Classic Vagueness of Classical Logic
This is my university essay from philosophy/computer science course of Logic*
AND, NOT and OR the pillars on which Classical Logic succeeds and falters. Purpose is to give formal definition to natural language and to prove arguments in a formal way. This is where classical logic succeeds in most cases but falters in borderline cases. Vagueness and ambiguity swoops through otherwise perfect logic system when borderline cases are present in proposition logic such as ‘heap’, ‘tall’ etc. ‘Heap of sand’ also known as Sorites paradox is a common example of vagueness in classical logic. First section dives a little deeper into the vagueness paradox as well as problems of vagueness. Then we move on to a discussion about different perspectives of dealing with the problem and in the final section we look at some approaches to solve the problem. Logic is a tool to prove and disprove arguments over a specified domain but also an asset to building computer architectures and control systems. Classical logic almost seems to be robotic as it clearly just follows rules without considering the actual proposition and the meaning behind it.
Conventional or classical logic has been proven by many academic resources that it is unfit for situations when the domain has gradual changes that accumulate over time. Sorites shows that classical logic is incapable of “precisifying” predicates as it only deals with binary truth values. For example, ambiguous and relative words such as “tall”, “hue”, “bald” etc. are common in the explanation of Sorites’ Paradox. In classical logic we always look at two extremes, for example either something exists and does not exist. Many philosophers seem to disregard or overlook the problem of vagueness and believe that conventional propositional logic has served scientists long enough and there is no need to introduce unnecessary complications with a third truth value for example an indefinite truth value for uncertainty or infinitely many truth values between 0 and 1 in “fuzzy logic”. From a computer science perspective, I feel there is a need for values to be considered which tell us the degree of correctness for a system. There is no reasoning to ignore such statements. In natural language we deal with vagueness and ambiguity daily, but we should not depend on a system which is not able to quantify vagueness like probability theory.
Let us define vagueness, in natural language we immediately think of uncertainty and imprecision. To increase specificity of a predicate we use two quantifiers such as FOR ALL and THERE EXISTS, these quantifiers help us restrict our predicate to a certain domain. Taking the example of radioactive decay of an atom — If we define a predicate D such that “For time T if we have an atom A, it still exists” and another predicate “There exists time T if we have an atom A ,it has decayed”. Truth value of 0 is defined for “atom in steady state” before radioactive decay starts and 1 for “atom after many half-lives” when it no longer exists. If we were to increment time T by 1 then we will get the following implications -
“If atom at T exists then at time T+1 an atom exists”
“If atom at T+1 exists then at time T+2 an atom exists
“If atom at 8T exists then at 8T+1 an atom exists”
The first predicate is true but for an atom the last predicate has a higher probability of being false. By 8T there should not an atom for decaying process to proceed but conventional logic does not have the basis for encoding states or intermediary values for a condition such as quarter of an atom or half of the atom. As we can see from predicates D and A that there is no way of defining intermediary states in conventional logic. Therefore, how can we conclude that scientists can overlook vagueness in logic.
Before researching ways of solving this problem we can look at the most common approach, first is introducing a new truth value and second is an entirely new system of logic. Usually the first step is to experiment with introducing a truth value of uncertainty, let us call it U. Now we have three possible inputs and outputs, 0, 1 and U. If we just look at a simple truth table of implication in this new system with three possible truth values, we will arrive at an antecedent which may be ambiguous and a conclusion which also has uncertainty associated with the truth value. What should be the resulting truth value of the implication? If we use U, the implication is essentially redundant and we need to build the old implication truth table as those are the only possible cases of uncertainty(true or false or both) and we reach the very beginning of trying to encapsulate vagueness under one truth value.
Another approach is to make an entirely new system of logic. After some research, majority of research papers tend to lean towards “fuzzy logic”, “supervaluationism” and “subvaluationism”. As we can see that if we just ignore the problem of vagueness, we are bound to get a false conclusion.
The most interesting practical solution to this problem is “fuzzy logic”. It is an alternative system to classical logic which has the same basic operators AND, OR and NOT but instead of having two definitive truth values of 0 and 1, there are infinite truth values between 0 and 1. For example for a predicate T — “a person is tall” can have a truth value of 0.8 assigned to it. 0.8 is close to 1 which shows the degree of correctness of being “tall”. This example clearly shows that being “tall” here is not necessarily vague but it “precisifies” the degree of membership.
Application of any system of logic conveys the importance and relevancy in the industry. Appliances such as vacuum cleaners, microwave ovens and video cameras can automatically adapt to different conditions because of fuzzy logic. In case of a video camera it changes exposure in the middle of recording according to the amount of light in a frame. Hypothetically, if it were to be based on classical logic there would be no way to define if at a state had a high or low exposure for light in a frame. “High” or “low” can only be defined by zeros and ones but here we want relatively high or low values of exposure according to the state of a frame or a scene in the viewfinder of the camera. This very reason of binary truth functional values is a boon and a bane for classical logic.
We have seen that classical logic tries to encapsulate vagueness, but it does not have the basic tools to do it. The argument of overlooking such a problem or merely accepting the result of Sorites paradox which turns out to be a false conclusion debunks in front of daily uses of fuzzy logic which seems to take a new approach on truth values. It is not to say that classical logic should not be preferred but fuzzy logic has become an extension of classical logic. The propositions which cannot be defined by classical logic should make use of alternatives.
Please check the resources below for a deeper explanation.
Resources
- Plato.stanford.edu. 2020. Fuzzy Logic (Stanford Encyclopedia Of Philosophy). [online] Available at: <https://plato.stanford.edu/entries/logic-fuzzy/>
- Plato.stanford.edu. 2020. Vagueness (Stanford Encyclopedia Of Philosophy). [online] Available at: <https://plato.stanford.edu/entries/vagueness/>
- 2020. Synthese Vol 30 №3/4, On the Logic Semantics of Vagueness (Apr. — May, 1975), pp. 265–300 (36 pages) [online] Available at:<https://www.jstor.org/stable/20115033?seq=1#metadata_info_tab_contents>
