Functional Logic • Inquiry and Analogy • 21

Inquiry and Analogy • Generalized Umpire Operators

To get a better handle on the space of higher order propositions and continue developing our functional approach to quantification theory, we’ll need a number of specialized tools. To begin, we define a higher order operator called the umpire operator, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result. Operators with optional numbers of arguments are called multigrade operators, typically defined as unions over function types. Expressing in that form gives the following formula.

In contexts of application, that is, where a multigrade operator is actually being applied to arguments, the number of arguments in the argument list tells which of the optional types is “operative”. In the case of the first and last arguments appear as indices, the one in the middle serving as the main argument while the other two arguments serve to modify the sense of the operation in question. Thus, we have the following forms.

The operation evaluates the proposition on each model of the proposition and combines the results according to the method indicated by the connective parameter In principle, the index may specify any logical connective on as many as arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums. By convention, each of the accessory indices is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition for the lower index and the continued conjunction or continued product operation for the upper index Taking the upper default value gives license to the following readings.

This means if and only if holds for all models of In propositional terms, this is tantamount to the assertion that or that

Throwing in the lower default value permits the following abbreviations.

This means if and only if holds for the whole universe of discourse in question, that is, if and only is the constantly true proposition The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Multigrade Operator
• Parametric Operator
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 20

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Table 21 provides a thumbnail sketch of the relationships discussed in this section.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 19

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.

John Dewey • How We Think

Tables 19 and 20 present the same information as Table 18, sorting the rows in different orders to reveal other symmetries in the arrays.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 18

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Last time we took up a fourfold scheme of quantified propositional forms traditionally known as a “Square of Opposition”, relating it to a quartet of higher order propositions which, depending on context, are also known as measures, qualifiers, or higher order indicator functions.

Table 18 develops the above ideas in further detail, expressing a larger set of quantified propositional forms by means of propositions about propositions.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 17

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Our excursion into the expanding landscape of higher order propositions has come round to the point where we can begin to open up new perspectives on quantificational logic.

Though it may be all the same from a purely formal point of view, it does serve intuition to adopt a slightly different interpretation for the two‑valued space we take as the target of our basic indicator functions. In that spirit we declare a novel type of existence-valued functions where is a pair of values indicating whether anything exists in the cells of the underlying universe of discourse. As usual, we won’t be too picky about the coding of those functions, reverting to binary codes whenever the intended interpretation is clear enough.

With that interpretation in mind we observe the following correspondence between classical quantifications and higher order indicator functions.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 16

Inquiry and Analogy • Extending the Existential Interpretation to Quantificational Logic

One of the resources we have for this work is a formal calculus based on C.S. Peirce’s logical graphs. For now we’ll adopt the existential interpretation of that calculus, fixing the meanings of logical constants and connectives at the core level of propositional logic. To build on that core we’ll need to extend the existential interpretation to encompass the analysis of quantified propositions, or quantifications. That in turn will take developing two further capacities of our calculus. On the formal side we’ll need to consider higher order functional types, continuing our earlier venture above. In terms of content we’ll need to consider new species of elemental or singular propositions.

Let us return to the 2‑dimensional universe A bridge between propositions and quantifications is afforded by a set of measures or qualifiers defined by the following equations.

A higher order proposition tells us something about the proposition namely, which elements in the space of type are assigned a positive value by Taken together, the operators give us a way to express many useful observations about the propositions in Figure 16 summarizes the action of the operators on the propositions of type

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 15

Inquiry and Analogy • Measure for Measure

Let us define two families of measures,

by means of the following equations:

Table 14 shows the value of each on each of the 16 boolean functions In terms of the implication ordering on the 16 functions, says that is above or identical to in the implication lattice, that is, in the implication ordering.

Table 15 shows the value of each on each of the 16 boolean functions In terms of the implication ordering on the 16 functions, says that is below or identical to in the implication lattice, that is, in the implication ordering.

