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Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of "information" to explain what it is that signs convey in the process.  By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn.  So let's take a moment to draw out a few of these characters.

Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of "information" to explain what it is that signs convey in the process.  By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn.  So let's take a moment to draw out a few of these characters.

[[Sign relation]]s, like any non-trivial brand of [[3-adic relation]]s, can become overwhelming to think about once the cardinality of the object, sign, and interpretant domains or the complexity of the relation itself ascends beyond the simplest examples.  Furthermore, most of the strategies that we would normally use to control the complexity, like neglecting one of the domains, in effect, projecting the 3-adic sign relation onto one of its 2-adic faces, or focusing on a single ordered triple of the form ‹ ''o'', ''s'', ''i'' › at a time, can result in our receiving a distorted impression of the sign relation's true nature and structure.

[[Sign relation]]s, like any non-trivial brand of [[3-adic relation]]s, can become overwhelming to think about once the cardinality of the object, sign, and interpretant domains or the complexity of the relation itself ascends beyond the simplest examples.  Furthermore, most of the strategies that we would normally use to control the complexity, like neglecting one of the domains, in effect, projecting the 3-adic sign relation onto one of its 2-adic faces, or focusing on a single ordered triple of the form <math>(o, s, i)\!</math> at a time, can result in our receiving a distorted impression of the sign relation's true nature and structure.

I find that it helps me to draw, or at least to imagine drawing, diagrams of the following form, where I can keep tabs on what's an object, what's a sign, and what's an interpretant sign, for a selected set of sign-relational triples.

I find that it helps me to draw, or at least to imagine drawing, diagrams of the following form, where I can keep tabs on what's an object, what's a sign, and what's an interpretant sign, for a selected set of sign-relational triples.

Here is how I would picture Peirce's example of equivalent terms:  ''v'' = ''p'', where "''v''" denotes the Vice-President of the United States, and "''p''" denotes the President of the Senate of the United States.

Here is how I would picture Peirce's example of equivalent terms, <math>\mathrm{v} = \mathrm{p}\!</math>, where <math>^{\backprime\backprime} \mathrm{v} ^{\prime\prime}</math> denotes the Vice-President of the United States, and <math>^{\backprime\backprime} \mathrm{p} ^{\prime\prime}</math> denotes the President of the Senate of the United States.

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Depending on whether we interpret the terms "''v''" and "''p''" as applying to persons who hold these offices at one particular time or as applying to all those persons who have held these offices over an extended period of history, their denotations may be either singular of plural, respectively.

Depending on whether we interpret the terms <math>^{\backprime\backprime} \mathrm{v} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \mathrm{p} ^{\prime\prime}</math> as applying to persons who hold these offices at one particular time or as applying to all those persons who have held these offices over an extended period of history, their denotations may be either singular of plural, respectively.

As a shortcut technique for indicating general denotations or plural referents, I will use the "elliptic convention" that represents these by means of figures like "o&nbsp;o&nbsp;o" or "o&nbsp;&hellip;&nbsp;o", placed at the object ends of sign relational triads.

As a shortcut technique for indicating general denotations or plural referents, I will use the ''elliptic convention'' that represents these by means of figures like "o&nbsp;o&nbsp;o" or "o&nbsp;&hellip;&nbsp;o", placed at the object ends of sign relational triads.

For a more complex example, here is how I would picture Peirce's example of an equivalence between terms that comes about by applying one of the distributive laws, for relative multiplication over absolute summation.

For a more complex example, here is how I would picture Peirce's example of an equivalence between terms that comes about by applying one of the distributive laws, for relative multiplication over absolute summation.