Pub Date : 2022-02-28DOI: 10.1017/9781108992756.008
Hongfei Fu, J. Hopcroft
A B –––––– A&B A&B –––––– A A&B –––––– B ~(A&B) =========== A → ~B vI vO ∼ ∼ ∼ ∼vI/O A –––––– A∨B B –––––– A∨B A∨B ∼A –––––– B A∨B ∼B –––––– A ~(A∨B) ∼A –––––– A→B B –––––– A→B A→B A –––––– B A→B ∼B –––––– ∼A ~(A→B) =========== A & ~B ↔ ↔ ↔ ↔I ↔ ↔ ↔ ↔O ∼ ∼ ∼ ∼↔ ↔ ↔ ↔I/O A→B B→A –––––– A↔B A↔B –––––– A→B A↔B –––––– B→A ~(A↔B) =========== ∼A↔B I O DN A ∼A –––– ––– A ~∼A ===== A Note: The ~O and ~I rules are combined, using a long equals sign '==='. Henceforth, any rule that is displayed with '===' is a bi-directional rule, which can be used both as an in-rule and as an out-rule.
A—B—————A&B A&B——————A A&B——————B ~ (A&B ) =========== A→B ~六vO∼∼∼∼vI O / A——————A∨B B——————A∨B A∨B A∼——————A∨B∼B——————~ (A∨B) A∼——————A→B, B——————A→B, A→B A——————A→B∼B——————∼~ (A→B ) =========== I A & B ~↔↔↔↔↔↔↔↔O∼∼∼∼↔↔↔↔I / O . B→A→B——————A↔A↔B——————A→B A↔B——————B→A ~ B (A↔ ) =========== DN A∼∼A↔B I O –––– ––– A∼~ = = = = = A .注:关于~ ~ O and I are rules,联合使用了长像用符号' = = = '。==地理==根据美国人口普查，这个县的面积为，其中土地面积为，其中土地面积为。
{"title":"Inference Rules","authors":"Hongfei Fu, J. Hopcroft","doi":"10.1017/9781108992756.008","DOIUrl":"https://doi.org/10.1017/9781108992756.008","url":null,"abstract":"A B –––––– A&B A&B –––––– A A&B –––––– B ~(A&B) =========== A → ~B vI vO ∼ ∼ ∼ ∼vI/O A –––––– A∨B B –––––– A∨B A∨B ∼A –––––– B A∨B ∼B –––––– A ~(A∨B) ∼A –––––– A→B B –––––– A→B A→B A –––––– B A→B ∼B –––––– ∼A ~(A→B) =========== A & ~B ↔ ↔ ↔ ↔I ↔ ↔ ↔ ↔O ∼ ∼ ∼ ∼↔ ↔ ↔ ↔I/O A→B B→A –––––– A↔B A↔B –––––– A→B A↔B –––––– B→A ~(A↔B) =========== ∼A↔B I O DN A ∼A –––– ––– A ~∼A ===== A Note: The ~O and ~I rules are combined, using a long equals sign '==='. Henceforth, any rule that is displayed with '===' is a bi-directional rule, which can be used both as an in-rule and as an out-rule.","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115328939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.018
{"title":"Anti-Objectivism About Set Theory","authors":"","doi":"10.1017/9781108992756.018","DOIUrl":"https://doi.org/10.1017/9781108992756.018","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"89 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126315170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.010
{"title":"Platonism or Nominalism?","authors":"","doi":"10.1017/9781108992756.010","DOIUrl":"https://doi.org/10.1017/9781108992756.010","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128307415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.023
{"title":"Archimedean and Rich Instantiation","authors":"","doi":"10.1017/9781108992756.023","DOIUrl":"https://doi.org/10.1017/9781108992756.023","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127267052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.002
{"title":"Actualist Set Theory","authors":"","doi":"10.1017/9781108992756.002","DOIUrl":"https://doi.org/10.1017/9781108992756.002","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127421367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.022
{"title":"Vindication of FOL Inference in Set Theory","authors":"","doi":"10.1017/9781108992756.022","DOIUrl":"https://doi.org/10.1017/9781108992756.022","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"177 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126023472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.013
{"title":"Explanatory Indispensability","authors":"","doi":"10.1017/9781108992756.013","DOIUrl":"https://doi.org/10.1017/9781108992756.013","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"46 9-10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131519956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.003
{"title":"Putnamian Potentialism: Putnam and Hellman","authors":"","doi":"10.1017/9781108992756.003","DOIUrl":"https://doi.org/10.1017/9781108992756.003","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"183 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132924785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively.
{"title":"Indispensability","authors":"A. Paseau, Alan Baker","doi":"10.2307/j.ctvgs09k1.10","DOIUrl":"https://doi.org/10.2307/j.ctvgs09k1.10","url":null,"abstract":"Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively.","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"170 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132436337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1017/9781108992756.014
{"title":"Physical Magnitude Statements and Sparsity","authors":"","doi":"10.1017/9781108992756.014","DOIUrl":"https://doi.org/10.1017/9781108992756.014","url":null,"abstract":"","PeriodicalId":294618,"journal":{"name":"A Logical Foundation for Potentialist Set Theory","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121637752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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