Abstract
The standard notion of formal theory, in logic, is in general biased exclusively towards assertion: it commonly refers only to collections of assertions that any agent who accepts the generating axioms of the theory should also be committed to accept. In reviewing the main abstract approaches to the study of logical consequence, we point out why this notion of theory is unsatisfactory at multiple levels, and introduce a novel notion of theory that attacks the shortcomings of the received notion by allowing one to take both assertions and denials on a par. This novel notion of theory is based on a bilateralist approach to consequence operators, which we hereby introduce, and whose main properties we investigate in the present paper.
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Notes
Consider for instance a set of sentences \(\mathbb {L}=\{p,q\}\), and a semantics containing only two canonical valuations, \(\nu _p\) and \(\nu _q\), such that \(\nu _x(y)=1\) iff \(x=y\). (This sort of problems related to the failure of absoluteness has been discussed as early as in Carnap (1943).)
For a straightforward class of examples (cf. Marcos 2007), let \(\nu _{\top \!\!\!\top }\) denote the the ‘dadaistic’ valuation on \(\mathbb {L}\) such that \({\mathbb {1}}_{\nu _{\top \!\!\!\top }}=\mathbb {L}\), consider a semantics \(\mathbf {V}\) such that \(\nu _{\top \!\!\!\top }\not \in \mathbf {V}\), and let \(\mathbf {V}^\star :=\mathbf {V}\cup \{\nu _{\top \!\!\!\top }\}\). Then \(\vartriangleright ^\mathsf {T}_{\mathbf {V}}\;=\;\vartriangleright ^\mathsf {T}_{\mathbf {V}^\star }\), and \(\mathbb {L}\vartriangleright ^\mathsf {S}_{\mathbf {V}}\varnothing \), while \(\mathbb {L}\blacktriangleright _{\mathbf {V}^\star }\varnothing \).
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The second author acknowledges that the work was done under the scope of Project UID/EEA/50008/2019 of Instituto de Telecomunicações, financed by the applicable framework (FCT/MEC through national funds and co-funded by FEDER-PT2020). The third author acknowledges partial funding by CNPq. The first author passed away on 26 Aug 2017.
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Blasio, C., Caleiro, C. & Marcos, J. What is a logical theory? On theories containing assertions and denials. Synthese 198 (Suppl 22), 5481–5504 (2021). https://doi.org/10.1007/s11229-019-02183-z
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DOI: https://doi.org/10.1007/s11229-019-02183-z