Applied to a given proposition the qualifiers and tell whether is above or below respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those which occupy the limiting positions in the Tables.

Expressed in terms of the propositional forms they value positively, is a wholly indifferent or indiscriminate measure, accepting every proposition whereas the measures and value the constant propositions and respectively, above all others.

Finally, in conformity with the use of fiber notation to indicate sets of models, it is natural to use notations like the following to denote sets of propositions satisfying the umpires in question.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 14

Inquiry and Analogy • Umpire Operators

The measures of type present a formidable array of propositions about propositions about 2‑dimensional universes of discourse. The early entries in their standard ordering define universes too amorphous to detain us for long on a first pass but as we turn toward the high end of the ordering we begin to recognize familiar structures worth examining from new angles.

Instrumental to our study we define a couple of higher order operators,

referred to as the relative and absolute umpire operators, respectively. If either operator is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.

Let be a two‑dimensional boolean space, generated by two boolean variables or logical features and

Given an ordered pair of propositions as arguments, the relative umpire operator reports the value if the first implies the second, otherwise it reports the value

Expressing it another way:

In writing this, however, it is important to observe that the appearing on the left side and the appearing on the right side of the logical equivalence have different meanings. Filling in the details, we have the following.

Writing types as subscripts and using the fact that it is possible to express this more succinctly as follows.

Finally, it is often convenient to write the first argument as a subscript. Thus we have the following equation.

The absolute umpire operator, also known as the umpire measure, is a higher order proposition defined by the equation In this case the subscript on the left and the argument on the right both refer to the constant proposition In most settings where is applied to arguments it is safe to omit the subscript since the number of arguments indicates which type of operator is meant. Thus, we have the following identities and equivalents.

The umpire measure is defined on boolean functions regarded as mathematical objects but can also be understood in terms of the judgments it induces on the syntactic level. In that interpretation recognizes theorems of the propositional calculus over giving a score of to tautologies and a score of to everything else, counting all contingent statements as no better than falsehoods.

One remark in passing for those who might prefer an alternative definition. If we had originally taken to mean the absolute measure then the relative measure could have been defined as

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 13

Inquiry and Analogy • Higher Order Propositional Expressions

Higher Order Propositions and Logical Operators (n = 2)

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let’s first consider the universe of discourse based on two logical features or boolean variables and

The universe of discourse consists of two parts, a set of points and a set of propositions.

The points of form the space:

Each point in may be indicated by means of a singular proposition, that is, a proposition which describes it uniquely. This form of representation leads to the following enumeration of points.

Each point in may also be described by means of its coordinates, that is, by the ordered pair of values in which the coordinate propositions and take on that point. This form of representation leads to the following enumeration of points.

The propositions of form the space:

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call on them again.

The next higher order universe of discourse built on is which may be developed in the following way. The propositions of become the points of and the mappings of the type become the propositions of In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form

To save a few words in the remainder of this discussion, I will use the terms measure and qualifier to refer to all types of higher order propositions and operators. To describe the present setting in picturesque terms, the propositions of may be regarded as a gallery of sixteen venn diagrams, while the measures are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not. In this way, each judge partitions the gallery of pictures into two aesthetic portions, the pictures that likes and the pictures that dislikes.

There are measures of the form Table 13 shows the first 24 of their number in the style of higher order truth table I used before. The column headed shows the value of the measure on each of the propositions for = 0 to 15. The arrangement of measures in the order indicated will be referred to as their standard ordering. In this scheme of things, the index of the measure is the decimal equivalent of the bit string in the corresponding column of the Table, reading the binary digits in order from bottom to top.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 12

Inquiry and Analogy • Higher Order Propositional Expressions

Interpretive Categories for Higher Order Propositions (n = 1)

Table 12 presents a series of interpretive categories for the higher order propositions in Table 11. I’ll leave those for now to the reader’s contemplation and discuss them when we get two variables into the mix. The lower dimensional cases tend to exhibit condensed or degenerate structures and their full significance will become clearer once we get beyond the 1‑dimensional case.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 11

