Abstract
Throughout Part I, we have been discussing sequent systems for particular logics, like classical logic and intuitionistic logic, and logical properties of these logics by proof-theoretic analysis of sequent systems for them. These results are sharp and deep, which are often obtained as consequences of cut elimination. On the other hand, cut elimination holds for only a limited number of sequent systems.
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- 1.
One may skip the rest of this section in her/his first reading, and go directly to Sect. 5.4.
- 2.
Superintuitionistic logics other than \(\varPhi \) are called intermediate logics , as they are intermediate between intuitionistic logic and classical.
- 3.
If we take a Hilbert-style system \(\mathbf HJ\) for \(\mathbf{Int}\) mentioned in Sect. 1.1, we can see that \(\mathbf HJ\) itself consists of finitely many axiom schemes. Thus, every finitely axiomatizable logic over \(\mathbf{Int}\) can be formalized in a Hilbert-style system with finitely many axiom schemes.
- 4.
Here \(\mathbf{FL}\), \(\mathbf{FL_{e}}\) etc. denote not only sequent systems, but also substructural logics determined by them, to ease a notational burden.
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Ono, H. (2019). Deducibility and Axiomatic Extensions. In: Proof Theory and Algebra in Logic. Short Textbooks in Logic. Springer, Singapore. https://doi.org/10.1007/978-981-13-7997-0_5
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DOI: https://doi.org/10.1007/978-981-13-7997-0_5
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