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Proof of the independence of the primitive symbols of Heyting's calculus of propositions
In this paper I shall show that no one of the four primitive symbols of Heyting's calculus of propositions is definable in terms of the other three. So as to make the paper self-contained, I begin by stating the rules and primitive sentences given by Heyting. The primitive symbols of the calcul...
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| Published in: | The Journal of symbolic logic 1939-12, Vol.4 (4), p.155-158 |
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| Format: | Article |
| Language: | English |
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| container_end_page | 158 |
| container_issue | 4 |
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| container_title | The Journal of symbolic logic |
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| creator | McKinsey, J. C. C. |
| description | In this paper I shall show that no one of the four primitive symbols of Heyting's calculus of propositions is definable in terms of the other three. So as to make the paper self-contained, I begin by stating the rules and primitive sentences given by Heyting.
The primitive symbols of the calculus are “⅂”, “∨”, “∧”, and “⊃”, which may be read, respectively, as “not,” “either…or,” “and,” and “if…then.” The symbol “⊃⊂”, which may be read “if and only if,” is defined in terms of these as follows:
The rule of substitution is assumed, and the rule that
S
2
follows from
S
1
and
S
1
⊃
S
2
; in addition it is assumed that
S
1
∧
S
2
follows from
S
1
and
S
2
. The primitive sentences are as follows: |
| doi_str_mv | 10.2307/2268715 |
| format | article |
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The primitive symbols of the calculus are “⅂”, “∨”, “∧”, and “⊃”, which may be read, respectively, as “not,” “either…or,” “and,” and “if…then.” The symbol “⊃⊂”, which may be read “if and only if,” is defined in terms of these as follows:
The rule of substitution is assumed, and the rule that
S
2
follows from
S
1
and
S
1
⊃
S
2
; in addition it is assumed that
S
1
∧
S
2
follows from
S
1
and
S
2
. The primitive sentences are as follows:</description><identifier>ISSN: 0022-4812</identifier><identifier>EISSN: 1943-5886</identifier><identifier>DOI: 10.2307/2268715</identifier><language>eng</language><publisher>New York, USA: Cambridge University Press</publisher><subject>Direct products ; Proof calculi</subject><ispartof>The Journal of symbolic logic, 1939-12, Vol.4 (4), p.155-158</ispartof><rights>Copyright 1939 Association for Symbolic Logic</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c2221-bb9509d0e4a0f2cd029de37eb93213b140fcecc81ddd76cbd5f56c97f799f563</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2268715$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2268715$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,780,784,885,27923,27924,58237,58470</link.rule.ids></links><search><creatorcontrib>McKinsey, J. C. C.</creatorcontrib><title>Proof of the independence of the primitive symbols of Heyting's calculus of propositions</title><title>The Journal of symbolic logic</title><description>In this paper I shall show that no one of the four primitive symbols of Heyting's calculus of propositions is definable in terms of the other three. So as to make the paper self-contained, I begin by stating the rules and primitive sentences given by Heyting.