Inquiry and Analogy • Higher Order Propositional Expressions

Higher Order Propositions and Logical Operators (n = 1)

A higher order proposition is a proposition about propositions. If the original order of propositions is a set of indicator functions then the next higher order of propositions consists of maps of type

For example, consider the case where There are exactly four propositions one can make about the elements of Each proposition has the concrete type and the abstract type From that beginning there are exactly sixteen higher order propositions one can make about the initial set of four propositions. Each higher order proposition has the abstract type

Table 11 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion.

• Columns 1 and 2 taken together present a form of truth table for the four propositions Column 1 displays the names of the propositions for = 1 to 4, while the entries in Column 2 show the value each proposition takes on the argument value listed in the corresponding column head.
• Column 3 displays one of the more usual expressions for the proposition in question.
• The last sixteen columns are headed by a series of conventional names for the higher order propositions, also known as the measures for = 0 to 15. The entries in the body of the Table show the value each measure assigns to each proposition

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 10

Inquiry and Analogy • Functional Conception of Quantification Theory

Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements. The mere act of writing quantified formulas like and involves a subscription to such notions, as shown by the membership relations invoked in their indices.

As we reflect more critically on the conventional assumptions in the light of pragmatic and constructive principles, however, they begin to appear as problematic hypotheses whose warrants are not beyond question, as projects of exhaustive determination overreaching the powers of finite information and control to manage.

Thus it is worth considering how the scene of quantification theory might be shifted nearer to familiar ground, toward the predicates themselves which represent our continuing acquaintance with phenomena.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Introduction to Inquiry Driven Systems
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 9

Inquiry and Analogy • Dewey’s “Sign of Rain” • An Example of Inquiry

We turn again to Dewey’s vignette, tracing figures of logic on grounds of semiotic.

A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something suggested. The pedestrian feels the cold; he thinks of clouds and a coming shower.

(John Dewey, How We Think, 6–7)

Inquiry and Inference

If we follow Dewey’s “Sign of Rain” example far enough to consider the import of thought for action, we realize the subsequent conduct of the interpreter, progressing up through the natural conclusion of the episode — the quickening steps, seeking shelter in time to escape the rain — all those acts form a series of further interpretants, contingent on the active causes of the individual, for the originally recognized signs of rain and the first impressions of the actual case. Just as critical reflection develops the associated and alternative signs which gather about an idea, pragmatic interpretation explores the consequential and contrasting actions which give effective and testable meaning to a person’s belief in it.

Figure 10 charts the progress of inquiry in Dewey’s “Sign of Rain” example according to the stages of reasoning identified by Peirce, focusing on the compound or mixed form of inference formed by the first two steps.

• Step 1 is Abductive, abstracting a Case from the consideration of a Fact and a Rule.
  • In the Current situation the Air is cool.
  • Just Before it rains, the Air is cool.
  • The Current situation is just Before it rains.

• The Current situation is just Before it rains.
• Just Before it rains, a Dark cloud will appear.
• In the Current situation, a Dark cloud will appear.

What precedes is nowhere near a complete analysis of Dewey’s example, even so far as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the inquiry process, but perhaps it will do for a start.

References

• Some passages adapted from:
  Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52. Archive. Journal. Online (doc) (pdf).
• Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY. Online.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Introduction to Inquiry Driven Systems • Inquiry
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 8

Inquiry and Analogy • Dewey’s “Sign of Rain” • An Example of Inquiry

To illustrate the role of sign relations in inquiry we begin with Dewey’s elegant and simple example of reflective thinking in everyday life.

A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something suggested. The pedestrian feels the cold; he thinks of clouds and a coming shower.

