# Conditional logic - how to apply a conditional with complex antecedent in tableaux?

I'm referring to the conditional logic of C+ as described Graham Priest in "An introduction to non-classical logic" chapter 5, where the strict conditional is enhanced with ceteris paribus, and a multiplicity of accessibility relations are defined for various formulae.

The simple tableaux rule for the conditional operator ">" is that:

``````** GIVEN: **
A > C, i      ** ie. "A > C" is the case at world i **
i r_A j       ** ie. world i A-accesses j **

** THEN I CAN ADD: **
A, j          **ie. A is the case at world j**
C, j          **ie. C is the case at world j**
``````

An example of an application of this rule involving a complex antecedent is:

``````** GIVEN: **
(A or B) > C
0 r_(A or B) 1     ** ie. world 0 (A or B)-accesses world 1 **

** THEN I CAN ADD: **
(A or B), 1
C, 1
``````

My question is whether I can apply the conditional using this rule in the case of a complex antecedent which is merely entailed by the formulae in the branch, rather than explicit. For example building on the example above:

``````** GIVEN: **
(A or B) > C
0 r_A 1     ** ie. world 0 A-accesses world 1 **

** QUESTION: CAN I ADD THIS, since A -> (A or B)? and 0 (A or B)-accesses 1? **
(A or B), 1
C, 1
``````

See : Graham Priest, An Introduction to Non-Classical Logic, Cambridge UP (2nd ed., 2008), page 86.

The rule for > is :

if we have A > B, i and i(r_A)j, then add B, j.

Having said tat, the rules for "decomposing" the formula must be applied one at each step and the rule to be applied must be chosen according to the "principal" connective.

In formula : (A or B ) > C the principal connective is > and thus your consideration is correct. We have to apply the rule for > first.

Thus, from :

(A or B) > C, 0 and 0 (r_A) 1 (i.e. world 0 A-accesses world 1)