Types of modality

Before you tackle this guide, you should be confident that you are familiar with the contents of the essential guide to modality and the guide to pure modal auxiliary verbs.
What follows is an analysis which can be used when planning to teach modal auxiliary verbs in English. It is based on some fundamental concepts in logic and the theory is that the categories of modality mirror the categories of human thought, making concepts clearer and more teachable.

4 shades of modality

epistemic modality: Epistemology is the study of theories of knowledge and the word epistemic means relating to knowledge.
This kind of modality is to do with the speaker's perception of the truth or otherwise of a proposition. So, for example, on hearing a knock at the door, the use of will in
    That will be the postman
signals the speaker's certainty about the proposition. Equally:
    That can't be the postman
signals the speaker's certainty of untruth and the use of might in
    That might be the postman
signals the speaker's uncertainty.
Modal auxiliary verbs of deduction fall into this category.
deontic modality: Deontology is the study of duty and deontic means relating to obligation and duty (or its lack). For example, the use of ought to in
    You ought to write to your mother
expresses the speaker's perception that it is the hearer's duty to write and the use of must in
    You must not write like that
expresses the speaker's view that the hearer is obliged not to write in that way.
The use of may in
    You may go
expresses the speaker's view that there is no longer a duty or obligation to stay.
Modal auxiliary verbs expressing any sense of obligation or its lack fall into this category.
dynamic modality: The term dynamic needs no explanation but here it refers to the fact that the modality is centred on the subject. For example, the use of could in
    I could swim well as a child
refers to the subject's own ability and not to externally imposed obligations or duties and the use of can in
    Can you help me with this?
refers to the hearer's ability alone and
    I'll get it for you
refers to the speaker's willingness to do something (volition). It is not, incidentally, a future tense form and does not necessarily refer to the future at all; it is an expression of current willingness to do something.
alethic modality: This term derives from the Greek word for truth [αλήθεια, aleethia] and refers to logical necessity (rather than deduction which concerns epistemic modality). For example, the use of must in
A square must have four sides
refers not to the speaker's perception and not to any form of obligation or deduction but to the fact that one of the necessary conditions of being a square is to have four sides and the use of can't in
Parallel lines can't meet
equally represents the truth of a proposition failing which, the lines cannot be parallel.

Subtler shades

This area of study is infested with a menagerie of terminology, some of which are concerned with quite subtle levels of analysis.
So, if you come across alternative expressions, here's a very short guide:

Epistemic modality is often singled out as being views of the truth based solely on the evidence available to the speaker and separated in that way from alethic modality which is not contingent on the speaker's setting but expresses truths in all possible worlds.
Deontic modality is sometimes distinguished from boulemic modality which expresses that which is necessary given a person's wants and desires. For example:
I must go now
You must not stay up later than 10
etc.
Deontic modality is also occasionally distinguished from teleological modality which expresses that which is necessary to achieve an aim as in, e.g.:
If you want to get into university you need to study hard
We must leave now or we'll miss the train
This form of modality is often expressed with conditional structures as above.
Dynamic modality is often referred to as ability modality (because that is what is mostly concerns). That is not compatible with the view taken in these sections of the site because we also consider expressions of willingness as falling into this category.
Alethic modality is sometimes referred to as circumstantial modality because it derives from the circumstances of the universe and expresses universal truths (which epistemic modality does not). For example:
The ignition temperature of paper is 233° C
which is true wherever you are and not dependent on when or where you state it.

For teaching purposes and for the analysis of language for planning, these subtleties are unnecessary, although some languages deal with the categories differently. However, if you are interested in the philosophy of language, they may have some utility.

An alternative view of modality

With this analysis in mind, we can try an alternative way to represent the diversity of modal auxiliary verbs in English. The usual analysis (used elsewhere on this site in the consideration of modal auxiliary verbs) is to take each verb in turn and then itemise and explain what functions they can perform. For example, from the guide to pure modal auxiliary verbs, we have something like:

might

Function	Example
present possibility	Careful. There might be a snake in the hall.
future possibility	It might rain tomorrow.
past possibility	He might have telephoned while I was out.
suggestion	You might try taking an aspirin.
permission	Might I talk to you?
complaint	You might have warned me!

