Logical Foundations of Induction

By Baqir As Sadr

Preface to Online Version (2)

Introduction (3)

Part 1:
Induction and Epistemology

Chapter 1. Aristotelian Induction (6)

Chapter 2. Criticism of Aristotelian Induction (19)

Chapter 3. Induction And Empiricism (29)

Part 2:
Induction And Probability

Chapter 1. Calculus of Probability (52)

Chapter 2. The Interpretation of Probability (59)

Chapter 3. The Deductive Phase Of Induction (82)

Chapter 4. Modern Theories of Probability (96)

Chapter 5. Induction and Certainty (113)

Part 3:
Human Knowledge And Probability

Chapter 1. Classes of Statements (139)

Chapter 2. Is There A priori Knowledge? (177)

Conclusion

Preface to Online Version (2)

The basic thesis of this book is that the same logic of induction on which scientific methodology is based can be used to prove the existence of God. The implication of this work is far reaching, for it attempts to layout a unifying, common basis of research in religion, social sciences, and natural sciences. “Our Philosophy” and “The Revealer, The Messenger, and the Message”, the two other books by the same author, are very relevant in this regard and useful for a wider understanding of author’s thesis.

Introduction (3)

In this book, we try to present a reformulation of the theory of knowledge in a scientific, philosophical and objective manner based on the theory of probability so as to fill the gap in the intellectual march of man.

Part 1:
Induction and Epistemology

Chapter 1. Aristotelian Induction (6)

Meaning of Induction; Aristotle’s Perfect Induction (7); Criticism of Perfect Induction; Recapitulation; Aristotle’s imperfect Induction (10); The Problem of Induction; The Formal Logic and Problem (12); Misunderstanding of Formal Logic (13);

Aristotelian Epistemology and Induction (14) ; Formal Logic and Chance; Need of Definite Formulation (16); The Crucial Power of Difference (17)

Aristotle did not distinguish between observation and experiment, and considered induction as any inference based on enumerating particular instances, consequently, he classified induction into perfect and imperfect, if the conclusion refers to all the particulars in question, induction is perfect, if it includes reference to some particular instances only, induction is imperfect [1].

Aristotle has considered perfect induction in a way different from his consideration of imperfect induction. Induction cannot be divided, in our view, into perfect and imperfect because induction in fact proceeds from particular to universal, whereas perfect induction does not do so, but its premises are general like its conclusion. Thus, we regard perfect induction as deduction not induction; and it is imperfect induction that is induction proper. (6)

Chapter 2. Criticism of Aristotelian Induction (19)

Indefinite Knowledge (19); Genesis of indefinite Knowledge (19); Aristotelian principle and indefinite knowledge (20); First Objection (21)

There are in our ordinary state of affairs instances of knowledge of indefinite rejection: we may know that this sheet of paper is not black (and that is knowledge of definite rejection), but we may know only that the sheet cannot be black and white at the same time (and this is knowledge of indefinite rejection). The sort of knowledge which rejects something in an indefinite (or exact) way may be called indefinite knowledge, and the sort of knowledge which involves a definite rejection of something may be called definite knowledge in consequence, the Aristotelian rejection of relative chance is an instance of indefinite knowledge. (19)

Chapter 3. Induction And Empiricism (29)

Certainty Attitude (29); On the First and Third Questions (30); Discussion (31); On the second question (31); Answer to that question (32);

Probability Attitude (34); Discussions (36); Psychological Attitude (37); Examination of psychological attitude (40); (1) Belief; (2) Causality and Reason (42); (3) Causality and Experience (44); (4) Concept of Causality (45); (5) Belief in causality (46); Physiological Explanation of Induction (49);

Inductive inference faces, as has already been noted, three main problems;

(1) why should we suppose a cause of (b), excluding absolute chance for its occurrence?

(2) if there is a cause of (b), why should we suppose that (a) is its cause being concomitant with it, and not supposing that (b) is connected with (c) by relative chance?

(3) If we could make sure, by inductive process, that (a) is the cause of (b), on what ground can we generalize the conclusion that all a’s would be causes of b’s?

Formal logic solved the first and third questions by appealing to certain a priori principles on rationalistic lines, and solved the second problem by supposing another a priori principle denying the systematic repetition of relative chance. (29)

Our comments on certainty attitude concerning induction are as follows. First, the author differs from both rationalistic and empiricist answers to the second question, namely, whether induction needs causality as a necessary postulate. Both schools, though different, answer that question in the positive, while the author will say no, owing to what will be maintained in the sequel. Secondly, we agree with certainty attitude that causal principle is itself reached by induction, and thus hold that induction needs no a priori postulates. But the impasse of empiricism in our view, is that it holds that induction is grounded upon causality postulate while it holds at the same time that causal principle is itself an inductive generalization. (30)

Part 2:
Induction And Probability

Chapter 1. Calculus of Probability (52)

Axioms of the theory (52); Rules of the Calculus (54); Bernoulli’s law of large numbers (56);

