Given that sentential classical logic relies on proofs in order to verify propositions, what logical conclusions can we draw from the use of the box and diamond symbols? This question is controversial as there is no one agreed upon system. Philosophers distinguish each of the classification systems on the basis of a logic type:

**K-logic (named after Saul Kripke, also known as ‘weak logic’)**

This is the most basic form of proof and is generally uncontested as far as modal logic goes. Both the box and the diamond act in a similar way to the ‘all’ and ‘some’ quantifiers. Necessitation dictates that if A is a theorem that is true, then ☐A is true. Importation dictates that if ☐(A -> B), then (☐A -> ☐B) is true.

**M-logic (or T-logic)**

This takes K-logic one step further. If ☐A is true, then A is true. Again, this is not a controversial principle

**S4-logic**

This can be represented by: ☐A -> ☐☐A. We are saying that if A is necessary, then it is necessary that A is necessary. Basically, this rule looks at removing all the superfluous modal operators. Lewis argued that this makes sense because anything necessary has to be necessary. However, a counter-argument is that many things we consider necessary don’t have to be necessary. If the Universe were created in a different way, gravity for instance might not work in the same way.

**S5-logic**

This is similar to S4 but deals with possibility. It can be represented as: ♢A -> ☐♢A. If A is possible, then it is necessary that A is possible. The S5 logic (in a slightly modified form known as B-logic: ♢☐A -> A) could be used to argue for the existence of God under the ontological argument. That is, it is possible that God exists as a necessary being. And given that whatever exists out of necessity does exist, then God must also exist.

A philosophical question that arises out of this is how we recognise whether something is ‘necessary’ or ‘possible’. Given that classical logic relies on the construction of truth tables to formally prove the validity of an argument, the operators of modal logic do not easily translate into a system of proof. One solution here is to rely on ‘possible worlds theory’. Classical logic only concerns itself with the actual world whereas modal logic can evaluate propositions based on possible worlds. If we want to determine if a statement is essential, we need to consider whether the statement is true in every possible world.

For instance, ‘2 + 2 = 4’ is true in every possible world, so it must be an essential truth. On the other hand, if we say that ‘unicorns exist’, there are possible worlds in which unicorns could have indeed existed, while in other words they do not exist. Therefore, if trying to create a proof for a sentential argument, a philosopher could use possible worlds to indicate whether a proposition is true or not in a variety of different worlds, only one of which will be the actual world. Overall truth is based on validity over the entire spectrum of possible worlds.

As the number of possible worlds could be infinite, philosophers have attempted to simplify things that we only need to consider possible worlds that are ‘accessible’ to another. However, this is not easy to define exactly. For instance, consider that you have a bicycle and are making some changes to it. After you are done, is it the same bicycle or a different one? You could argue that if you make three or less changes, it is still the same bicycle. In the actual world, you do make three changes to your bicycle, but in a closely related world, you start with a bicycle that already has had three changes made it (so is yet ‘unchanged’). Adding another three changes is valid according to the rule, so the bicycle would not be new in this case, even though six changes have taken place. However, since we are saying this scenario could exist in a possible world, then isn’t it possible that a bike could survive any number of changes, and still be considered an ‘old bike’? If we are saying that both these worlds are accessible, then we end up with a contradiction – the bike is ‘new’ in one world but ‘old’ in another.

David Lewis explores possible worlds further with his counterpart theory. This is based on two ideas. Firstly, any one object exists in one and only one world. So while people might have replicas in possible worlds, these replicas should not be seen as the same person but rather as counterparts. A counterpart is said to resemble an object more closely than anything else in that possible world. Therefore, as soon as you identify an object, you identify a world, as nothing exists in more than one world.

The second of Lewis’s ideas is that ‘truth’ is limited to the circumstances in the one possible world alone. It does not have to be true for all possible worlds.

To handle his counterpart theory, Lewis suggests that classical logic be expanded with a number of additional constructs such as ‘Wx’ (x is a possible world), ‘Ixy’ (x is in possible world y), ‘Ax’ (x is actual), and ‘Cxy’ (x is a counterpart of y).

According to Lewis, counterpart theory also helps to clarify the concept of ‘identity’. In classical logic, we can say a=b -> ☐a=b. If a and b are identical, it is necessary for them to be identical. However, under counterpart theory an object may have more than one counterpart, so it is not necessary for anything to be same as another object.

Finally, I will conclude this series of posts on modal logic with some objections. One is that the whole idea of ‘possibility’ is vague. Regardless of whether you use possible worlds or not, you still have to have some idea of what you mean by ‘possible’. How closely connected does the possible scenario have to be to the real world? A response to this is that provided by Lewis and his counterpart theory. Because each world is a self-contained entity, we do not need to be concerned with comparing possibilities across different worlds. As long as the proposition is true in the possible world, this is all that matters.

An empiricist would also criticise modal logic on the basis that it attempts to provide evidence of a priori truths that have not been obtained via direct evidence. The anti realist would also argue that the modal is unknowable because no counterfactual can be directly proven, so there is nothing to know. However, Melia argues that modal logic fits well into English because we are often speaking in a hypothetical way as part of everyday speech.