Multiple-conclusion logic

A multiple-conclusion logic is one in which <a href="/facts/Logical_consequence/AyDXp4dn">logical consequence</a> is a <a href="/facts/Relation_(mathematics)/HjblDaMy">relation</a>, 
 
 
 
 ⊢
 
 
 {\displaystyle \vdash }
 
, between two <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> of <a href="/facts/Sentence_(mathematical_logic)/lnsLzvTx">sentences</a> (or <a href="/facts/Proposition_(mathematics)/f2Pe1FMD">propositions</a>). 
 
 
 
 Γ
 ⊢
 Δ
 
 
 {\displaystyle \Gamma \vdash \Delta }
 
 is typically interpreted as meaning that whenever each element of 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 is true, some element of 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 is true; and whenever each element of 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 is false, some element of 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 is false.
This form of logic was developed in the 1970s by D. J. Shoesmith and <a href="/facts/Timothy_Smiley/lc7ElP4r">Timothy Smiley</a> but has not been widely adopted.
Some <a href="/facts/Logician/8JnpJLtt">logicians</a> favor a multiple-conclusion consequence relation over the more traditional single-conclusion relation on the grounds that the latter is <a href="/facts/Asymmetry/U009nepR">asymmetric</a> (in the informal, non-mathematical sense) and favors truth over falsity (or assertion over denial).

Multiple-conclusion logic open-in-new

Multiple-conclusion logic