Biconditional Introduction/Proof Rule

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Proof Rule

The Rule of Biconditional Introduction is a valid deduction sequent in propositional logic.

As a proof rule it is expressed in the form:

If we can conclude both $\phi \implies \psi$ and $\psi \implies \phi$, then we may infer $\phi \iff \psi$.


It can be written:

$\displaystyle { {\phi \implies \psi \qquad \psi \implies \phi} \over \phi \iff \psi} \iff_i$


Thus it is used to introduce the biconditional operator into a sequent.


Tableau Form

Let $\phi \implies \psi$ and $\psi \implies \phi$ be two conditional statements involving the two propositional formulas $\phi$ and $\psi$ in a tableau proof.

Biconditional Introduction is invoked for $\phi$ and $\psi$ in the following manner:

Pool:    The pooled assumptions of each of $\phi \implies \psi$ and $\psi \implies \phi$             
Formula:    $\phi \iff \psi$             
Description:    Biconditional Introduction             
Depends on:    Both of the lines containing $\phi \implies \psi$ and $\psi \implies \phi$             
Abbreviation:    $\mathrm {BI}$ or $\iff \mathcal I$             


Also known as

Some sources refer to the Biconditional Introduction as the rule of Conditional-Biconditional.


Technical Note

When invoking Biconditional Introduction in a tableau proof, use the {{BiconditionalIntro}} template:

{{BiconditionalIntro|line|pool|statement|first|second}}

or:

{{BiconditionalIntro|line|pool|statement|first|second|comment}}

where:

line is the number of the line on the tableau proof where Biconditional Introduction is to be invoked
pool is the combined pool of assumptions of each of the constituents (comma-separated list)
statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
first is the first of the two lines of the tableau proof upon which this line directly depends
second is the second of the two lines of the tableau proof upon which this line directly depends
comment is the (optional) comment that is to be displayed in the Notes column.


Sources