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Formative Experience

Three Different Kinds of Inference

We have then three different kinds of inference:

Deduction or inference à priori,

Induction or inference à particularis, and

Hypothesis or inference à posteriori.

(C.S. Peirce, "On the Logic of Science" (1865), 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.

(C.S. Peirce, "On the Logic of Science" (1865), CE 1, 180).

In the roughest terms, abduction is what we use to generate a likely hypothesis or an initial diagnosis in response to a phenomenon of interest or a problem of concern, while deduction is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis, and induction is used to test the sum of the predictions against the sum of the data. These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in the knowledge or skills, in other words, an augmentation in the competence or performance, of the agent or community engaged in the inquiry. In the pragmatic way of thinking every thing has a purpose, and the purpose of any thing is the first thing that we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call 'knowledge' or 'certainty'. It needs to be appreciated that the three kinds of inference, insofar as they contribute to the end of inquiry, describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others. For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and that induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since it is not just any wild guess at explanation that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of inference realizes its purpose only in accord with its proper role in the whole cycle of inquiry. No matter how much it may be necessary to study these processes in abstraction from each other, the integrity of inquiry places strong limitations on the effective modularity of its principal components. If we then think to inquire, 'What sort of constraint, exactly, does pragmatic thinking place on our guesses?', we have asked the question that is generally recognized as the problem of 'giving a rule to abduction'. Peirce's way of answering it is given in terms of the so-called 'pragmatic maxim', and this in turn gives us a clue as to the central role of abductive reasoning in Peirce's pragmatic philosophy.

Peirce's Three Kinds of Inference

We have then three different kinds of inference:

Deduction or inference à priori,

Induction or inference à particularis, and

Hypothesis or inference à posteriori.

| We have then three different kinds of inference:
|
| Deduction  or inference 'à priori',
|
| Induction  or inference 'à particularis',
|
| Hypothesis or inference 'à posteriori'.
|
| C.S. Peirce, Chronological Edition, 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'.
|
| C.S. Peirce, Chronological Edition, CE 1, 180
|
| Charles Sanders Peirce, "Harvard Lectures On the Logic of Science (1865)",
| Writings of Charles S. Peirce:  A Chronological Edition, Volume 1, 1857-1866,
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

o---------------------------------------------------------------------o
|                                                                     |
|  D ("done by a wise man")                                           |
|   o                                                                 |
|    \*                                                               |
|     \ *                                                             |
|      \  *                                                           |
|       \   *                                                         |
|        \    *                                                       |
|         \     *                                                     |
|          \      * A ("a wise act")                                  |
|           \       o                                                 |
|            \     /| *                                               |
|             \   / |   *                                             |
|              \ /  |     *                                           |
|               .   |       o B ("benevolence", a certain character)  |
|              / \  |     *                                           |
|             /   \ |   *                                             |
|            /     \| *                                               |
|           /       o                                                 |
|          /      * C ("contributes to charity", a certain conduct)   |
|         /     *                                                     |
|        /    *                                                       |
|       /   *                                                         |
|      /  *                                                           |
|     / *                                                             |
|    /*                                                               |
|   o                                                                 |
|  E ("earlier today", a certain occasion)                            |
|                                                                     |
o---------------------------------------------------------------------o
Figure 1.  A Thrice Wise Act

