TY - JOUR
T1 - On permutation in simplified semantics
AU - Restall, Greg
AU - Roy, Tony
N1 - Funding Information:
Acknowledgements Thanks to Graham Priest for conversations on this topic. This research is supported by the Australian Research Council, through grant dp0556827. See http://consequently.org/writing/permutation for the latest version of the paper, to post comments and to read comments left by others.
PY - 2009/6
Y1 - 2009/6
N2 - This note explains an error in Restall's 'Simplified Semantics for Relevant Logics (and some of their rivals)' (Restall, J Philos Logic 22(5):481-511, 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) (there called c6) and permutation (A → (B → C)) → (B → (A → C)) (there called c7). We show that the modelling conditions for assertion and permutation proposed in 'Simplified Semantics' overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the "base point" is inconsistent. This problem is not restricted to 'Simplified Semantics.' The techniques of that paper are used in Graham Priest's textbook An Introduction to Non-Classical Logic (Priest, 2001), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in 'Simplified Semantics' and propose two different corrections.
AB - This note explains an error in Restall's 'Simplified Semantics for Relevant Logics (and some of their rivals)' (Restall, J Philos Logic 22(5):481-511, 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) (there called c6) and permutation (A → (B → C)) → (B → (A → C)) (there called c7). We show that the modelling conditions for assertion and permutation proposed in 'Simplified Semantics' overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the "base point" is inconsistent. This problem is not restricted to 'Simplified Semantics.' The techniques of that paper are used in Graham Priest's textbook An Introduction to Non-Classical Logic (Priest, 2001), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in 'Simplified Semantics' and propose two different corrections.
KW - Completeness
KW - Frame semantics
KW - Inclusion
KW - Relevant logic
UR - http://www.scopus.com/inward/record.url?scp=65549142177&partnerID=8YFLogxK
U2 - 10.1007/s10992-009-9104-z
DO - 10.1007/s10992-009-9104-z
M3 - Article
AN - SCOPUS:65549142177
SN - 0022-3611
VL - 38
SP - 333
EP - 341
JO - Journal of Philosophical Logic
JF - Journal of Philosophical Logic
IS - 3
ER -