(John Dewey, How We Think, 6–7)

Inquiry and Interpretation

In Dewey’s narrative we can identify the characters of the sign relation as follows. Coolness is a Sign of the Object rain, and the Interpretant is the thought of the rain’s likelihood. In his description of reflective thinking Dewey distinguishes two phases, “a state of perplexity, hesitation, doubt” and “an act of search or investigation” (p. 9), comprehensive stages which are further refined in his later model of inquiry.

Reflection is the action the interpreter takes to establish a fund of connections between the sensory shock of coolness and the objective danger of rain by way of the impression rain is likely. But reflection is more than irresponsible speculation. In reflection the interpreter acts to charge or defuse the thought of rain (the probability of rain in thought) by seeking other signs this thought implies and evaluating the thought according to the results of that search.

Figure 9 shows the semiotic relationships involved in Dewey’s story, tracing the structure and function of the sign relation as it informs the activity of inquiry, including both the movements of surprise explanation and intentional action. The labels on the outer edges of the semiotic triple suggest the significance of signs for eventual occurrences and the correspondence of ideas with external orientations. But there is nothing essential about the dyadic role distinctions they imply, as it is only in special or degenerate cases that their shadowy projections preserve enough information to determine the original sign relation.

References

• Some passages adapted from:
  Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52. Archive. Journal. Online (doc) (pdf).
• Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY. Online.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Introduction to Inquiry Driven Systems • Inquiry
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 7

Inquiry and Analogy • Peirce’s Formulation of Analogy • Version 2

C.S. Peirce • “A Theory of Probable Inference” (1883)

The formula of the analogical inference presents, therefore, three premisses, thus:

are a random sample of some undefined class of whose characters are samples,

We have evidently here an induction and an hypothesis followed by a deduction; thus:

(Peirce, CP 2.733, with a few changes in Peirce’s notation to facilitate comparison between the two versions)

Figure 8 shows the logical relationships involved in the above analysis.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 6

Inquiry and Analogy • Peirce’s Formulation of Analogy • Version 1

C.S. Peirce • “On the Natural Classification of Arguments” (1867)

The formula of analogy is as follows:

are taken at random from such a class that their characters at random are such as

Such an argument is double. It combines the two following:

Owing to its double character, analogy is very strong with only a moderate number of instances.

(Peirce, CP 2.513, CE 2, 46–47)

Figure 7 shows the logical relationships involved in the above analysis.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 5

Inquiry and Analogy • Aristotle’s “Paradigm” • Reasoning by Analogy

Aristotle examines the subject of analogical inference or “reasoning by example” under the heading of the Greek word παραδειγμα, from which comes the English word paradigm. In its original sense the word suggests a kind of “side-show”, or a parallel comparison of cases.

We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third.

E.g., let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, e.g., that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad.

Aristotle, “Prior Analytics” 2.24, Hugh Tredennick (trans.)

Figure 6 shows the logical relationships involved in Aristotle’s example of analogy.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 4

Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning

Peirce’s notion of abductive reasoning is derived from Aristotle’s treatment of it in the Prior Analytics. Aristotle’s discussion begins with an example which may seem incidental but the question and its analysis are echoes of the investigation pursued in one of Plato’s Dialogue, the Meno. It concerns nothing less than the possibility of knowledge and the relationship between knowledge and virtue, or between their objects, the true and the good. It is not just because it forms a recurring question in philosophy, but because it preserves a close correspondence between its form and its content, that we shall find this example increasingly relevant to our study.

We have Reduction (απαγωγη, abduction): (1) when it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet nevertheless is more probable or not less probable than the conclusion; or (2) if there are not many intermediate terms between the last and the middle; for in all such cases the effect is to bring us nearer to knowledge.

(1) E.g., let A stand for “that which can be taught”, B for “knowledge”, and C for “morality”. Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if BC is not less probable or is more probable than AC, we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that AC is true.