Tabula Rasa

Simple Case

{| 
| Cell 1, row 1 
| Cell 2, row 1 
|- 
| Cell 1, row 2 
| Cell 2, row 2 
|}

and

{| 
| Cell 1, row 1 || Cell 2, row 1 
|- 
| Cell 1, row 2 || Cell 2, row 2 
|}

both generate

Cell 1, row 1	Cell 2, row 1
Cell 1, row 2	Cell 2, row 2

Simple Table

Table A.
abc	def	ghi
jkl	mno	pqr
stu	vwx	yz

Table B.
abc	def	ghi
jkl	mno	pqr
stu	vwx	yz

Multiplication Table

Multiplication Table
×	1	2	3
1	1	2	3
2	2	4	6
3	3	6	9
4	4	8	12
5	5	10	15

Multiplication Table
×	1	2	3
1	1	2	3
2	2	4	6
3	3	6	9
4	4	8	12
5	5	10	15

Sign Relation Table

Table 1. Sign Relation L
Object	Sign	Interpretant
jkl	mno	pqr
stu	vwx	yz

Table 2. Sign Relation L
Object	Sign	Interpretant
o_1	s_1	i_1
o_2	s_2	i_2

Table 3. Sign Relation L
Object	Sign	Interpretant
o₁	s₁	i₁
o₂	s₂	i₂
o₃	s₃	i₃
o₄	s₄	i₄
o₅	s₅	i₅

Table 4. Sign Relation L
Object	Sign	Interpretant
o₁	s₁	i₁
o₂	s₂	i₂
o₃	s₃	i₃
o₄	s₄	i₄
o₅	s₅	i₅

Table 5. Sign Relation L
Object	Sign	Interpretant
o₁	s₁	i₁
o₂	s₂	i₂
o₃	s₃	i₃
o₄	s₄	i₄
o₅	s₅	i₅

Table 6. Sign Relation L
Object	Sign	Interpretant
o₁	s₁	i₁
o₂	s₂	i₂
o₃	s₃	i₃
o₄	s₄	i₄
o₅	s₅	i₅

**Table 7. Sign Relation L**
Object	Sign	Interpretant
o₁	s₁	i₁
o₂	s₂	i₂
...	...	...
o_m	s_m	i_m

Relational Data Table

Table 1. Relational Data Table L
Domain 1	Domain 2	...	Domain k
x₁₁	x₁₂	...	x_1k
x₂₁	x₂₂	...	x_2k
...	...	...	...
x_m1	x_m2	...	x_mk

Table 2. Relational Data Table L
Domain 1	Domain 2	...	Domain k
x₁₁	x₁₂	...	x_1k
x₂₁	x₂₂	...	x_2k
...	...	...	...
x_m1	x_m2	...	x_mk

**Table 3. Relational Database**
Domain 1	Domain 2	...	Domain k
x₁₁	x₁₂	...	x_1k
x₂₁	x₂₂	...	x_2k
...	...	...	...
x_m1	x_m2	...	x_mk

**Table 4. Relational Database**
Domain 1	Domain 2	...	Domain j	...	Domain k
x₁₁	x₁₂	...	x_1j	...	x_1k
x₂₁	x₂₂	...	x_2j	...	x_2k
...	...	...	...	...	...
x_i1	x_i2	...	x_ij	...	x_ik
...	...	...	...	...	...
x_m1	x_m2	...	x_mj	...	x_mk

**Table 5. Relational Database**
Domain 1	Domain 2	...	Domain j	...	Domain k
x₁₁	x₁₂	...	x_1j	...	x_1k
x₂₁	x₂₂	...	x_2j	...	x_2k
...	...	...	...	...	...
x_i1	x_i2	...	x_ij	...	x_ik
...	...	...	...	...	...
x_m1	x_m2	...	x_mj	...	x_mk

Cross-Tabs Table

$A{\underline {\oplus }}B$		A
$A{\underline {\oplus }}B$		T	F
B	T	T	F
B	F	F	T

Six ways of looking at a sign relation

So in a triadic fact, say, the example

A gives B to C

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:

A gives B to C	A benefits C with B
B enriches C at expense of A	C receives B from A
C thanks A for B	B leaves A for C

These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).

Cell 1, row 1	Cell 2, row 1
Cell 1, row 2	Cell 2, row 2

Cell 1, row 1	Cell 2, row 1
Cell 1, row 2	Cell 2, row 2

Junkyard

$proj_{XY}(L)$ = $L_{XY}$ = {(x, y) in X × Y : (x, y, z) in L for some z in Z},

$proj_{XZ}(L)$ = $L_{XZ}$ = {(x, z) in X × Z : (x, y, z) in L for some y in Y},

$proj_{YZ}(L)$ = $L_{YZ}$ = {(y, z) in Y × Z : (x, y, z) in L for some x in X}.