Logic: Classical, Quantum, Modal

In class last week, one of the professors brought up the notion of different types of logic. I saw the words binary, quantum, and modal written on the board so I tried to find out a bit more about them. Here is a quick summary of what I found.

Classical logic forms its basis on the principle of bivalence. Basically, every declarative can either be true or false. This type of logic is also referred to as Boolean logic in the STEM world (0 for false, 1 for true.) In this type of logic, only the maximum (1) and minimum (0) values are allowed. Nothing in-between exists in this logic world. Classical logic also introduced many properties of logic that are used today, such as the law of the excluded middle, commutativity of conjunction, and the De Morgan duality. A brief example of classical logic statements are as follows; “All Ps are Q, Some Ps are Q, No Ps are Q, and Some Ps are not Q“

Quantum logic was developed in 1936 by Garrett Birkhoff and John von Neumann, who had several problems with using classical logic in the quantum era. Basically, the introduction of complementary variables in quantum mechanics, which are variables that cannot be measured accurately at the same time (think electron position and momentum), made classical logic inconsistent. In quantum logic, values can range in between the maximum and minimum values. For example, one variable may range from 0 to 1, while another ranges from 1 to 3 and yet another 0 to 1/6.

Modal logic was formally developed in the 1960’s. The basic goal of modal logic was to extend classical (binary logic) to include modality operators. These qualifiers come in several different forms, including alethic, temporal, denotic, epistemic, or doxastic. With modal logic, we turn statements like “the electron is here” to “the electron is usually here”. In Alethic logic (aka classical modal logic), a proposition can be possible, necessary, or contingent, instead of just the true and false values denoted by binary logic. On the other hand, temporal logic uses statements qualified by time like “I am always hungry”, “I will eventually be hungry” or “I will be hungry until I eat something.”

To me, the paradigm shift between classical and quantum logic was straightforward. Scientists realized that there was not a yes or no answer to every question they were asked, especially when some sort of distributive logic was used. For example, if asked about an electrons momentum and velocity, only probabilities exist. You simply cannot know both parts of this complementary pair. On the other hand, it seemed like modal logic was just an extension of quantum logic that was easier to apply to non-science matters. Spin numbers on an electron ranging between 0 and some small fraction may make sense to scientists, but qualifiers such as “sometimes” and “probably” relate to a broader audience.

I also thought these links from Stanford were interesting and at times easier to read than Wikipedia/other sources:

Quantum logic: http://plato.stanford.edu/entries/qt-quantlog/#2

Modal Logic: http://plato.stanford.edu/entries/logic-modal/

Classical Logic: http://plato.stanford.edu/entries/logic-classical/