##### K. P. Mohanan

**CONJECTURES/PREDICTIONS**

A knowledge claim is a proposition that we believe to be true, but have not proved yet. To be considered as part of Academic knowledge, knowledge claims must be accompanied by proofs that demonstrate the truth of the claims.

If a knowledge claim is expected to be proved by demonstrating that it is deducible from the theoretical premises/assumptions of a theory (its axioms/laws, and definitions) it is called a conjecture. In mathematics, a conjecture that has been proved is called a theorem, and in science, it is called a prediction. Whether we use the term theorem or prediction to denote a true conjecture, it is important is that the alleged theorem/prediction (= conjecture) be proved to be a true.

To prove that a conjecture is true, we need to demonstrate that it is a logical consequence of the premises of the theory, by providing a valid proof, that is, by deriving it from the premises through a set of steps that we call a derivation, where each step is sanctioned by the previous steps through a rule of inference in deductive logic.

**OBSERVATIONAL GENERALISATIONS**

Unlike mathematical theories, scientific theories are empirical. That means that scientific theories must be proved to be true on the basis of observational generalisations which in turn must be proved to be true on the basis of observational reports.

An observational Generalisation on a population is a proposition that holds on the entire population. When the population is extremely large or new members can be added to it (as in the case of the population of human beings), we cannot take every member in the population and show that the generalisation is true for that member. Hence, the logic used for proving alleged observational generalisations is that of sample-to-population inductive logic from a finite sample (not to be confused with mathematical induction which is a method of deductive proof.)

**PROVING THEORIES**

One of the epistemic norms of mathematics is that for a knowledge claim in a theory to be called be called a theorem, we must prove it by deducing it from the premises of the theory in question. That is to say, its conjectures must be proved. There is no requirement that theories be proved to be true.

In contrast, there are three kinds of knowledge claims in science.

Type 1: Knowledge claim as a conjecture/prediction within a given theory.

Type 2: Knowledge claim as an observational generalisation.

Type 3: Knowledge claim as a theoretical premise/assumption from which the predictions are derived.

That means there are three kinds of proofs in science, corresponding to each type of proof. The proof for type 1 claims uses deductive logic, the proof for type 2 claims uses inductive logic, and the proof for type 3 claims uses abductive logic. The epistemic norm for proving a theory to be true calls for the following demonstrations:

1) There exist a set of alleged predictions, and prove them to be true.

2) there exist a set of alleged observational generalisations and prove them to be true.

3) the true predictions and the true observational generalisations are not logically inconsistent, and

4) the theory we wish to defend is better than its alternatives.

There is something that is potentially confusing in the use of the expression â€˜true predictionâ€™ in the context of both (1) and (3). In the context of (1), an alleged prediction is true iff it is derived from the premises of the theory through valid deductive reasoning. In the context of (3), a prediction is true iff there are corresponding observational generalisations which have been proved to be true, and the two are not logically contradictory. We are using the same term â€˜trueâ€™ for two distinct concepts of truth.

To avoid this confusion, it might be useful to say valid prediction for a prediction derived through a valid derivation from the premises (requirement 1), and correct prediction for a valid prediction that agrees with a true observational generalisation (requirement 3).

A theorem/prediction that has been proved to be valid cannot be disproved unless it is shown that the alleged proof is not valid. In contrast, a scientific theory that has been proved to be true through a valid proof can be disproved subsequently by showing

that there exist better alternatives (step 4) or

that it is defective/flawed by proving that some of its predictions are incorrect (step 3).

Needless to say, a flaw in the derivation of any of these proofs also lowers our confidence in the theory, theorem/prediction, and observational generalisation under scrutiny.

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