This is clear and quite concise but, for teaching purposes, is a flawed way to see the analysis. Speakers of the language do not proceed from a consideration of the various functions of the verb and then make meaning using it. What they do is to conceive a meaning and then select a modal expression which suits their purposes.

For example, I may have in mind the fact that I do not believe Mary when she says:
    Hey, I saw a unicorn on my way back from the pub last night!
For my purposes, I might select any number of expressions of modality depending on my degree of certainty, formality, setting and so on. I may consider, for example:
    It's an impossibility that you saw a unicorn
    You can't have seen a unicorn
    It is impossible that you saw a unicorn
    I suppose you might have seen a unicorn, but I doubt it
    You couldn't have seen a unicorn; they don't come round here
and so on.

These examples are forms of epistemic modality (i.e., to do with the speaker's view of the truth of a proposition) although the verbs and expressions themselves can be used to express other types of modality. They all express a degree of certainty about the central proposition.

Now we can revisit our original 11 pure modal auxiliaries (from the other guide) and re-analyse them from the point of view of these four categories.

An alternative analysis of modality

The following is not meant to be exhaustive but, for most English-language teaching purposes, it will suffice as a starting point when planning to teach modality based on mental processes rather than form.
The alternative modal expressions are suggestive only and include modal adverbials, nouns and adjectives.

epistemic modality

That must be the teacher's desk

Modal auxiliary verb	Example	Alternative modal expressions
might	If there's too little current, the pump might not work. He might have telephoned while I was out.	It's possible the pump isn't working. Possibly, he's already gone out. I'm assuming the pump isn't broken. My assumption is that that's the postman. I'm sure that is what he said. It's obviously his brother. It's clear they are related. It seems likely that it's him. I doubt she is there.
could	It could rain tomorrow. He couldn't have gone out this early.
can	The pump can't be broken. It's new.
should	Ah, that should be the postman now.
would	Well, he would, wouldn't he? It's in his interests. That would be his mother you saw.
must	That must be his brother. Aren't they alike?
may	Well it may be his brother. I don't know.
will	That will be the postman at the door.
ought to	She ought to be there by now.

For more on how epistemic modality is expressed in English, see the guide to expressing (un)certainty and the fuller guide to epistemic modality.

deontic modality

They shall not pass.

Modal auxiliary verb	Example	Alternative modal expressions
might	Might I ask a question?	Is there a chance of asking a question? Do you have a moment? Seeing a doctor would be advisable. I forbid you to speak to me like that. Is he allowed to come, too? Do you have permission to be here? I don't imagine this work will take long. They are advised to get here early. If I were you, I'd see a dentist. It's her duty to write. Registration is compulsory. I advise you to leave soon.
could	Could I just say something?
can	Can I see you for a minute?
should	You should not speak to me like that. You should see a doctor.
must	You must finish the work before 6.
need	This work needn't take long They need to get here early
may	May he come with us, please?
will	You will not speak to me so rudely again.
ought to	She ought to write to her father.
have (got) to	You have (got) to register online.
had better	Hadn't you better leave soon?
shall	You shall not speak to your father like that

In some of the above, of course, there is an implied reference to possibility (i.e., epistemic modality). For example,
    Do you have a moment?
can be interpreted as a question about a current state but is more likely to mean
    Do you give me permission to interrupt your day?
and
    Registration is compulsory
can be interpreted simply as a statement of fact but is more likely to mean
    You have to register.

The verb need can also imply alethic modality as in, e.g.:
He needs to get to work before 6
which implies a simple truth rather than an obligation per se.
The verb also has a non-modal use as a lexical verb meaning require.

For more on how deontic modality is expressed in English, see the guide to expressing obligation and the fuller guide to deontic and alethic modality.

dynamic modality

We can do this!

Some analyses of dynamic modality include uses which are better seen as either deontic (obligation or its lack) or epistemic (to do with the truth of a proposition) so are excluded from this list.
Strictly speaking, dynamic modality is confined to expressions of ability or willingness. That will limit the number of verbs quite severely to can, could, will, would and be able to for the most part.

