Rescuing the Excluded Middle

This is a continuation of “She wore a blue dress“, in which we introduced to the concepts of imprecision and uncertainty. I will now turn the focus back on the imprecise value ‘blue’ and make that imprecision a bit more formal. In the works of Brouwer related to intuitionism an imprecise value can be thought of as a mapping. I will introduce the notation >blue< for such a mapping of the imprecise value ‘blue’. The mapping >blue< would then be:

>blue< : x ⟶ [0,1]

In other words, for any color x it evaluates to either 1 for it being fully considered as blue or 0 if it cannot be considered blue. However, according to Brouwer any value in between is also allowed. It could be 0.5 for half blue, which is also known as a fuzzy impecise value. Allowing these will confuse the with imprecision codependent concept of uncertainty. I will therefore restrict imprecise values, such as blue to:

>blue< : x ⟶ {true, false}

The reasoning is that subjectivity enters already in the evaluation of this mapping. In the terminology of transitional modeling, it is when asserting the statement “She wore a blue dress” that the asserter evaluates the actual color of the dress against the value ‘blue’. As such, the posit will be crisp from the asserter’s point of view. Given that the dress was acceptably ‘blue’ enough, the asserter can determine their certainty towards the posit. Values can therefore be said to be crisp imprecise values, but only relative a subject.

If we assume that the occasion when she wore a dress took place on the 1st of April 2020 and this is used as the appearance time in the posit, then it is also an imprecise value. Most of us will take this as the precise interval from midnight to midnight on the following day. At some point in that crisp interval, the dress was put on. Even so, putting on a dress is not an instantaneous event and time cannot be measured with infinite precision, so regardless of how precisely that time is presented, appearance time will remain imprecise.

With finer detail, the appearance time could, for example have been expressed as at two minutes to midnight on the 1st of April 2020. But, here we start to see the fallacy of taking some time range for granted though. With the same reasoning as before we would assume that to refer to the interval between two minutes and one minute to midnight. However, there is no way of knowing that a subject will always interpret it this way. So, we need the mapping once again:

>two minutes to midnight on the 1st of April 2020< : x ⟶ {true, false}

It seems as if the evaluation of this mapping is not only subjective, but also contextual. If we know that it could have taken more than a minute to put on the dress in question, then maybe this allows for both tree and one minute to midnight evaluating to true. Even when such a range is possible to specify it is almost never available in the information we consume, so we often have to deal with evaluations like these. We have, however, become so used to evaluating the imprecision that we do so more or less subconsciously.

But, didn’t we lose a whole field of applicability in the restriction of Brouwer’s mapping? That fuzziness is actually not all lost. I believe that what assertions do in transitional modeling is to fill that gap, while paying respect to subjectivity and contextuality. It is not possible to capture the exact reasoning behind the assertion, but we can at least capture its result. Recall that an assertion is someone expressing a degree of certainty towards a posit, here exemplified by “She wore a blue dress”. An example of an assertion is: “Archie thinks it likely that she wore a blue dress”. With time involved this becomes: “On the 2nd of April Archie thinks it likely that she wore a blue dress two minutes to midnight on the 1st of April”. Even more precisely and closer to a formal assertion: “Since the >2nd of April< the value >likely< appears for (Archie, certainty) in relation to ‘since the >1st of April< the value >blue< appears for (she, dress color)'”.

As can be seen, assertions can themselves be formulated as posits. Given the example assertion, it’s value is also imprecise, with a mapping:

>likely< : x ⟶ {true, false}

We have however, in transitional modeling, decided that certainty is better expressed using a numerical value. Certainty is taken from the range [-1, 1], with 1 being 100% certain, -1 being 100% certain of the opposite, and 0 for complete uncertainty. Certainties in between represent beliefs to some degree. We have to ask Archie, when you say ‘likely’, how certain is that given as a percentage? Let’s assume it is 80%. That means the corresponding mapping becomes:

>0.8< : x ⟶ {true, false}

Certainty is just another crisp imprecise value, but relative a subject who has performed a contextual evaluation of the imprecise values present in a posit with the purpose of judging their certainty towards it. An asserter (the subject) made an assertion (the evaluation and judgement), in transitional modeling terminology.

The interesting aspect of crisp imprecise values are that they respect “tertium non datur”, which is Latin for “no third is given”, more commonly known as the law of the excluded middle. In propositional logic it can be written as (P ∨ ¬P), basically saying that no statement can be both true and not true. An asserter making an assertion, evaluating whether the actual color of the dress can be said to be blue, obeys this law. It can either be said to be blue or it cannot. This law does not hold for fuzzy imprecise values. If something can be half blue, then neither “the dress was blue” nor “the dress was not blue” is fully true.

Fuzziness is not lost in transitional modeling though. Since certainty is expressed in the interval [-1, 1], it encompasses that of fuzzy values. The difference is that fuzziness comes from uncertainty and not from imprecision. Uncertainty is subjective and contextual, whereas fuzzy imprecise values are assumed objective and universal. I believe that this makes for a richer and truer to life, albeit more complex, foundation. It also rescues the excluded middle. Statements are either true or false with respect to crispness, but it is possible to express subjective doubt. Thanks to the subjectivity of doubt, contradicting opinions can be expressed, but that is the story of my previous articles, starting with “What needs to be agreed upon“.

As a consequence of the reasoning above, a posit is open for evaluation with respect to its imprecisions. Such imprecisions are evaluated in the act of performing an assertion, but an assertion is also a posit. In other words, the assertion is open for evaluation with respect to its imprecisions (the >certainty< and >since when< this certainty was stated). This can be remedied by someone asserting the assertion, but then those assertions will remain open, so someone has to assert the new assertions asserting the first assertions. But then those remain open, so someone has to assert the third level assertions asserting the second level assertions asserting the first level assertions, and so on…

Rather than having “turtles all the way down“, in transitional modeling there are posits all the way down, but for practical purposes it’s likely impossible to capture more than a few levels. The law of the excluded middle holds, within a posit and even if imprecise, but only in the light of subjective asserters performing contextual evaluations resulting in their judgments of certainty. To some extent, the excluded middle has been rescued!

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Lars Rönnbäck

Co-developer of the Anchor Modeling technique. Programmer of the online modeling tool. Site maintainer. Presenter and trainer.

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