Definition:Logical Implication

From ProofWiki
Jump to navigation Jump to search


In a valid argument, the premises logically imply the conclusion.

If the truth of one statement $p$ can be shown in an argument directly to cause the meaning of another statement $q$ to be true, then $q$ follows from $p$ by logical implication.

We may say:

$p$, therefore $q$ and write $p \vdash q$.
$q$, because $p$ and write $q \dashv p$.

Also known as

The term logical consequence can often be seen as a synonym for logical implication.

In symbolic logic, the concept of logical consequence occurs in the form of semantic consequence and provable consequence.

In the context of proofs of a conventional mathematical nature on $\mathsf{Pr} \infty \mathsf{fWiki}$, the notation:

$p \leadsto q$

is preferred, where $\leadsto$ can be read as leads to.

Semantic Consequence

Let $\mathscr M$ be a formal semantics for a formal language $\LL$.

Let $\FF$ be a collection of WFFs of $\LL$.

Let $\map {\mathscr M} \FF$ be the formal semantics obtained from $\mathscr M$ by retaining only the structures of $\mathscr M$ that are models of $\FF$.

Let $\phi$ be a tautology for $\map {\mathscr M} \FF$.

Then $\phi$ is called a semantic consequence of $\FF$, and this is denoted as:

$\FF \models_{\mathscr M} \phi$

That is to say, $\phi$ is a semantic consequence of $\FF$ if and only if, for each $\mathscr M$-structure $\MM$:

$\MM \models_{\mathscr M} \FF$ implies $\MM \models_{\mathscr M} \phi$

where $\models_{\mathscr M}$ is the models relation.

Note in particular that for $\FF = \O$, the notation agrees with the notation for a $\mathscr M$-tautology:

$\models_{\mathscr M} \phi$

The concept naturally generalises to sets of formulas $\GG$ on the right hand side:

$\FF \models_{\mathscr M} \GG$

if and only if $\FF \models_{\mathscr M} \phi$ for every $\phi \in \GG$.

Provable Consequence

Let $\mathscr P$ be a proof system for a formal language $\LL$.

Let $\FF$ be a collection of WFFs of $\LL$.

Denote with $\map {\mathscr P} \FF$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\FF$ as axioms.

Let $\phi$ be a theorem of $\map {\mathscr P} \FF$.

Then $\phi$ is called a provable consequence of $\FF$, and this is denoted as:

$\FF \vdash_{\mathscr P} \phi$

Note in particular that for $\FF = \O$, this notation agrees with the notation for a $\mathscr P$-theorem:

$\vdash_{\mathscr P} \phi$

Distinction between Logical Implication and Conditional

It is important to understand the difference between:

$A \implies B$: If we assume the truth of $A$, we can deduce the truth of $B$


$A \leadsto B$: $A$ is asserted to be true, therefore it can be deduced that $B$ is true

When $A$ is indeed true, the distinction is less important than when the truth of $A$ is in question, but it is a bad idea to ignore it.

Compare the following:

\(\text {(1)}: \quad\) \(\ds x > y\) \(\implies\) \(\ds \paren {x^2 > x y \text { and } x y > y ^2}\)
\(\ds \) \(\implies\) \(\ds x^2 > y^2\)

\(\text {(2)}: \quad\) \(\ds x\) \(>\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds x^2\) \(>\) \(\ds x y\)
\(\, \ds \text { and } \, \) \(\ds x y\) \(>\) \(\ds y^2\)
\(\ds \leadsto \ \ \) \(\ds x^2\) \(>\) \(\ds y^2\)

We note that $(1)$ is a conditional statement of the form:

$A \implies B \implies C$

This can mean either:

$\paren {A \implies B} \implies C$


$A \implies \paren {B \implies C}$

instead of what is actually meant:

$\paren {A \implies B} \text { and } \paren {B \implies C}$

Hence on $\mathsf{Pr} \infty \mathsf{fWiki}$ we commit to using the form $A \leadsto B$ rigorously in our proofs.

The same applies to $\iff$ and $\leadstoandfrom$ for the same reasons.

Also see

  • Results about logical implication can be found here.