Definition:Logical Implication
Definition
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:
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:
and:
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:
- $\paren {A \implies B} \implies C$
or:
- $A \implies \paren {B \implies C}$
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.
Sources
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 1$: The Logic of Statements $(1)$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.7$: Tableaus
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logic
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logic