The primitive symbols of the calculus are “⅂”, “∨”, “∧”, and “⊃”, which may be read, respectively, as “not,” “either…or,” “and,” and “if…then.” The symbol “⊃⊂”, which may be read “if and only if,” is defined in terms of these as follows:
The rule of substitution is assumed, and the rule that
S
2
follows from
S
1
and
S
1
⊃
S
2
; in addition it is assumed that
S
1
∧
S
2
follows from
S
1
and
S
2
. The primitive sentences are as follows:</description><subject>Direct products</subject><subject>Proof calculi</subject><issn>0022-4812</issn><issn>1943-5886</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1939</creationdate><recordtype>article</recordtype><recordid>eNp9kF9LwzAUxYMoOKf4FQoKPlXzp22SN2XophRUmCK-hDZJNbVratKK-_a2ru5RuOSGc3-ce7gAHCN4jgmkFxgnjKJ4B0wQj0gYM5bsggmEGIcRQ3gfHHhfQghjHrEJeHlw1hZBX-27DkytdKP7p5b6T2ucWZnWfOnAr1e5rfwwWOh1a-q3Mx_IrJJd1f2qjbON9T1sa38I9oqs8vpo7FOwvLlezhZhej-_nV2locQYozDPeQy5gjrKYIGlgpgrTajOOcGI5CiChdRSMqSUoonMVVzEieS0oJz3PzIFlxvbfnepZas7WRklhtCZWwubGTF7Skd1bKWvBEKMEJZQjHuLk63FZ6d9K0rbuboPLRDmkEbxEGUKzjaUdNZ7p4vtDgTFcHgxHr4nTzdk6Vvr_sHCDWZ8q7-3WOY-REIJjUUyfxQsTfnr891cLMkPscSQug</recordid><startdate>19391201</startdate><enddate>19391201</enddate><creator>McKinsey, J. C. 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C.</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Periodicals Index Online Segment 01</collection><collection>Periodicals Index Online Segment 04</collection><collection>Periodicals Index Online Segment 29</collection><collection>Periodicals Index Online</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - West</collection><collection>Primary Sources Access (Plan D) - International</collection><collection>Primary Sources Access & Build (Plan A) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Midwest</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Northeast</collection><collection>Primary Sources Access (Plan D) - Southeast</collection><collection>Primary Sources Access (Plan D) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Southeast</collection><collection>Primary Sources Access (Plan D) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - UK / I</collection><collection>Primary Sources Access (Plan D) - Canada</collection><collection>Primary Sources Access (Plan D) - EMEALA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - International</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - International</collection><collection>Primary Sources Access (Plan D) - West</collection><collection>Periodicals Index Online Segments 1-50</collection><collection>Primary Sources Access (Plan D) - APAC</collection><collection>Primary Sources Access (Plan D) - Midwest</collection><collection>Primary Sources Access (Plan D) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Canada</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - EMEALA</collection><collection>Primary Sources Access & Build (Plan A) - APAC</collection><collection>Primary Sources Access & Build (Plan A) - Canada</collection><collection>Primary Sources Access & Build (Plan A) - West</collection><collection>Primary Sources Access & Build (Plan A) - EMEALA</collection><collection>Primary Sources Access (Plan D) - Northeast</collection><collection>Primary Sources Access & Build (Plan A) - Midwest</collection><collection>Primary Sources Access & Build (Plan A) - North Central</collection><collection>Primary Sources Access & Build (Plan A) - Northeast</collection><collection>Primary Sources Access & Build (Plan A) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - Southeast</collection><collection>Primary Sources Access (Plan D) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - APAC</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - MEA</collection><jtitle>The Journal of symbolic logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>McKinsey, J. C. C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Proof of the independence of the primitive symbols of Heyting's calculus of propositions</atitle><jtitle>The Journal of symbolic logic</jtitle><date>1939-12-01</date><risdate>1939</risdate><volume>4</volume><issue>4</issue><spage>155</spage><epage>158</epage><pages>155-158</pages><issn>0022-4812</issn><eissn>1943-5886</eissn><abstract>In this paper I shall show that no one of the four primitive symbols of Heyting's calculus of propositions is definable in terms of the other three. So as to make the paper self-contained, I begin by stating the rules and primitive sentences given by Heyting.
The primitive symbols of the calculus are “⅂”, “∨”, “∧”, and “⊃”, which may be read, respectively, as “not,” “either…or,” “and,” and “if…then.” The symbol “⊃⊂”, which may be read “if and only if,” is defined in terms of these as follows:
The rule of substitution is assumed, and the rule that
S
2
follows from
S
1
and
S
1
⊃
S
2
; in addition it is assumed that
S
1
∧
S
2
follows from
S
1
and
S
2
. The primitive sentences are as follows:</abstract><cop>New York, USA</cop><pub>Cambridge University Press</pub><doi>10.2307/2268715</doi><tpages>4</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-4812 |
| ispartof | The Journal of symbolic logic, 1939-12, Vol.4 (4), p.155-158 |
| issn | 0022-4812 1943-5886 |
| language | eng |
| recordid | cdi_projecteuclid_primary_oai_CULeuclid_euclid_jsl_1183386722 |
| source | JSTOR Archival Journals and Primary Sources Collection |
| subjects | Direct products Proof calculi |
| title | Proof of the independence of the primitive symbols of Heyting's calculus of propositions |
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