions). We just pay attention to formula presented. If it is material implication then antecedent need not be related to consequent e.g. "If I am human, then pigs can fly." – Ubi.B Jul 02 '21 at 11:01
  • Also, how would you explain F→T (material implication case) in that example? That is Sufficient condition is false and Necessary condition is true? Resulting in event to occur successfully with sufficient condition being false. It is counter intuitive! – Ubi.B Jul 02 '21 at 11:15

1 Answers1

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Extra-large comment

With reference to the linked post, let assume for the current discussion the theory of arithmetic with the well-known Peano axioms.

We can formalize the theory in predicate logic with some primitive symbols (expressing the corresponding notions): $0, S, +, \times$, where $s$ is the successor function.

With them we can define new relations, like $<$ and $\le$.

Finally, we introduce the following definition:

$\text {Div}(n,m) =_{\text{def}} \exists k (0 < k \land n=k \times m)$,

expressing the fact that "$m$ Divides $n$".

Now, using the $\mathsf {PA}$ axioms and already proved theorems, we can prove the following result: if $\text {Div}(n,6)$, then $\text {Div}(n,2)$, for a number $n$ whatever.

The converse obviously does not hold, because e.g. $\text {Div}(4,2)$ holds but $\text {Div}(4,6)$ does not.

In a more formal way, we have proved that "if $\text {Div}(n,2)$, then $\text {Div}(n,6)$" is a theorem of arithmetic, i.e. that:

$\mathsf {PA} \vdash \text {Div}(n,2) \to \text {Div}(n,6)$, where the symbol $\vdash$ expresses derivability.

Due to Completeness of predicate logic this is equivalent to: $\mathsf {PA} \vDash \text {Div}(n,2) \to \text {Div}(n,6)$, where the symbol $\vDash$ expresses consequence.

Due to this fact, we can read the above result in an ambiguous way using "implies":

in arithmetic, divisibility by $2$ implies divisibility by $6$,

and if we want to use a symbol for "implies", some author uses $\implies$.

But the use of the symbol does not add anything to the result.

See e.g.: Ethan Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics (Springer, 2nd ed., 2011), page 87:

In the final write-up of a proof, do not use logical symbols, such as $\land, \lor, \exists, \forall$ and $\Rightarrow$, as abbreviations for words. Unless you are writing about logic, where logical symbols are necessary, the use of logical symbols makes proofs harder for others to read. Of course, you may use any symbols you want in your scratch work.

Bad: $\forall$ distinct real numbers $x \land y$, if $x < y \Rightarrow \exists$ rational $q$ such that $x<q<y$.

Good: For all distinct real numbers $x$ and $y$ if $x < y$ then there exists a rational number $q$ such that $x < q < y$.

Mauro ALLEGRANZA
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