Logical impossibilities

What does impossible mean? Other than ‘not-possible’. Logical impossibility is the topic of this essay. More specifically we’re going to explore the two distinct meanings of logical impossible that I briefly mentioned in a previous essay A journey into possibility land.1

The two logical impossibilities

The two kinds of logical impossibility are absolute and relative (aka. hypothetical) logical impossibility. Absolute logical impossibility means a contradiction or something that implies a contradiction and I contend that this was the first meaning of logical impossibility. The second I contend arose out of confusion and perhaps errors using modal terms and modal reasoning. Why do I contend this? It seems like a good theory to explain the usage of relative logical impossibility.

Consider a simple reductio argument:

n Proposition Explanation
1 P Assumption
2 P→Q Premise
3 P→¬Q Premise
4 Q From 1, 2, MP
5 ¬Q From 1, 3, MP
6 Q∧¬Q From 4, 5, conj. Contradiction.
7 ¬◊P Reductio

Such a reductio that ends with the assumption being proven impossible is quite common, but in what sense is P is impossible? Certainty not in the absolute sense since P itself does not imply a contradiction. What does, then, imply a contradiction (and is thus impossible)? It is the the conjunction of all the premises and the assumption (call this a superconjunction) that implies a contradiction and is impossible. Formally:

F1. (P∧[P→Q]∧[P→¬Q])→(Q∧¬Q)

No single premise or assumption in the argument implies a contradiction and thus no single premise or assumption is impossible. The conclusion in such typical reductio arguments does not follow. What does follow in the above reductio is ¬P.

What happened, then, in our reasoning since we moved the modal operator ‘impossible’ from the superconjunction to the assumption? I don’t know. I suppose people were just sloppy with their reasoning. But people have persisted and persist to move this modal operator from the superconjunction to the assumption and a whole new meaning of logically ‘impossible’ has arisen.

To recap, the theory is this: Originally logically impossible was used for contradictions and things that implied contradictions. Then people were sloppy with their reasoning with modal operators, and ‘impossible’ acquired a new meaning, that is, is a proposition in an impossible (absolute sense) superconjunction (like the one above). Thus, the concept relative logical impossibility is defined via the absolute logical impossibility concept.

I showed that sloppy modal reasoning is what happened when the logical problem of evil proved that God cannot exist.2 It doesn’t follow that God is impossible in the absolute sense only in the relative sense but people got confused and thought they have proven God to be absolutely logically impossible when they have only showed that if there is evil, then there is no God.

What use is the concept of relative logical impossibility?

Since this concept arose out of confusion or sloppy reasoning I think we should discard it. It serves no useful purpose that I can see other than to introduce more confusion because there are, if accepted, two meanings of ‘logical impossible’.

This usage of ‘impossible’ is like using ‘means’ to mean ‘the meaning of . . . is’ and ‘implies’. Such ambiguous words should not be used for serious reasoning.

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