Asserting the truth of an implication vs material conditional

Question

For context, consider the following notation proposed by user ryang in their answers 1, 2, 3, 4, 5, 6, from which I quote below.

• material conditional $\left(\to\right)$
• implication$\left(\Rightarrow\right):$
  $\quad\to$ is true (perhaps in an axiom system) in the current interpretation
• logical implication / (semantic) logical entailment $\left(\models\right):$
  $\quad\to$ is true regardless of interpretation
• derivability / syntactic logical entailment $\left(\vdash\right):$
  $\quad\to$ can be proven true regardless of interpretation

(⊢, ⊨, ⇒ are metalanguage symbols, while → is in the object language.)

P.S. Symbolic logic is an area rife with conflicting notation, terminology and even notions;

P.P.S. To be clear: although I distinguish analytical and synthetic implication ⇒ from logical entailment ⊨, in practice I do frequently use ⇒ (which is better recognised) even when I specifically mean the latter.

In the given formulation $$|x-c|<\delta\;\; \textbf{implies} \;\;|f(x)-f(c)|<\varepsilon,$$ “implies” is not the material conditional $\large\rightarrow\normalsize$ per se, but rather mathematical implication $\large\Rightarrow\normalsize;$ it analytically (from mathematical axioms and a given context) asserts that its right side can be derived from its left.

I believe that the above notation is being used in the following way.

We can apply it to propositional logic. For example, letting $P$ be "Alice is in Europe" and $Q$ be "Bob is in the USA", $$P \to Q$$ is a compound proposition. Now, one could assert $(P \to Q)$ to be True, which in symbols is $$P \Rightarrow Q.$$ This means that for the truth table of $(P \to Q),$ we can discard the row where $P$ is True and $Q$ is False.

If we now move to predicate logic in the context of mathematics (in $\mathbb{R}$), we can for example let $P(x)$ and $Q(x)$ be $x=x$ and $x^2 \geq x$ respectively, and write $$\forall x \ P(x) \to \forall x \ Q(x).$$ This compound sentence is a material conditional of two sentences that have truth values depending on the interpretation of the symbols. Following the same idea as before it is possible make the assertion $$\forall x \ P(x) \Rightarrow \forall x \ Q(x),$$ where we read ($\Rightarrow$) as mathematical implication by considering the standard interpretation of the symbols "$=, ^2, \geq$" and axioms of mathematics. A more common sentence in mathematics is the universal conditional statement. For example, letting $P(x)$ and $Q(x)$ be "$x \geq 2$" and "$x^2 \geq 4$" respectively, we can write $$ \forall x (P(x) \to Q(x)). $$ In mathematics it is common to make use of implicit quantification, so the above sentence is often abbreviated to $$P(x) \to Q(x).$$ Like before we can assert that this universal conditional statement is True, and write $$P(x) \Rightarrow Q(x),$$ where we read ($\Rightarrow$) as universal mathematical implication by considering the standard interpretation of the symbols "$2, 4, ^2, \geq$" and axioms of mathematics, meaning that if $P(x)$ is true for some value of $x$, then $Q(x)$ is true for the same value of $x$. This last example gives me trouble, because it seems that we are mixing meta-logical symbols and logic ones. If we think of the universal quantifier as infinitely many ($\land$), I would parse it as
 "$P(x_1) \to Q(x_1)$" is True $\land$ "$P(x_2) \to Q(x_2)$" is True $\land$ $\dots.$

Finally, here's my question: how can we read/interpret

$$\forall\epsilon{>}0 \; \exists\delta{>}0 \; \forall x{\in}D \;\big(0<|x - c| < \delta \implies |f(x) - L| < \epsilon \big),$$

in this answer, considering that it is neither an implication between closed formulas (the quantifiers are outside), nor an implicit universal implication (the quantifiers are explicitly there)?

Answer 1

1. Your summary is mostly fine. However, in the object language, implicit quantification is not a thing, and ∀x (Px→Qx) is never shortened as Px→Qx; only informally—for example, in Mathematics—is Px⟹Qx understood to mean ∀x (Px⟹Qx). Also, most first-order logics have = (equality/identity) as a logic symbol, in which case it is not considered an arithmetical symbol.

2. Your question actually has only a tenuous connection to its long preamble, and can be distilled to this:

   • if ⟹ is understood as mathematical implication, then how to read the statement $$∀x \;(x>2 ⟹ x>1);$$ in particular, is the unquantified string x>2 ⟹ x>1 technically meaningful?

   Firstly, observe that your question is not really about mathematical implication, as it equally applies to the the non-mathematical statement $$∀x \;(\text{$x$ is in June ⟹ $x$ is not Christmas Day}).$$

   In any case, these two statements are parsed in the same manner: under the relevant interpretation and axioms (for example, Christmas is in December), the open formulae $$x>2 → x>1$$ and $$\text{$x$ is in June → $x$ is not Christmas Day}$$ are true whichever the object x; in particular, notice that the string x>2 ⟹ x>1 is not being offered in isolation and its truth in the context of the entire statement is always with respect to some object $x$ as the quantification is being considered.

In practice, → and ⟹ are interchangeble (in fact, they are also merely different choices of symbols to mean material implication), and it is always clear anyway that statements like the above are assertions of truth in a given context, rather than any sort of language-independent logical entailment.