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Deductions Not Necessarily Based on Propositions

A deduction is usually taken to be a transition, a passage, from the premises to the conclusion. To simplify matters, let us assume that the premises, which are finitely many in numbers, have all been collected with the help of a connective like conjunction into a single one. Henceforth, until the last section, we will speak only about deductions that are a transition from one premise to one conclusion. Such deductions can mimic all the others.

The terms transition and passage in the preceding paragraph are far from being completely clear, and we shall return to them later, at the beginning of the third section. For the time being, let us concentrate on the premise and the conclusion. These are usually taken to be propositions, and by that is meant pieces of language that can be asserted. So it seems that our, rather common, characterization of deduction presupposes that we have propositions. Hence deducing presupposes asserting.

Could one imagine a deduction where one would pass from something that is not a proposition as a premise to something that is not a proposition as a conclusion? A deduction from a command to a command, or a deduction from a question to a question? Or, non-uniformly, a deduction from a proposition to a command, or from a command to a question? (In [24] and references therein one may find a defence of deductions where commands occur as premises and conclusions together with propositions. For the logic of questions, one may consult [16], and references therein; I don't know however of a reference dealing explicitly with deductions where questions occur as premises and conclusions, together perhaps with propositions or commands.)

Let us take a brief look at the uniform deductions, from a command to a command, or from a question to a question. Kolmogorov's contribution in [17] to the interpretation of intuitionistic logic that bears his name, besides those of Brouwer and Heyting, suggests that we should understand a deduction in constructive mathematics as taking us not from a proposition to a proposition, but from a problem to a problem. A problem however is something that does not seem to be necessarily tied with asserting. When the solution of a problem is expressed by a proposition, the statement of that problem may be a command, or a question. If the solution of our

problem is “For x , y and z being respectively 3, 4 and 5 we have x 2 + y2 = z2”, then

the problem could be the command “Find three natural numbers x , y and z such that x 2 + y2 = z2!” or the question “Are there three natural numbers x , y and z such that x 2 + y2 = z2?”.

Kolmogorov's examples of problems in [17] are expressed by commands, but it seems that they could equally well be expressed by questions. It is not however clear in these examples that the solution should be expressed by a proposition rather than by producing one or several objects, i.e. by naming them.

From x 2 + y2 = z2 one can deduce (z+x )(zx ) = y2. Can't we say therefore that from the command “Find three natural numbers x , y and z such that x 2 + y2 = z2!” as a premise one can deduce the command “Find three natural numbers x , y and z such that (z + x )(zx ) = y2!” as a conclusion? Making one command would yield making the other. The second command would follow from the first, it could be inferred from it. And can't we say that from the question “Are there three natural numbers x , y and z such that x 2 + y2 = z2?” as a premise one can deduce the question “Are there three natural numbers x , y and z such that (z + x )(zx ) = y2?” as a conclusion? Making one question would yield making the other. The second question would follow from the first, it could be inferred from it.

To make such a deduction with commands it is not necessary to assume that the command in the premise is actually made, as in deductions with propositions it is not necessary to assume that the premise is actually asserted. Analogously, to make such a deduction with questions it is not necessary to assume that the question in the premise is actually put.

To make a deduction with propositions it does not matter whether the premise is true or not. The premise being false does not invalidate the deduction. It would invalidate it as a proof, if the deduction was proposed as a proof of the conclusion. As a deduction simpliciter, it is however perfectly legitimate with a false premise. Analogously, to make a deduction with commands it would not matter whether the premise can be fulfilled or not. The premise being impossible to fulfil would not invalidate the deduction. With propositions the deduction may serve to show that the premise is false because it yields a false conclusion, as in reductio ad absurdum. With commands, the deduction might serve to show that the premise cannot be fulfilled, because it yields a conclusion that cannot be fulfilled.

Commands here are assumed to have two fulfilment values: can be fulfilled and cannot be fulfilled, but it is not clear that therefore the logic of commands should be taken as fulfilment-functional and two-valued. The negation of a problem p need not be interpreted as it is not possible to fulfil p, but as from the assumption that p can be fulfilled one can derive a contradiction, which is in tune with intuitionistic logic (see [17]). The implication of that logic may be tied to deduction, and hence it would be intuitionistic.

Can the notion of deduction be widened so as to cover also non-uniform deductions involving propositions, commands and questions, like those mentioned above? Not all of these deductions need make sense. It is indeed not easy to see what would be a deduction from a command to a question. It could however again be a transition from a problem to a problem, as suggested by Kolmogorov. Deductions from propositions to commands, and vice versa, from commands to propositions, are easier to conceive, and have been examined in [24].

We shall next consider a matter that would extend even more the range of the application of the word deduction, and go beyond the linguistic sphere. We would thereby transcend its widest application in this sphere.

Can one make deductions involving non-linguistic entities as premises or conclusions? Could one take as a premise the perception of something small a and something big b, and deduce from that as a conclusion the proposition that a is smaller than b? Can this transition from a perception to a proposition be called a deduction? And can one deduce from a proposition a perception, not of something external, but a mental image? And can one deduce one mental image from another mental image? Why should this widening of the application of the word deduction to the non-linguistic sphere represent a danger for the mathematical theory of deduction, which it is the duty of logic to formulate and investigate, and which we will consider in the next two sections?

We will not go so far as to claim that the premise and conclusion of a deduction can be anything. It seems that one could take a name as a premise or a conclusion of a deduction only in an elliptical sense. From the context one can find the proposition involving the name for which the name stands. Without that context, from a pure name, it is not clear that one could deduce anything. (Kolmogorov's solutions that are objects, which we mentioned above, could be taken as being solutions in an analogous elliptical sense.)

It does not seem unreasonable to claim however that premises and conclusions can be other things than propositions. Formulae with free variables are tied to propositions, but are not strictly speaking propositions[1]. These things, and things like commands and questions, may perhaps be tied in some way with propositions— though they are not propositions, they are somehow in the same field. On the other hand, the connection with propositions in the case of perceptions and mental images becomes less clear. Are they too in the field of propositions?

  • [1] I am grateful to Thomas Piecha for suggesting this
 
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