(2) Or again we have reduction if there are not many intermediate terms between B and C; for in this case too we are brought nearer to knowledge. E.g., suppose that D is “to square”, E “rectilinear figure”, and F “circle”. Assuming that between E and F there is only one intermediate term — that the circle becomes equal to a rectilinear figure by means of lunules — we should approximate to knowledge.

Aristotle, “Prior Analytics” 2.25, Hugh Tredennick (trans.)

A few notes on the reading may be helpful. The Greek text seems to imply a geometric diagram, in which directed line segments AB, BC, AC indicate logical relations between pairs of terms taken from A, B, C. We have two options for reading the line labels, either as implications or as subsumptions, as in the following two paradigms for interpretation.

In the latter case, is read as that is, or

The method of abductive reasoning bears a close relation to the sense of reduction in which we speak of one question reducing to another. The question being asked is “Can virtue be taught?” The type of answer which develops is as follows.

If virtue is a form of understanding, and if we are willing to grant that understanding can be taught, then virtue can be taught. In this way of approaching the problem, by detour and indirection, the form of abductive reasoning is used to shift the attack from the original question, whether virtue can be taught, to the hopefully easier question, whether virtue is a form of understanding.

The logical structure of the process of hypothesis formation in the first example follows the pattern of “abduction to a case”, whose abstract form is diagrammed and schematized in Figure 5.

The sense of the Figure is explained by the following assignments.

Abduction from a Fact to a Case proceeds according to the following schema.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu (1) (2)
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 3

Inquiry and Analogy • Comparison of the Analyses

The next two Figures will be of use when we turn to comparing the three types of inference as they appear in the respective analyses of Aristotle and Peirce.

Types of Reasoning in Transition

Types of Reasoning in Peirce

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu (1) (2)
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 2

Inquiry and Analogy • Three Types of Reasoning

Types of Reasoning in C.S. Peirce

Peirce gives one of his earliest treatments of the three types of reasoning in his Harvard Lectures of 1865 “On the Logic of Science”. There he shows how the same proposition may be reached from three directions, as the result of an inference in each of the three modes.

We have then three different kinds of inference.

• Deduction or inference à priori,
• Induction or inference à particularis,
• Hypothesis or inference à posteriori.

(Peirce, CE 1, 267).

• If I reason that certain conduct is wise because it has a character which belongs only to wise things, I reason à priori.
• If I think it is wise because it once turned out to be wise, that is, if I infer that it is wise on this occasion because it was wise on that occasion, I reason inductively [à particularis].
• But if I think it is wise because a wise man does it, I then make the pure hypothesis that he does it because he is wise, and I reason à posteriori.

(Peirce, CE 1, 180).

Suppose we make the following assignments.

Recognizing a little more concreteness will aid understanding, let us make the following substitutions in Peirce’s example.

The converging operation of all three reasonings is shown in Figure 2.

The common proposition concluding each argument is AC, contributing to charity is wise.

• Deduction could have obtained the Fact AC from the Rule AB, benevolence is wisdom, along with the Case BC, contributing to charity is benevolent.
• Induction could have gathered the Rule AC, contributing to charity is exemplary of wisdom, from the Fact AE, the act of earlier today is wise, along with the Case CE, the act of earlier today was an instance of contributing to charity.
• Abduction could have guessed the Case AC, contributing to charity is explained by wisdom, from the Fact DC, contributing to charity is done by this wise man, and the Rule DA, everything wise is done by this wise man. Thus, a wise man, who does all the wise things there are to do, may nonetheless contribute to charity for no good reason and even be charitable to a fault. But on seeing the wise man contribute to charity it is natural to think charity may well be the mark of his wisdom, in essence, that wisdom is the reason he contributes to charity.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • 1

Inquiry and Analogy • Three Types of Reasoning

Types of Reasoning in Aristotle

Figure 1 gives a quick overview of traditional terminology I’ll have occasion to refer to as discussion proceeds.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Functional Logic • Inquiry and Analogy • Preliminaries

Functional Logic • Inquiry and Analogy

This report discusses C.S. Peirce’s treatment of analogy, placing it in relation to his overall theory of inquiry. We begin by introducing three basic types of reasoning Peirce adopted from classical logic. In Peirce’s analysis both inquiry and analogy are complex programs of logical inference which develop through stages of these three types, though normally in different orders.