We have already said that induction, in its first stage, is a sort of inference; and we shall show in this part that induction in this stage does not proceed from particular to universal and that inductive inference does not give certainty but the highest degree of probability. Thus induction in its first stage is related to the theory of probability, and it may then be well to begin with the latter. (52)

Chapter 2. The Interpretation of Probability (59)

(A) Fundamental Definition (59); The First problem (60); The Second Problem (61); (B) Probability in the Finite Frequency Theory (62); Real and Hypothetical Probabilities (63); Does this definition exhaust all probabilities? (64);

New Definition of Probability (66); A. The axioms of the new definition (68); Difficulties of our definition (70); The new definition and the calculus (70); The new definition and inverse probability (71); The definition and the Bags – example (71); Our definition and Bernoulli’s law (72); Completeness of our definition (73);

New axioms (74); Ground of Dominance Axiom (76); Categorical and Hypothetical indefinite knowledge (77); Conditional knowledge that is real (78); Recapitulation (80);

Now we come to our new definition of probability: Probability capable of determined value is always one of a class of probabilities represented in indefinite knowledge, its value is always equal to a number of items of indefinite knowledge. (67)

Chapter 3. The Deductive Phase Of Induction (82)

Causality (82); First Application (83); Rule of multiplication (85);

Induction has two phases: deductive and subjective, and we have still been concerned with the former. By deductive phase here is meant that inductive inference aims at generalizations, and in order that these be effected, induction finds help in the study of probability. But we get the highest degree of probability of our generalization in a deductive manner, that is, deduction from certain axioms and postulates. (82)

Chapter 4. Modern Theories of Probability (96)

Now, the pioneers of the theory of probability seem to have taken already the same course as we did, with the difference that they defended the deductive phase of induction even without presupposing anything concerning causality.

Laplace is one such pioneer. In what follows we shall discuss Laplace’s position and then compare it with ours. (96)

Chapter 5. Induction and Certainty (113)

This credibility is expressed by greater probability value arising from collecting a greater number of cases concerning the principle of causality. Now, the deductive conclusion in induction shows a degree of credibility of the statement, “A causes B”, and not of the principle of causality itself. Such credibility would approximate, but does not reach, certainty. (113)

Part 3:
Human Knowledge And Probability

Chapter 1. Classes of Statements (139)

Now, all inference depending on certain statements is called demonstration, but when inference depends on commonsensical and acceptable statements it is called dialectic; and when inference is arrived at from probable and authoritative statements it is called rhetorical, and when it uses false statements it is called fallacy. Thus demonstration is the only inference that is certain and always true. If we examine the principles of inference, referred to above, we shall find that most of them are not really principles but derivatives. (141)

Chapter 2. Is There A priori Knowledge? (177)

Empirical Statements (177); Formal Statements (178); Logical Positivism (180); Criticism (182); Empiricism and Meaning of Statements (183); Has knowledge Necessarily A Beginning? (186); Reichenbach’s Position (187); Russell’s Objection (187); Discussion (188);

Now, we must have before us a criterion by means of which we can compare and evaluate rationalism and empiricism. This criterion may be reached by pointing the minimal degree of belief in the truth of both formal and empirical statements. And any theory of knowledge disclaiming such criterion is doomed to failure, while it is approved when is consistent with such criterion. Now what is the minimal degree of credulity in formal and empirical statements? (177)

Therefore we have to reject empiricism owing to its failure to explain the necessity of formal statements, in favour of rationalism in this respect. Further, the empiristic dictum that sense experience is the sole source of human knowledge is not itself a logical truth, not is it itself derived from experience. It remains that this dictum is obtained a priori; if true, then empiricism admits a priori knowledge; and if it is empirical and a priori then it is probable. This implies that rationalism is probably true for empiricism [???]. (183)

CONCLUSION (190)

The object of our study in this book is two fold. [1] First, we are concerned to show the logical foundations of inductive inference which embraces all scientific inferences based on observation and experiment. In this context we have offered a new explanation of human knowledge based on inductive inference. [2] Secondly, we are interested to show certain conclusions connected with religious beliefs based on our study of induction. That is, the logical foundations of all scientific inferences based on observation and experiment are themselves the logical foundations on which a proof of the existence of God can be based. This proof is a version of the argument from design, and is inductive in its character.

Now, we have to choose the whole scientific knowledge or reject it, and then an inductive proof of the existence of God would be on the same footing as any scientific inference. Thus, we have found that science and religion are connected and consistent, having the same logical basis; and cannot be divorced. Such logical connection between the methods of science and the method of proving God’s existence may be regarded as the ground of understanding the divine direction, in the Koran, the Holy Book of Muslims, to observe the workings of the natural world.

The Koran is encouraging people to scientific knowledge on empirical grounds. And in this sense, the argument from design is preferred in Koran to other proofs of the existence of God, being akin to sense and concreteness and far from abstractions and sheer speculations.