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 invokedpool
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$ ... $
delimitersfirst
is the first of the two lines of the tableau proof upon which this line directly dependssecond
is the second of the two lines of the tableau proof upon which this line directly dependscomment
is the (optional) comment that is to be displayed in the Notes column.
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction: Exercises $1.6: \ 7$