natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
An implication may be either an entailment or a conditional statement; these are closely related but not quite the same thing.
Entailment is a preorder on propositions within a given context in a given logic.
We say that $p$ entails $q$ syntactically, written as a sequent $p \vdash q$, if $q$ can be proved from the assumption $p$.
We say that $p$ entails $q$ semantically, written $p \vDash q$, if $q$ holds in every model in which $p$ holds.
(These relations are often equivalent, by various soundness? and completeness theorems.)
A conditional statement is the result of a binary operation on propositions within a given context in a given logic. If $p$ and $q$ are propositions in some context, then so is the conditional statement $p \to q$, at least if the logic has a notion of conditional.
Notice that $p$, $q$, and $p \to q$ are all statements in the object language (the language that we are talking about), whereas the hypothetical judgements $p \vdash q$ and $p \vDash q$ are statements in the metalanguage (the language that we are using to talk about the object language).
Depending on what logic one is using, $p \to q$ might be anything, but it's probably not fair to consider it a conditional statement unless it is related to entailment as follows:
If, in some context, $p$ entails $q$ (either syntactically or semantically), then $p \to q$ is a theorem (syntactically) or a tautology (semantically) in that context, and conversely.
In particular, this holds for classical logic and intuitionistic logic.
You can think of entailment as being an external hom (taking values in the poset of truth values) and the conditional as being an internal hom (taking values in the poset of propositions). In particular, we expect these to be related as in a closed category:
where $\top$ is an appropriate constant statement (often satisfying $p \vdash \top$, although not always, as in linear logic with $\multimap$ for $\to$ and $1$ for $\top$).
Most kinds of logic used in practice have a notion of entailment from a list of multiple premises; then we expect entailment and the conditional to be related as in a closed multicategory.
Just as we may identify the internal and external hom in Set, so we may identify the entailment and conditional of truth values. In the $n$Lab, we tend to write this as $\Rightarrow$, a symbol that is variously used by other authors in place of $\vdash$, $\vDash$, and $\rightarrow$.
Although Heyting algebras were first developed as a way to discuss intuitionistic logic, they appear in other contexts; but their characterstic feature is that they have an operation analogous to the conditional operation in logic, usually called Heyting implication and denoted $\rightarrow$ or $\Rightarrow$. If you use $\to$ and replace $\vdash$ above with the Heyting algebra's partial order $\leq$, then everything above applies.
In type theory
a conditional statement is, under propositions-as-types a function type $p \to q$ (or the bracket type thereof).
an entailment is a hypothetical judgement/sequent.
basic symbols used in logic
$\phantom{A}$symbol$\phantom{A}$ | $\phantom{A}$meaning$\phantom{A}$ |
---|---|
$\phantom{A}$$\in$ | $\phantom{A}$element relation |
$\phantom{A}$$\,:$ | $\phantom{A}$typing relation |
$\phantom{A}$$=$ | $\phantom{A}$equality |
$\phantom{A}$$\vdash$$\phantom{A}$ | $\phantom{A}$entailment / sequent$\phantom{A}$ |
$\phantom{A}$$\top$$\phantom{A}$ | $\phantom{A}$true / top$\phantom{A}$ |
$\phantom{A}$$\bot$$\phantom{A}$ | $\phantom{A}$false / bottom$\phantom{A}$ |
$\phantom{A}$$\Rightarrow$ | $\phantom{A}$implication |
$\phantom{A}$$\Leftrightarrow$ | $\phantom{A}$logical equivalence |
$\phantom{A}$$\not$ | $\phantom{A}$negation |
$\phantom{A}$$\neq$ | $\phantom{A}$negation of equality / apartness$\phantom{A}$ |
$\phantom{A}$$\notin$ | $\phantom{A}$negation of element relation $\phantom{A}$ |
$\phantom{A}$$\not \not$ | $\phantom{A}$negation of negation$\phantom{A}$ |
$\phantom{A}$$\exists$ | $\phantom{A}$existential quantification$\phantom{A}$ |
$\phantom{A}$$\forall$ | $\phantom{A}$universal quantification$\phantom{A}$ |
$\phantom{A}$$\wedge$ | $\phantom{A}$logical conjunction |
$\phantom{A}$$\vee$ | $\phantom{A}$logical disjunction |
$\phantom{A}$$\otimes$$\phantom{A}$ | $\phantom{A}$multiplicative conjunction$\phantom{A}$ |
$\phantom{A}$$\oplus$$\phantom{A}$ | $\phantom{A}$multiplicative disjunction$\phantom{A}$ |