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In the book, the author uses ($\to$) to denote material conditional, which he uses when formalizing English sentences, and uses ($\Rightarrow$) to denote implication in various contexts.

The following are three examples of how we uses the notation:

Example 1: On page 73 we can see how he employs the notation when formalizing mathematical statements:

In calculus, we learn the meaning of “the function $f$ converges to $L$ as $x$ approaches $a$”: $$ \forall \varepsilon(\varepsilon>0 > \rightarrow \exists \delta(\delta>0 \wedge \forall x(|x-a|<\delta > \rightarrow|f x-L|<\varepsilon))) $$ This is, apart from notational matters, a formula of the sort we are interested in, using a predicate symbol for ordering, function symbols for $f$, subtraction, and absolute values, and constant symbols for $0$, $a$, and $L$.

Example 2: On page 178 we have an example of how he writes mathematical English, by mixing English and logical symbols.

PROOF OF EQUIVALENCE WITH THE ORDINARY DEFINITION. First suppose that $F$ converges at $a$ to $b$ in the ordinary sense. That is, for any $ε > 0$ there is a $δ > 0$ such that $$0 \neq |x − a| < δ ⇒ |b − F(x)| < ε \quad \textrm{for any} \; x.$$

Example 3: On page 131 the author uses ($\Rightarrow$) to connect two meta-statements about first order languages

In this section we establish two major theorems: the soundness of our deductive calculus $(\Gamma \vdash \varphi \Rightarrow \Gamma \vDash \varphi)$ ...

So regarding symbol (⇒), on page 1 he writes:

A sentence “If . . . , then . . .” will sometimes be abbreviated “. . . ⇒ . . . .” We also have “⇐” for the converse implication (for the peculiar way the word “implication” is used in mathematics). For “if and only if” we use the shorter “iff” (this has become part of the mathematical language) and the symbol “⇔”.

Question: What does the author mean by:

for the peculiar way the word “implication” is used in mathematics

Additionally, are Examples 2 and 3 instances of the notation suggested in this answer? Where example 2 is mathematical implication, and example 3 is some meta implication, i.e., assertions that the conditional is True?

Mauro ALLEGRANZA
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fire-bee
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  • I understand that there is the object level and the meta-level. I'm asking about the author's remark and additionally if when we write the meta-implications we are asserting them – fire-bee Jul 04 '23 at 18:25
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    $\Rightarrow$ is a metalogical symbol – RyRy the Fly Guy Jul 04 '23 at 20:32
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    A mathematical text (and a math log textbook is a math text) uses a jargon made of natural language + symbols. How we read the statement: "for any ε>0 there is a δ>0 such that $0≠|x−a|<δ ⇒ |b−F(x)|<ε$ for any x" ? We can read it in two ways: either as (i) "for any ε>0 there is a δ>0 such that: if the hypotheses $0≠|x−a|<δ$ holds, then...", and thus in a formal setting the symbol means $\to$ [see page 73], or as (ii) "for any ε>0 there is a δ>0 such that: from the hypotheses $0≠|x−a|<δ$ it follows that...", and thus in a formal setting the symbol means $\vDash$. – Mauro ALLEGRANZA Jul 05 '23 at 06:30
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    The symbol is deliberately ambiguous, but there is no harm. – Mauro ALLEGRANZA Jul 05 '23 at 06:31
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    The second example above [page 178] is an example from a mathematical text, using math jargon, and not an example of formal language of predicate logic. The third example [page 131] is again an example of math text: it is a (meta-logic) statement about two basic formal relations: $\vdash$, i.e. derivability in the proof system, and $\vDash$, i.e. logical consequence. Both are rigorously defined mathematical concepts formalizing (from two different but linked points of view) the intuitive concept: "it follows from". – Mauro ALLEGRANZA Jul 05 '23 at 06:40
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    As you can see, the mathematical jargon: "($\Gamma \vdash \varphi \Rightarrow \Gamma \vDash \varphi)$" is expressed exactly as "if $\Gamma \vdash \varphi$, then $\Gamma \vDash \varphi$" – Mauro ALLEGRANZA Jul 05 '23 at 06:41
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    In conclusion, if we want to use a symbol for the intuitive concept of consequence ("implication"), as per the answer linked above, we have to take care not to consider it a "formal definition" in addition to the already existing ones: $\vDash$ and $\vdash$, because there is no "third" relation in addition to them. – Mauro ALLEGRANZA Jul 05 '23 at 06:45
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    Additionally, are Examples 2 and 3 instances of the notation suggested in this answer? $\quad$ Eg 2 yes; Eg 3 no, because no specific interpretation is being considered, instead think of the ⇒ here as an informal meta-metalogical symbol (Amen to Mauro's second comment and Peter's linked answer, which I upvoted years ago and which complements the answer that you linked). – ryang Jul 05 '23 at 06:49
  • @MauroALLEGRANZA Thanks for your answers. Just one more thing, if one author uses ⇒ instead of natural language, how can one differentiate between the "if ... then..." and "from ... it follows that ..."? – fire-bee Jul 05 '23 at 13:39
  • @fire-bee - it depends... Consider a rigorous math textbook that do not deal with math logic: Ethan Bloch, The real numbers and real analysis (Springer, 2011), Lemma 1.4.5, page 21: "Let $x,y,z \in \mathbb Z$. If $x+z = y+z$, then $x = y$ (Cancellation Law for Addition)." It's fine. – Mauro ALLEGRANZA Jul 05 '23 at 13:52
  • Does this answer your question? What's the difference between $P \to Q$ and $P \implies Q$? – Another User Jul 12 '23 at 23:22

1 Answers1

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On Enderton's passing remark. Consider these entries for "implication":

The Cambridge Dictionary: "a suggestion of something that is made without saying it directly".

The Oxford Learner's Dictionary: "a possible effect or result of an action or a decision; something that is suggested or indirectly stated (= something that is implied)".

Merriam Websters, starts: "something implied: such as (a) a possible significance (b) suggestion; a close connection (especially, an incriminating involvement) ..."

Eventually Websters does get round to the special maths/logic usage: "logical relation between two propositions that fails to hold only if the first is true and the second is false". But the dictionaries do rather suggest that this isn't the principal or most common meaning of "implication". I guess when Enderton calls it "peculiar" he doesn't mean "odd" or "strange", so much as "special-to-maths-and-logic".

And for the more important point about $\to$ vs $\Rightarrow$, see also this answer of mine:

Peter Smith
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  • Thanks for your answer (and book). The "special-to-maths-and-logic" vs common use perspective makes sense. – fire-bee Jul 05 '23 at 13:27
  • In your linked answer, when you write "As for '⇒', this -- like the informal use of 'implies' -- seems to be used (especially by non-logicians), in different contexts for any of these three.", which of the three would you say that are being used in my examples 2 and 3? – fire-bee Jul 05 '23 at 13:33