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A WEAK QUANTUM-LIKE INDETERMINACY PRINCIPLE IN LOGIC

We resort to a logical phenomenon related to paradoxes and possibly to other logical facts, like limitation theorems and transfinite set theory, to shed light upon the meaning of Heisenberg’s Indeterminacy Principle. Our aim is to show how critical realism as opposed to naïve realism might be consistent with this principle and its empirical and theoretical consequences. Call (1) the sentence-token “(1) expresses no true proposition”. (1) has no truth value, so it describes no state-of-affairs. We partly state this by means of the sentence-token (2) which says “(1) expresses no true proposition”. (2) describes an actual state-of-affairs and is true. This is possible only because (1) and (2) are uttered in different logical contexts. Since (2) is based upon a previous assessment of (1), we say that (2) stands in a logically posterior instant. There is a state-of-affairs that is an available state-of-affairs in the logical instant to which (2) belongs but is not yet such in the logical instant in which (1) stands. Only the introduction of this kind of logical temporality makes the set of available states-of-affairs relative to the logical context, thereby rendering possible the solution to the paradox. This reveals that some logical objects are distributed along some logical temporality. The reason of this relativity resides in a fact that can be couched in the terms of Husserl’s phenomenology: no intentional act can be contained in its own intentional object. For the thinker who thinks that p, his being thinking that p is no possible intentional object and hence no possible state-of-affairs, even if it is so for any thinker that stands in a logically posterior instant. Thus, there is in logic an indeterminacy phenomenon, ultimately due to some principle of no intentional selfreference (PNS). This indeterminacy is, however, not absolute but relative to the logical contexts in which thinkers stand. We argue that the Indeterminacy Principle in physics stems from a situation akin to that of logical paradoxes and that consequently it is relative to the observer’s cognitive context. From the standpoint of a broad-sense critical realism (whose keystone is the distinction between objective reality and its phenomenal appearance) indeterminacy is only pertinent to the general way in which phenomena have to appear to certain observers, so that it implies nothing about objective reality itself. This is what we call a weak indeterminacy principle. It is often assured that the experiments revealing violations of Bell’s inequalities force us to renounce at least one of these two tenets: realism, localism. We argue that the reasoning only holds if reality could be translated without holes into its phenomenal appearance and this cannot be the case in presence of PNS. So critical realism together with PNS could make those tenets consistent with the known empirical or mathematical outcomes. In conclusion, it is suggested that an appropriate distinction between reality and phenomena, along with PNS, could turn quantum oddities more intuitive as well as compatible with critical realism. So, our main contribution could be the proposal of a reason why objective reality could not be completely reproduced in its translation into phenomenal appearance. This might in turn be of interest in the field of logic. I. Assume that Heisenberg’s Indeterminacy Principle only asserts we cannot simultaneously measure with arbitrary accuracy a particle’s momentum and position; then the principle refers to our possible knowledge and it is what I shall call a weak indeterminacy principle. Assume, on the contrary, it asserts that a particle cannot have objective momentum and position at the same time. Then it refers to the particle itself and it is what I shall call a strong indeterminacy principle. Let’s now assume that Quantum Mechanics is correct and let’s assume locality and usual logical induction as well. Then there exist some results that seem to force us to