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| − | The concept of a sign relation is typically extended as a set L c OxSxI. Because this extensional representation of a sign relation is one of the most natural forms that it can take up, along with being one of the most important forms that it is likely to be encountered in, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.
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| − | For the purposes of this discussion, let it be supposed that each set Q, that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set X, one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment. In a setting like this it is possible to make a number of useful definitions, to which I now turn.
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| − | The "negation" of a sentence z, written as "-(z)-" and read as "not z", is a sentence that is true when z is false, and false when z is true.
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| − | The "complement" of a set Q with respect to the universe X is denoted by "X-Q", or simply by "~Q" when the universe X is determinate, and is defined as the set of elements in X that do not belong to Q, that is:
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| − | ~Q = X-Q = {x in X : -(x in Q)- }.
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| − | The "relative complement" of P in Q, for two sets P, Q c X, is denoted by "Q-P" and defined as the set of elements in Q that do not belong to P, that is:
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| − | Q-P = {x in X : x in Q and -(x in P)- }.
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| − | The "intersection" of P and Q, for two sets P, Q c X, is denoted by "P |^| Q" and defined as the set of elements in X that belong to both P and Q.
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| − | P |^| Q = {x in X : x in P and x in Q }.
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| − | The "union" of P and Q, for two sets P, Q c X, is denoted by "P |_| Q" and defined as the set of elements in X that belong to at least one of P or Q.
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| − | P |_| Q = {x in X : x in P or x in Q }.
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| − | The "symmetric difference" of P and Q, for two sets P, Q c X, is denoted by "P ± Q" and is defined as the set of elements in X that belong to just one of P or Q.
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| − | P ± Q = {x in X : x in P-Q or x in Q-P }.
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| | The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a "sentence" is, they all rely on the reader's native intuition of what a "set" is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance, that just about everybody has of the logical connectives "not", "and", "or", as these are expressed in natural language terms. | | The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a "sentence" is, they all rely on the reader's native intuition of what a "set" is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance, that just about everybody has of the logical connectives "not", "and", "or", as these are expressed in natural language terms. |
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