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| | =====1.3.12.2. Derived Equivalence Relations===== | | =====1.3.12.2. Derived Equivalence Relations===== |
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| − | The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion:
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| − | : Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.
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| − | A few of the many different expressions for this concept are recorded in Definition 9.
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| − | <pre>
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| − | Definition 9
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| − | If R c OxSxI,
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| − | then the following are identical subsets of IxS:
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| − | D9a. RIS
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| − | D9b. RSI^
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| − | D9c. ConR^
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| − | D9d. Con(R)^
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| − | D9e. PrIS(R)
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| − | D9f. Conv(Con(R))
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| − | D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
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| − | </pre>
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| − | Recall the definition of Den(R), the denotative component of R, in the following form:
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| − | : Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.
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| − | Equivalent expressions for this concept are recorded in Definition 10.
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| − | <pre>
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| − | Definition 10
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| − | If R c OxSxI,
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| − | then the following are identical subsets of OxS:
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| − | D10a. ROS
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| − | D10b. DenR
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| − | D10c. Den(R)
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| − | D10d. PrOS(R)
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| − | D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
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| − | </pre>
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| − | The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion:
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| − |
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| − | : Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.
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| − |
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| − | A few of the many different expressions for this concept are recorded in Definition 11.
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| − | <pre>
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| − | Definition 11
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| − | If R c OxSxI,
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| − | then the following are identical subsets of SxO:
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| − | D11a. RSO
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| − | D11b. ROS^
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| − | D11c. DenR^
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| − | D11d. Den(R)^
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| − | D11e. PrSO(R)
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| − | D11f. Conv(Den(R))
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| − | D11g. {< s, o> C SxO : <o, s, i> C R for some i C I}
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| − | </pre>
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| − | The "denotation of x in R", written "Den(R, x)", is defined as follows:
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| − | : Den(R, x) = {o C O : <o, x> C Den(R)}.
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| − | In other words:
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| − | : Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
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| − |
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| − | Equivalent expressions for this concept are recorded in Definition 12.
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| − | <pre>
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| − | Definition 12
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| − | If R c OxSxI,
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| − | and x C S,
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| − | then the following are identical subsets of O:
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| − | D12a. ROS.x
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| − | D12b. DenR.x
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| − | D12c. DenR|x
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| − | D12d. DenR(, x)
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| − | D12e. Den(R, x)
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| − | D12f. Den(R).x
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| − | D12g. {o C O : <o, x> C Den(R)}
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| − | D12h. {o C O : <o, x, i> C R for some i C I}
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| − | </pre>
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| | Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms. | | Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms. |