Note on notation. The discussion to follow uses logical conjunctions, expressed in the form of concatenated tuples and minimal negation operations, expressed in the form of bracketed tuples as the principal expression-forming operations of a calculus for boolean-valued functions, that is, for propositions. The expressions of this calculus parse into data structures whose underlying graphs are called cacti by graph theorists. Hence the name cactus language for this dialect of propositional calculus.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

Information = Comprehension × Extension • Comment 7

Let’s stay with Peirce’s example of inductive inference a little longer and try to clear up the more troublesome confusions tending to arise.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

One final point needs to be stressed. It is important to recognize the disjunctive term itself — the syntactic formula “neat, swine, sheep, deer” or any logically equivalent formula — is not an index but a symbol. It has the character of an artificial symbol which is constructed to fill a place in a formal system of symbols, for example, a propositional calculus. In that setting it would normally be interpreted as a logical disjunction of four elementary propositions, denoting anything in the universe of discourse which has any of the four corresponding properties.

The artificial symbol “neat, swine, sheep, deer” denotes objects which serve as indices of the genus herbivore by virtue of their belonging to one of the four named species of herbivore. But there is in addition a natural symbol which serves to unify the manifold of given species, namely, the concept of a cloven‑hoofed animal.

As a symbol or general representation, the concept of a cloven‑hoofed animal connotes an attribute and connotes it in such a way as to determine what it denotes. Thus we observe a natural expansion in the connotation of the symbol, amounting to what Peirce calls the “superfluous comprehension”, the information added by an “ampliative” or synthetic inference.

In sum we have sufficient information to motivate an inductive inference, from the Fact and the Case to the Rule

Reference

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

• This Blog • Survey of Pragmatic Semiotic Information
• OEIS Wiki • Information = Comprehension × Extension
• C.S. Peirce • Upon Logical Comprehension and Extension

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Ma^thstodon • Academia.edu
cc: Research Gate

#Abduction #CSPeirce #Comprehension #Deduction #Extension #Hypothesis #IconIndexSymbol #Induction #Inference #InformationComprehensionExtension #Inquiry #Intension #Logic #PeirceSCategories #PragmaticSemioticInformation #Pragmatism #ScientificMethod #Semiotics #SignRelations

Information = Comprehension × Extension • Comment 6

Re: Information = Comprehension × Extension • Comment 2

Returning to Peirce’s example of inductive inference, let’s try to get a clearer picture of why he connects it with disjunctive terms and indicial signs. At this point in time I can’t say I’m entirely satisfied with my understanding of the relationship between disjunctive terms, indicial signs, and inductive inferences as presented by Peirce in his early accounts. What follows is just one of the simplest and least question‑begging attempts at rational reconstruction I’ve been able to devise.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

If there is any distinguishing feature shared by all the instances under the disjunctive description “neat, swine, sheep, deer” then sign users may take that feature as a predictor of being herbivorous, precisely because all the things under the disjunctive description are herbivorous. But everything under the disjunctive description is cloven‑hoofed, so the cases under the disjunctive description serve to indicate, support, or witness the utility of the induction from cloven‑hoofed to herbivorous.

Reference

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

• This Blog • Survey of Pragmatic Semiotic Information
• OEIS Wiki • Information = Comprehension × Extension
• C.S. Peirce • Upon Logical Comprehension and Extension

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Ma^thstodon • Academia.edu
cc: Research Gate

#Abduction #CSPeirce #Comprehension #Deduction #Extension #Hypothesis #IconIndexSymbol #Induction #Inference #InformationComprehensionExtension #Inquiry #Intension #Logic #PeirceSCategories #PragmaticSemioticInformation #Pragmatism #ScientificMethod #Semiotics #SignRelations

Inquiry Into Inquiry

2024-10-12 12:45:16

Information = Comprehension × Extension • Comment 2

Let’s examine Peirce’s second example of a disjunctive term — neat, swine, sheep, deer — within the style of lattice framework we used before.

