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Revision as of 19:40, 9 November 2016
, 19:40, 9 November 2016update
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| [[File:ICE Comment 1 Alt 3.jpg|center]]
| [[File:ICE Figure 1.jpg|center]]
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| height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math>
| height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math>
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| [[File:ICE Comment 2.jpg|center]]
| [[File:ICE Figure 2.jpg|center]]
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| height="20px" | <math>\text{Figure 2.} ~~ \text{Disjunctive Term}~ u, ~\text{Taken as Subject}\!</math>
| height="20px" | <math>\text{Figure 2.} ~~ \text{Disjunctive Term}~ u, ~\text{Taken as Subject}\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:ICE Comment 3a.jpg|center]]
| [[File:ICE Figure 3.jpg|center]]
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| height="20px" | <math>\text{Figure 3.} ~~ \text{Conjunctive Predicate}~ z, ~\text{Abduction of Case}~ \texttt{(} x \texttt{(} y \texttt{))}\!</math>
| height="20px" | <math>\text{Figure 3.} ~~ \text{Conjunctive Predicate}~ z, ~\text{Abduction of Case}~ \texttt{(} x \texttt{(} y \texttt{))}\!</math>
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| [[File:ICE Comment 3b.jpg|center]]
| [[File:ICE Figure 4.jpg|center]]
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| height="20px" | <math>\text{Figure 4.} ~~ \text{Disjunctive Subject}~ u, ~\text{Induction of Rule}~ \texttt{(} v \texttt{(} w \texttt{))}\!</math>
| height="20px" | <math>\text{Figure 4.} ~~ \text{Disjunctive Subject}~ u, ~\text{Induction of Rule}~ \texttt{(} v \texttt{(} w \texttt{))}\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:ICE Comment 1 Alt 3.jpg|center]]
| [[File:ICE Figure 1.jpg|center]]
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| height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math>
| height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math>
The relationship between conjunctive terms and iconic signs may be understood as follows. If there is anything that has all the properties described by the conjunctive term — ''spherical bright fragrant juicy tropical fruit'' — then sign users may use that thing as an icon of an orange, precisely by virtue of the fact that it shares those properties with an orange. But the only natural examples of things that have all those properties are oranges themselves, so the only thing that can serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange.
The relationship between conjunctive terms and iconic signs may be understood as follows. If there is anything that has all the properties described by the conjunctive term — ''spherical bright fragrant juicy tropical fruit'' — then sign users may use that thing as an icon of an orange, precisely by virtue of the fact that it shares those properties with an orange. But the only natural examples of things that have all those properties are oranges themselves, so the only thing that can serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange.
===Discussion 5===
Let's stay with Peirce's example of abductive inference a little longer and try to clear up the more troublesome confusions that tend to arise.
Figure 1 shows the implication ordering of logical terms in the form of a ''lattice diagram''.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:ICE Figure 1.jpg|center]]
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| height="20px" | <math>\text{Figure 1.} ~~ \text{Conjunctive Term}~ z, ~\text{Taken as Predicate}\!</math>
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One thing needs to be stressed at this point. It is important to recognize that the conjunctive term itself — namely, the syntactic string “spherical bright fragrant juicy tropical fruit” — is not an icon but a symbol. It has its place in a formal system of symbols, for example, a propositional calculus, where it would normally be interpreted as a logical conjunction of six elementary propositions, denoting anything in the universe of discourse that has all six of the corresponding properties. The symbol denotes objects that may be taken as icons of oranges by virtue of bearing those six properties. But there are no objects denoted by the symbol that aren't already oranges themselves. Thus we observe a natural reduction in the denotation of the symbol, consisting in the absence of cases outside of oranges that have all the properties indicated.
The above analysis provides another way to understand the abductive inference that reasons from the Fact <math>x \Rightarrow z\!</math> and the Rule <math>y \Rightarrow z\!</math> to the Case <math>x \Rightarrow y.\!</math> The lack of any cases that are <math>z\!</math> and not <math>y\!</math> is expressed by the implication <math>z \Rightarrow y.\!</math> Taking this together with the Rule <math>y \Rightarrow z\!</math> gives the logical equivalence <math>y = z.\!</math> But this reduces the Case <math>x \Rightarrow y\!</math> to the Fact <math>x \Rightarrow z\!</math> and so the Case is justified.
Viewed in the light of the above analysis, Peirce's example of abductive reasoning exhibits an especially strong form of inference, almost deductive in character. Do all abductive arguments take this form, or may there be weaker styles of abductive reasoning that enjoy their own levels of plausibility? That must remain an open question at this point.
===Discussion 6===
Figure 2 shows the implication ordering of logical terms in the form of a ''lattice diagram''.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[File:ICE Figure 2.jpg|center]]
|-
| height="20px" | <math>\text{Figure 2.} ~~ \text{Disjunctive Term}~ u, ~\text{Taken as Subject}\!</math>
|}
===Selection 7===
===Selection 7===
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===Discussion 5===
===Discussion 7===
If you dreamed that this inquiry had come full circle then I inform you of what you already know, that there are always greater circles. I revert to Peirce's Harvard University Lectures of the year before, to pick up additional background material and a bit more motivation.
If you dreamed that this inquiry had come full circle then I inform you of what you already know, that there are always greater circles. I revert to Peirce's Harvard University Lectures of the year before, to pick up additional background material and a bit more motivation.
