This is a subtle issue. It goes to the heart of the difference between math and physics.
That A implies B does not entail that B implies A. One encounters frequently the errant reasoning that it does even among engineering students in the university (yesterday a student told me that because a matrix was diagonalizable it must be symmetric).
However, that A implies B and one has observed B provides evidence for believing A. This statement is somehow the basis for the scientific method.
In the example given in the original post, that Maria bought ice-cream today does provide some evidence that she might have gone to the beach today.
The difference is that between deductive reasoning and inference.
Inferential reasoning is more common, more natural, and more powerful (why does anyone believe Euclid's axioms?).
I quote from V. I. Arnold (Translation of the V. I. Arnold paper “From Superpositions to KAM Theory” (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740)), who in his inimitably polemical manner explains the issue better than I can anyway:
Now it became possible to apply the techniques developed in the
problem of adiabatic invariants. As soon as I accomplished that,
Kolmogorov suggested that I should submit the paper on perpetual
adiabatic invariance to ZhETF, the main physical journal in the
USSR. A few weeks later, M. A. Leontovich (who was, as far as I
remember, a deputy to the editor-in-chief of ZhETF) invited me to
his home (near the Atomic Energy Institute of the USSR Academy of
Sciences) to discuss the manuscript. Having fed me, as usual, by
boiled buckwheat and calling me, as usual, “Dimka” (M. A. called me
in such a way until his death), Mikhail Aleksandrovich explained to me
that the paper could not be published in ZhETF due to the following
- The manuscript contained the words “theorem” and “proof” forbidden in ZhETF.
- The manuscript claimed that “A implies B” while every physicist knew examples showing that B does not imply A.
- The manuscript used the unintelligible terms “Lebesgue measure”, “invariant tori”, “Diophantine conditions”. Mikhail Aleksandrovich
therefore proposed that I should rewrite the paper.
Now I realize how
right he was in defending a physical journal from the Bourbaki-like
mathematical jargon. For instance, indeed, while claiming that “A
implies B” the author must point out explicitly whether the converse
holds, otherwise any reader not spoiled by the mathematical slang
would understand the claim as “A is equivalent to B”.
The issue is that mathematical deductive reasoning is rarely if ever applicable outside of mathematics where it should be replaced by the Bayesian paradigm of what Polya called plausible inference, of which it is an extreme case.
The moral for teaching is that the difference between necessary and sufficient conditions is not something to be passed over lightly and that confusion in regards to it is not necessarily evidence of stupidity. When one teaches that all symmetric matrices are diagonalizable one must remind students that not all diagonalizable matrices are symmetric. Moreover, all orthogonally diagonalizable matrices are symmetric ...