Hence if we find out that neat are herbivorous, swine are herbivorous, sheep are herbivorous, and deer are herbivorous; we may be sure that there is some class of animals which covers all these, all the members of which are herbivorous. (468–469).

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol. But suppose we know of no symbol for neat, swine, sheep, and deer except cloven‑hoofed animals. (469).

This is apparently a stock example of inductive reasoning Peirce is borrowing from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omnivores.

In view of the analogical symmetries the disjunctive term shares with the conjunctive case, we can run through this example in fairly short order. We have the following four terms.

Suppose is the logical disjunction of the above four terms.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Here we have a situation which is dual to the structure of the conjunctive example. There is a gap between the logical disjunction in lattice terminology, the least upper bound of the disjoined terms, and what we might regard as the natural disjunction or natural lub of those terms, namely, cloven‑hoofed.

Once again, the sheer implausibility of imagining the disjunctive term would ever be embedded exactly as such in a lattice of natural kinds leads to the evident naturalness of the induction to the implication namely, the rule that cloven‑hoofed animals are herbivorous.

Reference

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

• This Blog • Survey of Pragmatic Semiotic Information
• OEIS Wiki • Information = Comprehension × Extension
• C.S. Peirce • Upon Logical Comprehension and Extension

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Ma^thstodon • Academia.edu
cc: Research Gate

#Abduction #CSPeirce #Comprehension #Deduction #Extension #Hypothesis #IconIndexSymbol #Induction #Inference #InformationComprehensionExtension #Inquiry #Intension #Logic #PeirceSCategories #PragmaticSemioticInformation #Pragmatism #ScientificMethod #Semiotics #SignRelations

Jon Awbrey (@Inquiry@mathstodon.xyz)

Information = Comprehension × Extension • Preamble • https://inquiryintoinquiry.com/2024/10/04/information-comprehension-x-extension-preamble/ Eight summers ago I hit on what struck me as a new insight into one of the most recalcitrant problems in P…

^Mathstodon

Information = Comprehension × Extension • Comment 5

Let’s stay with Peirce’s example of abductive inference a little longer and try to clear up the more troublesome confusions tending to arise.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

One thing needs to be stressed at this point. It is important to recognize 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 with all six of the corresponding properties. The symbol denotes objects which may be taken as icons of oranges by virtue of their bearing those six properties in common with oranges. But there are no objects denoted by the symbol which 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 which have all the properties indicated.

The above analysis provides another way to understand the abductive inference from the Fact and the Rule to the Case The lack of any cases which are and not is expressed by the implication Taking this together with the Rule gives the logical equivalence But this reduces the Case to the Fact 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 which enjoy their own levels of plausibility? That must remain an open question at this point.

Reference

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

• This Blog • Survey of Pragmatic Semiotic Information
• OEIS Wiki • Information = Comprehension × Extension
• C.S. Peirce • Upon Logical Comprehension and Extension

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Ma^thstodon • Academia.edu
cc: Research Gate

#Abduction #CSPeirce #Comprehension #Deduction #Extension #Hypothesis #IconIndexSymbol #Induction #Inference #InformationComprehensionExtension #Inquiry #Intension #Logic #PeirceSCategories #PragmaticSemioticInformation #Pragmatism #ScientificMethod #Semiotics #SignRelations

Information = Comprehension × Extension • Comment 4

Re: Information = Comprehension × Extension • Comment 3

Many things still puzzle me about Peirce’s account at this point. The question marks I added to the Figures of the previous post indicate the node labels I have remaining doubts about. For example, in Figure 3, is really an icon of object Again, in Figure 4, is really an index of object There is nothing for it but returning to Peirce’s text and trying again to follow his reasoning.

