Modal logic can be seen as an extension to classical logic, or whatever other ‘language’ of logic that a philosopher uses (the schema itself will depend on the logical constants).
One of the criticism of classical logic is that it does not take ‘possibility’ into account. Classical logic only concerns itself with states of affairs as they exist in the current moment. For instance, a statement such as ‘if I drop the glass on the floor it will break’ is difficult to capture in classical logic since the glass hasn’t actually been dropped yet (in fact you may never have dropped a glass in your life), but the statement still sounds truthful based on our intuitions. Modal logic attempts to deal with the ‘what if’ scenario.
Exploring this idea further, Joseph Melia notes that classical logic attempts to be ‘categorical’ by defining every proposition in strict logical form (true or false). However, this fails to capture the nature of ‘truth’, which can differ depending on why the proposition is true. Melia notes two ways in which classical logic is unclear or incomplete.
Firstly, classical logic does not make any distinction between essential properties and accidental properties. Two statements such as ‘John is tall’ and ‘John is human’ might sound similar (and be noted in the same way in classical logic) however the information content is not exactly the same. The fact that John is human is essential: there is no escaping this fact. However, John may not necessarily have been tall. He could have had an accident, for instance, that caused him to lose his legs. So the fact that he is tall might be true, but it is ‘accidental’ and not true in the same sense as John being human. Melia notes that drawing a distinction between essential and accidental properties is an example of ‘de re’ modality.
In addition to properties, Melia also notes that propositions can be true in different ways. He offers another two examples: ‘All bachelors are unmarried’ and ‘All emeralds are green’. The first is essentially true as it is logically impossible for a married bachelor to exist. However, the second is only contingently true. There is no logical reason why a non-green emerald couldn’t be discovered. So a second form of modal distinction is necessary truths and contingent truths. When the modality attaches to the proposition as a whole (as opposed to a specific property, as per the previous category), we can describe this as ‘de dicto’ modality.
There is a direct relationship between the two modes just described. If something has an accidental property of being tall, then this means that any proposition based on that property will be contingent (ie ‘John is tall’ is a contingent truth). However, this doesn’t work the other way around. Even if we have an essential truth (‘all bachelors are unmarried’), we do not have any essential properties here (no person is married or unmarried by necessity).
Modal logic attempts to provide a mechanism where we can clarify logical ‘truth’, drawing a distinction between essential truths and accidental ones. This allows us to explore the world of the possible and of counterfactual situations. The classical logic construct of ‘if A then B’ is automatically true when A is false, which does not help us if we are trying to describe a counterfactual situation (‘if Germany won the war…’). Classical logic is also of limited use when we want to describe a tendency of something, such as the breaking glass example.
The most basic form of modal logic adds two additional operators to classical logic. This is described as ‘QML’ (quantified modal logic). The two operators are ☐ (the box symbol that stands for ‘necessity’) and ♢ (the diamond symbol that stands for ‘possibility’). QML also uses the predicate ‘E’ to indicate that something exists. By using ☐, we can indicate that a proposition happens out of necessity, while using ♢ shows that an outcome may happen, but this is not guaranteed. We can also define possibility in terms of necessity:
♢A is equivalent to ~☐~A
If something is possible (‘A’ might be true’), we are saying that it not necessary for A to be false. We are recognising that A might be true or it might not be true, but neither result is certain or necessary.
Modal logic can help to clarify statements that are ambiguous in classical logic. For instance, consider the statement ‘all footballers are footballers’. This could be interpreted in two ways:
- Necessarily, any footballer is a footballer: ☐Vx(Mx -> Mx)
- Any footballer is necessarily a footballer: Vx(Mx -> ☐Mx)
Arguably, it is the first statement that we are trying to assert here, given that there is no logical reason why a footballer has to be a footballer – he/she could easily have pursued another profession. So the exact way in which we use the ‘box’ symbol will help to clarify the meaning of the proposition.
Another example that displays the opposite effect is the proposition ‘something exists that is human’. Again this has two interpretations:
- Necessarily, something is human: ☐Ex(Hx)
- Something is such that it is necessarily human Ex☐(Hx)
In this example, it is the second type of ‘truth’ we are trying to capture here, as there is no reason why human life was ‘necessary’ on Planet Earth.
We can distinguish these two examples on the de dicto / de re classification. De re modality occurs when the box symbol is used in ‘narrow scope’ (at the property level), whereas de dicto modality uses the box in wide scope (at the proposition level).
The next question to consider is the term ‘necessity’. By itself, this is a vague concept in English. We could be referring to a number of different ways in which something is necessary. For instance, an act might be morally necessary: ‘if you find a wallet on the street, you must hand this in to the police’. Or an act might be advisable in the circumstances: ‘make sure you arrive at the airport in good time for your flight’. Or it might be necessary under physical laws: ‘if you drop a glass, it will break on a concrete floor’.
However, in terms of modal logic, it is generally only logical necessity that is considered important (‘A is equal to A’). As indicated above, exactly what we mean by logical necessity isn’t clear as this depends on what language of logic we wish to follow (which in turn is driven by what we consider are the ‘correct’ logical constants). However, for the remainder of these posts, I will assume that we are using the language of classical logic.