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It is also inconsistent over much weaker logics such as minimal logic and partial logic.
Field (2008) and others criticised theories based on partial logic for the absence of a 'proper' conditional and bi-conditional.
For this reason Halbach and Horsten (2006) and Horsten (2011) explore an axiomatization of Kripke's theory with partial logic as inner and outer logic.
A reason for preferring a paraconsistent logic over a partial logic is that paradoxical sentences such as the liar can then be modelled as true contradictions (dialetheia) rather than truth-value gaps.
Unlike Field's and his own 1984 theory, Feferman's (2008) theory DT is formulated in classical logic, but it's internal logic is again a partial logic with a strong conditional.
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In fuzzy logic, partial antecedents result in partial implications.
Section 5 samples applications to theories of description, logics of partial or non-strict functions, logics with Kripke semantics, logics of fiction and logics that are in a certain sense Meinongian.
In order to implement partial reconfiguration of logic resources, this paper puts forward dynamic reconfiguration instruction-DRP, and designs the JTAG circuit of accomplishing dynamic reconfiguration.
Classic early references on partial and multivalent logics include Bochvar (1939) and Kleene (1952), while Blamey (1989) provides a useful overview of such logics.
In 1925, Andrei Kolmogorov, at the age of 22, published the first (partial) formalization of intuitionistic logic, and also made an extensive comparison with formalized classical logic, in a paper called "On the principle of the excluded middle".
In applications of free logic involving partial functions, the existence predicate 'E!' is often replaced by the postfix definedness predicate '↓'.
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