Multiple-conclusion logic

A multiple-conclusion logic is one in which logical consequence is a relation, ${\displaystyle \vdash }$, between two sets of sentences (or propositions). ${\displaystyle \Gamma \vdash \Delta }$ is typically interpreted as meaning that whenever each element of ${\displaystyle \Gamma }$ is true, some element of ${\displaystyle \Delta }$ is true; and whenever each element of ${\displaystyle \Delta }$ is false, some element of ${\displaystyle \Gamma }$ is false. Such a reading is related to Gerhard Gentzen's interpretation of the multiple-succedent sequent calculus LK, though Gentzen interprets his sequents ${\displaystyle \Gamma \vdash \Delta }$ as formulae ${\displaystyle (\bigwedge \Gamma )\supset (\bigvee \Delta )}$.^[1]

This form of logic was developed in the 1970s by D. J. Shoesmith and Timothy Smiley^[2] but has not been widely adopted.

Some logicians (for example, Greg Restall^[3]) favor a multiple-conclusion consequence relation over the more traditional single-conclusion relation on the grounds that the latter is asymmetric (in the informal, non-mathematical sense) and favors truth over falsity (or assertion over denial).

See also

• Sequent calculus

References

1. G. Gentzen, 'Investigations into logical deduction'. American Philosophical Quarterly 1(4):288 - 306, 1964. [Translation of 'Untersuchungen über das logische Schliessen', Mathematische Zeitschrift 39:167 - 221, 1935]
2. D. J. Shoesmith and T. J. Smiley, Multiple Conclusion Logic, Cambridge University Press, 1978
3. G. Restall, 'Multiple conclusions', in P. Hájek et al. (ed.), Logic, Methodology, and Philosophy of Science, College Publications, 2005. Also available at https://consequently.org/papers/multipleconclusions.pdf