Let’s go back to Peirce’s example of abductive inference and try to get a clearer picture of why he connects it with conjunctive terms and iconic signs.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

The relationship between conjunctive terms and iconic signs may be understood along the following lines. If there is anything with 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 because it shares those properties with an orange. But the only natural examples of things with all those properties are oranges themselves, so the only thing qualified to serve as a natural icon of an orange by virtue of those very properties is that orange itself or another orange.

Reference

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

• This Blog • Survey of Pragmatic Semiotic Information
• OEIS Wiki • Information = Comprehension × Extension
• C.S. Peirce • Upon Logical Comprehension and Extension

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Ma^thstodon • Academia.edu
cc: Research Gate

#Abduction #CSPeirce #Comprehension #Deduction #Extension #Hypothesis #IconIndexSymbol #Induction #Inference #InformationComprehensionExtension #Inquiry #Intension #Logic #PeirceSCategories #PragmaticSemioticInformation #Pragmatism #ScientificMethod #Semiotics #SignRelations

Inquiry Into Inquiry

2024-10-13 16:36:58

Information = Comprehension × Extension • Comment 3

Peirce identifies inference with a process he describes as symbolization. Let us consider what that might imply.

I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information. (467).

Even if it were only a rough analogy between inference and symbolization, a principle of logical continuity, what is known in physics as a correspondence principle, would suggest parallels between steps of reasoning in the neighborhood of exact inferences and signs in the vicinity of genuine symbols. This would lead us to expect a correspondence between degrees of inference and degrees of symbolization extending from exact to approximate (non‑demonstrative) inferences and from genuine to approximate (degenerate) symbols.

For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.

In the first place there are likenesses or copies — such as statues, pictures, emblems, hieroglyphics, and the like. Such representations stand for their objects only so far as they have an actual resemblance to them — that is agree with them in some characters. The peculiarity of such representations is that they do not determine their objects — they stand for anything more or less; for they stand for whatever they resemble and they resemble everything more or less.

The second kind of representations are such as are set up by a convention of men or a decree of God. Such are tallies, proper names, &c. The peculiarity of these conventional signs is that they represent no character of their objects.

Likenesses denote nothing in particular; conventional signs connote nothing in particular.

The third and last kind of representations are symbols or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all words and all conceptions. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol. (467–468).

In addition to Aristotle, the influence of Kant on Peirce is very strongly marked in these earliest expositions. The invocations of “conceptions of the understanding”, the “use of concepts” and thus of symbols in reducing the manifold of extension, and the not so subtle hint of the synthetic à priori in Peirce’s discussion, not only of natural kinds but also of the kinds of signs leading up to genuine symbols, can all be recognized as pervasive Kantian themes.

In order to draw out these themes and see how Peirce was led to develop their leading ideas, let us bring together our previous Figures, abstracting from their concrete details, and see if we can figure out what is going on.

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

Reference

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

• This Blog • Survey of Pragmatic Semiotic Information
• OEIS Wiki • Information = Comprehension × Extension
• C.S. Peirce • Upon Logical Comprehension and Extension

cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Ma^thstodon • Academia.edu
cc: Research Gate

#Abduction #CSPeirce #Comprehension #Deduction #Extension #Hypothesis #IconIndexSymbol #Induction #Inference #InformationComprehensionExtension #Inquiry #Intension #Logic #PeirceSCategories #PragmaticSemioticInformation #Pragmatism #ScientificMethod #Semiotics #SignRelations

Jon Awbrey (@Inquiry@mathstodon.xyz)

Information = Comprehension × Extension • Preamble • https://inquiryintoinquiry.com/2024/10/04/information-comprehension-x-extension-preamble/ Eight summers ago I hit on what struck me as a new insight into one of the most recalcitrant problems in P…

^Mathstodon