Understanding Modal Logic


/ published 6 years ago

Understanding Modal Logic

In simplest terms, modal logic is just an extension of the pure logic, allowing people to use modal operators that can express modality like possibility and necessity

When you talk casually with someone, it is just impossible to use logic to defend every argument of yours. At least, pure logic for that matter. Before the 19th century, classic and pure mathematical logic was used to defend almost every argument. Every premise was either true or false. For example, you could say “I will burn if I lie in the sun for too long”. And this is either true or false. When you use modal logic, this premise gets possibility or impossibility of something happening. The example is “I might get burned if I lie in the sun for too long”. See the difference?

In simplest terms, modal logic is just an extension of the pure logic, allowing people to use modal operators that can express modality like possibility and necessity. Necessity and possibility are the two simplest modal operators. That is the narrow sense of understanding modal logic. The wider includes expression of past and future truths as well. So, while classic logic works perfectly in mathematics, modal logic allows for analysis of arguments and daily language.

Development of modal logic

You can say modal logic dates back to the same time when pure or mathematical logic was used. Aristotle developed certain modalities, but they never got the popularity as pure logic.

The founder for modal logic is actually C.I. Lewis, who founded in 1910 in his Harvard thesis. After his Harvard thesis, Lewis published a series of articles on the topic in 1912, and then the book “Symbolic Logic” in 1932.

And while Lewis set the pillars for modal logic, it was Ruth C. Barcan who developed the system for using it. In 1959, Saul Kripke, who was only 19 years old at the time, introduced the Kripke semantics for using of modal logic. It was then that contemporary/modern era of modal logic usage began. In 1957, A. N. Prior added “eventually” and “previously” to the previously set of modal operators. In years to follow, different scientists added different views on modal logic, and named them with names like linear temporal logic, propositional dynamic logic, computational tree logic and similar. In essence, they were all the same.

The Model Theory

To fully understand modal logic, one needs to study it for several years. But we can always break it down to simplest terms. For one sentence to be modally logical, the first step is to define a frame. The frame is a non-empty set, called “G”. Members of the sentence are generally possible words. Then, you have a binary relation, or “R”, that is placed between the possible words G. The relation is called accessibility relation, and the example is “w R u”, meaning that “u” is accessible from “w”. The result is a pair, “G,R”. Sometimes, we can a constant term, a part of the sentence that is called “actual world”, and it symbolized as “w”. Next step is to extend the model to one where truth-values are specified for all the propositions at each of the worlds in “G”. This is possible only by defining the relation between “v” and all possible words and literals. In others words, if there is “w” that v(w, P), then “P” is true at “w”.

Modal Logic Variations

As mentioned, there are many variations of modal logic, using more modal operators. Let’s take a look at some of the variations.

The first one is alethic logic, with the use of modal operators for possibility and necessity. They are called special modalities. In this aspect, something is possible “if and only if it is not necessarily false”. And something is necessary “if and only if it is not possibly false”. In other words, alethic logic allows for something to be “possible but not true”. Think of multiple possible words, in one of which the premise is possible, but it is not true for the current world. A simple example is the kitchen, where your tea is placed on the lower shelf and the coffee on the top shelf. Think that it might be possible that both are on the lower shelf, or both are on the upper shelf, or their position is rearranged. This is not true for the current world, but it is possible in another.

Next is epistemic logic, a type of modal logic that deals with certainty of things. The simple explanation is that “for everything X knows, it may be true that”. Let’s take a person for example, and we call him Joe. When Joe makes a claim that “The Loch Ness monster is real, I am certain of that”. In this case Joe claims that given the information he has, there is no question in his mind that Nessy is real. Joe can also make a claim that “the modal theory is possibly true”, but also it is possible that it is false by saying “if it is true, then it is necessarily true, not possibly false”. For all that Joe knows, the theory is true and there is no proof that the theory might be false. But Joe leaves the opportunity open. In simplest terms, epistemic logic is making a claim in which “for all we know, the premise is true”.

Temporal logic is logic that is true for only some time. For example, we have accepted that “1+1=2”, and that premise is true at all times. However, if we make a claim that “Joe is happy”, this is true only for some time. There might be a time when John is not happy. For temporal logic, we always have to use two modal operators, one for the past and one for the future.

Doxastic logic puts the focus on the belief. For example, when we say “it is believed that God exists”, we use a set of agents that can make the premise true.

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