**by Axel Arturo Barceló Aspeitia**

Let M and E be *means* and *ends, *and let C be the property of there being a *connection *between ends and means such that whether the means were appropriate or not (at least partially) explains why the end was achieved or not. In other words, C means that no luck was involved, i.e., whether the end was achieved or not was not a matter of luck.

## The Metaphysical Space

*This *is easily represented by a simple formula (well, actyally, by many equivalent formulas) in propositional logic:

B [Basic Law] = C ⊃ (M ≡ E)

Since this formula is equivalent to the diagram, this synthesizes the metaphysical relations between means, ends and luck necessary to model falibility, Gettier cases, etc. and, as therefore, it should be considered the basic law of fallible systems. However, it is very difficult to paraphrase in simple English. What it says is that the issue of whether a proper connection exists between ends and means such that whether the means were appropriate or not (at least partially) explains why the end was achieved or not only occurs either when the end is achieved and the means are appropriate or when the end is not achieved and the ends were not the appropriate ones. This is not only a mouthful, but a very complex sentence (and I doubt I have made a good work of conveying *B* in English at all!). However, looking at the formula (and the table) above, we see that the relation is actually quite simple (once properly represented).

Thus if we take the formula above is the only axiom, we can get as simple theorems most of the basic properties of falible systems. For example: take processes of belief formation to be the means we get to reach the truth. Under this assumption,we can interpret the variables, *M*, *E* to mean *epistemic justification* and *truth* respectively (and consequently, *C* would be the absence of epistemic luck). Thus, from the above fundamental axiom of falible epistemic justification), we easily get fallibility as theorem:

B, M ⊬ E

i.e., that justification is fallible. Furthermore, if we adopt an anti-luck definition of knowledge, i.e., that knowledge is justified true belief plus the absence of luck,

K = C ∧ M ∧ E

we also get the Gettier theorem, i.e., that knowledge is not justified true belief :

B, M, E ⊬ K

and that Gettier implies luck, i.e., that in all Gettier cases, some luck is involved:

B ⊢(B ∧ M, ∧ E ∧ ¬ K) ⊃ (¬C).

Furthermore, we also get Zagzebski's theorem, i.e., that no other falible condition can be added to justification and truth that would yield knowledge.

If B, F ⊬ E, then B, M, F, E ⊬ K

i.e., justified true belief plus any falible property is not enough for knowledge either.

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