Inference in propositional logic
Biconditional introductionType | Rule of inference |
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Field | Propositional calculus |
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Statement | If is true, and if is true, then one may infer that is true. |
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Symbolic statement | ![{\displaystyle {\frac {P\to Q,Q\to P}{\therefore P\leftrightarrow Q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b69370e2f8eda3ff6803418125c3dc722cb47a) |
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Transformation rules |
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Propositional calculus |
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Rules of inference |
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- Implication introduction / elimination (modus ponens)
- Biconditional introduction / elimination
- Conjunction introduction / elimination
- Disjunction introduction / elimination
- Disjunctive / hypothetical syllogism
- Constructive / destructive dilemma
- Absorption / modus tollens / modus ponendo tollens
- Negation introduction
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Rules of replacement |
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- Associativity
- Commutativity
- Distributivity
- Double negation
- De Morgan's laws
- Transposition
- Material implication
- Exportation
- Tautology
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Predicate logic |
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Rules of inference |
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- Universal generalization / instantiation
- Existential generalization / instantiation
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In propositional logic, biconditional introduction[1][2][3] is a valid rule of inference. It allows for one to infer a biconditional from two conditional statements. The rule makes it possible to introduce a biconditional statement into a logical proof. If
is true, and if
is true, then one may infer that
is true. For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive". Biconditional introduction is the converse of biconditional elimination. The rule can be stated formally as:
![{\displaystyle {\frac {P\to Q,Q\to P}{\therefore P\leftrightarrow Q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b69370e2f8eda3ff6803418125c3dc722cb47a)
where the rule is that wherever instances of "
" and "
" appear on lines of a proof, "
" can validly be placed on a subsequent line.
The biconditional introduction rule may be written in sequent notation:
![{\displaystyle (P\to Q),(Q\to P)\vdash (P\leftrightarrow Q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04a8bb1ec19705969d2b71f91d64a31daa6ffe29)
where
is a metalogical symbol meaning that
is a syntactic consequence when
and
are both in a proof;
or as the statement of a truth-functional tautology or theorem of propositional logic:
![{\displaystyle ((P\to Q)\land (Q\to P))\to (P\leftrightarrow Q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4dd69dbc58367e359eb9735158b7b1e7daac6c2)
where
, and
are propositions expressed in some formal system.
References
- ^ Hurley
- ^ Moore and Parker
- ^ Copi and Cohen