Biconditional Elimination/Proof Rule
Theorem
The rule of biconditional elimination is a valid argument in types of logic dealing with conditionals $\implies$ and biconditionals $\iff$.
This includes classical propositional logic and predicate logic, and in particular natural deduction.
As a proof rule it is expressed in either of the two forms:
- $(1): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\phi \implies \psi$.
- $(2): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\psi \implies \phi$.
It can be written:
- $\ds {\phi \iff \psi \over \phi \implies \psi} {\iff}_{e_1} \qquad \text{or} \qquad {\phi \iff \psi \over \psi \implies \phi} {\iff}_{e_2}$
Thus it is used to introduce the biconditional operator into a sequent.
Tableau Form
Let $\phi \iff \psi$ be a well-formed formula] in a tableau proof whose main connective is the biconditional operator.
Biconditional Elimination is invoked for $\phi \iff \psi$ in either of the two forms:
- Form 1
Pool: | The pooled assumptions of $\phi \iff \psi$ | ||||||||
Formula: | $\phi \implies \psi$ | ||||||||
Description: | Biconditional Elimination | ||||||||
Depends on: | The line containing $\phi \iff \psi$ | ||||||||
Abbreviation: | $\mathrm {BE}_1$ or $\iff \EE_1$ |
- Form 2
Pool: | The pooled assumptions of $\phi \iff \psi$ | ||||||||
Formula: | $\psi \implies \phi$ | ||||||||
Description: | Biconditional Elimination | ||||||||
Depends on: | The line containing $\phi \iff \psi$ | ||||||||
Abbreviation: | $\mathrm {BE}_2$ or $\iff \EE_2$ |
Also known as
Some sources refer to the Biconditional Elimination as the rule of Biconditional-Conditional.
Also see
- This is a rule of inference of the following proof systems:
Technical Note
When invoking Biconditional Elimination in a tableau proof, use the {{BiconditionalElimination}}
template:
{{BiconditionalElimination|line|pool|statement|depend|1 or 2}}
or:
{{BiconditionalElimination|line|pool|statement|depend|1 or 2|comment}}
where:
line
is the number of the line on the tableau proof where Biconditional Elimination is to be invokedpool
is the pool of assumptions (comma-separated list)statement
is the statement of logic that is to be displayed in the Formula column, without the$ ... $
delimitersdepend
is the line of the tableau proof upon which this line directly depends1 or 2
should hold 1 forBiconditionalElimination_1
, and 2 forBiconditionalElimination_2
comment
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 $1.6: \ 7$