# Encyclopedia of Systems Biology

2013 Edition
| Editors: Werner Dubitzky, Olaf Wolkenhauer, Kwang-Hyun Cho, Hiroki Yokota

# Proof, Logic

• C. Maria Keet
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-9863-7_735

## Definition

A proof is a convincing argument expressed in the language of logic and mathematics.

### Example

A sample computation to prove automatically if the propositional formula ((p ∨ (q ∧ r)) →((p ∨ q) ∧ (p ∨ r)) is valid or not is included in Fig.  1 and Table  1, using tableau reasoning ( ). With respect to systems biology, one could substitute p, q, and r for, e.g., “produces glucose,” “produces carbon dioxide,” and “releases water,” respectively. The tableau method is a decision procedure that checks the existence of a model (i.e., that it can be instantiated). It exhaustively looks at all the possibilities, so that it can eventually prove that no model could be found for unsatisfiable formulas (if it is satisfiable, we have found a counterexample). This is done by decomposing the formula in top-down fashion after it has been translated into negation normal form (i.e., all the negations have been pushed inside), which can be achieved using equivalences. Further, if a model...
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## References

1. Hedman S (2004) A first course in logic – an introduction to model theory, proof theory, computability, and complexity. Oxford University Press, Oxford, UKGoogle Scholar
2. Tree Proof Generator. (http://www.umsu.de/logik/trees/