Pub Date : 2022-01-01DOI: 10.1007/978-3-030-67396-3_8
Odysseus Makridis
{"title":"Proof-Theoretical System for Predicate Logic: ∏πφ=","authors":"Odysseus Makridis","doi":"10.1007/978-3-030-67396-3_8","DOIUrl":"https://doi.org/10.1007/978-3-030-67396-3_8","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85761907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1007/978-3-030-67396-3_4
Odysseus Makridis
{"title":"Sentential Logic Languages ∑","authors":"Odysseus Makridis","doi":"10.1007/978-3-030-67396-3_4","DOIUrl":"https://doi.org/10.1007/978-3-030-67396-3_4","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86797616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1007/978-3-030-67396-3_7
Odysseus Makridis
{"title":"Semantic Models for ∏: ∏⧉","authors":"Odysseus Makridis","doi":"10.1007/978-3-030-67396-3_7","DOIUrl":"https://doi.org/10.1007/978-3-030-67396-3_7","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75829023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1007/978-3-030-67396-3_3
Odysseus Makridis
{"title":"Formal Logic of Sentences, Sentential Logic (also called Sentential Logic and Statement Logic)","authors":"Odysseus Makridis","doi":"10.1007/978-3-030-67396-3_3","DOIUrl":"https://doi.org/10.1007/978-3-030-67396-3_3","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75000212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1007/978-3-030-67396-3_2
Odysseus Makridis
{"title":"Concepts of Deductive Reasoning","authors":"Odysseus Makridis","doi":"10.1007/978-3-030-67396-3_2","DOIUrl":"https://doi.org/10.1007/978-3-030-67396-3_2","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81608783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1007/978-3-030-67396-3_6
Odysseus Makridis
{"title":"Translations from English into ∏πφ= (also called Symbolizations, Formalizations)","authors":"Odysseus Makridis","doi":"10.1007/978-3-030-67396-3_6","DOIUrl":"https://doi.org/10.1007/978-3-030-67396-3_6","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87015830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1007/978-3-030-67396-3_10
Odysseus Makridis
{"title":"Basics of Set Theory","authors":"Odysseus Makridis","doi":"10.1007/978-3-030-67396-3_10","DOIUrl":"https://doi.org/10.1007/978-3-030-67396-3_10","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81123581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The article approaches cut elimination from a new angle. On the basis of an arbitrary inference relation among logically atomic formulae, an inference relation on a language possessing logical operators is defined by means of inductive clauses similar to the operator-introducing rules of a cut-free intuitionistic sequent calculus. The logical terminology of the richer language is not uniquely specified, but assumed to satisfy certain conditions of a general nature, allowing for, but not requiring, the existence of infinite conjunctions and disjunctions. We investigate to what extent structural properties of the given atomic relation are preserved through the extension to the full language. While closure under the Cut rule narrowly construed is not in general thus preserved, two properties jointly amounting to closure under the ordinary structural rules, including Cut, are.Attention is then narrowed down to the special case of a standard first-order language, where a similar result is obtained also for closure under substitution of terms for individual parameters. Taken together, the three preservation results imply the familiar cut-elimination theorem for pure first-order intuitionistic sequent calculus.In the interest of conceptual economy, all deducibility relations are specified purely inductively, rather than in terms of the existence of formal proofs of any kind.
{"title":"Preservation of Structural Properties in intuitionistic Extensions of an Inference Relation","authors":"Tor Sandqvist","doi":"10.1017/BSL.2017.26","DOIUrl":"https://doi.org/10.1017/BSL.2017.26","url":null,"abstract":"The article approaches cut elimination from a new angle. On the basis of an arbitrary inference relation among logically atomic formulae, an inference relation on a language possessing logical operators is defined by means of inductive clauses similar to the operator-introducing rules of a cut-free intuitionistic sequent calculus. The logical terminology of the richer language is not uniquely specified, but assumed to satisfy certain conditions of a general nature, allowing for, but not requiring, the existence of infinite conjunctions and disjunctions. We investigate to what extent structural properties of the given atomic relation are preserved through the extension to the full language. While closure under the Cut rule narrowly construed is not in general thus preserved, two properties jointly amounting to closure under the ordinary structural rules, including Cut, are.Attention is then narrowed down to the special case of a standard first-order language, where a similar result is obtained also for closure under substitution of terms for individual parameters. Taken together, the three preservation results imply the familiar cut-elimination theorem for pure first-order intuitionistic sequent calculus.In the interest of conceptual economy, all deducibility relations are specified purely inductively, rather than in terms of the existence of formal proofs of any kind.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88313875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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