Modal auxiliary verb	Example	Alternative modal expressions
can	I can speak Spanish well.	She has the ability to speak 4 languages. I haven't the strength to do it. I am happy to accept. I volunteer to do that. Let me go; I know the way. Did you manage to find what you wanted?
could	I couldn't possible lift it. She could read well at the age of 4.
will	I'll get the door.
would	I would love to come.
able to	Were you able to find what you wanted?

alethic modality

The blue bit can't go here.

Modal auxiliary verb	Example	Alternative modal expressions
must	A quintet must contain 5 players.	Unless it has five players, it's not a quintet. Having more or fewer than 8 players disqualifies it as an octet. It is necessary to have 4 sides in a square.
can't	An octet can't have more or fewer than 8 players.
need	A square needs 4 sides.

When expressing alethic modality, the verb must is negated in standard English as can't or couldn't (as it is for epistemic modality). In some dialect forms, however, mustn't is used to express impossibility rather than negative obligation.
The verb need is somewhat rarer in this sense.

It is also quite rare for alethic modality to be expressed in the past because, obviously, it usually refers to timeless facts. It is occasionally encountered when a fact has been superseded and is no longer true, for example:
    At that time a battle had to be fought on foot.
in such cases, the normal rule of replacing must with have to and can't with couldn't applies.
However, the verb need is not negated in the past when it is used to express alethic modality, so, e.g.:
    At that time armies needed to move solely on foot
cannot be negated as
    At that time armies didn't need to move solely on foot
or
    At that time armies needn't have moved solely on foot
because that implies a lack of obligation rather than a negative fact.

The guide to expressing degrees of likelihood includes considerations of alethic modality and considers the overlap between it and epistemic modality.

Notes

In some analyses, the modal auxiliary verb must is classified as representing intrinsic or even dynamic modality because it is seen as personal to the speaker rather than an externally imposed duty and have to appears under extrinsic deontic modality because, the theory has it, that it expresses an externally imposed duty. However, where it is not obvious which is intended, native speakers use them in free variation.
It is also the case that it is difficult to separate the external from the internal so, for example:
I have a toothache and have to go to the dentist
can just as well be expressed as:
I have a toothache and must go to the dentist
because it is unclear whether a toothache represents an internal or external phenomenon.
The strength of both deontic and epistemic modality is often determined by intonation and stress rather than depending on the modal expression chosen by the speaker.
In this analysis, logical deduction and logical necessity are distinguished (as epistemic and alethic modalities) but, for most teaching purposes, can be conflated.
The verb need is treated here as a central modal although it has semi-modal characteristics in many cases.
Teaching modal expressions using this kind of analysis presumes that you start from the type of modality your learners need to express and work on a limited range of ways to realise the mental processes in English.
There is a guide on this site to teaching modality.

Summary

Here's a summary with examples using pure (or central) modal auxiliary verbs only. Semi-modal auxiliary verbs and marginal modal auxiliary verbs can also express different types of modality. All four types of modality can also be expressed without modal auxiliary verbs at all.
For teaching purposes, the summary is useful because you can see at a glance whether the lesson you planned mixes the same verb expressing different forms of modality. That can confuse, especially at lower levels.

examples of modality

To check your understanding, try a multiple choice quiz on this area.

Related guides
essential guide to modality	a simpler guide in the initial training section
modality map	a way to find your way around a complex area
pure modal auxiliary verbs	a traditional view taking each modal in turn and identifying its function
semi-modal auxiliary verbs	which also considers marginal modal auxiliary verbs such as seem, tend, be about to etc.
non-modal auxiliary verb modality	which considers adverbials, verbs, adjectives and nouns used to express modality
epistemic modality	modality for expressing the speaker's view of the truth or a proposition, i.e., likelihood
expressing uncertainty	a functional approach to epistemic modality
deontic modality	modality for expressing the speaker's view of obligation or its lack with some consideration of alethic modality with which it overlaps
dynamic modality	modality for expressing ability and willingness
complex tenses	which also considers complex tenses in relation to modality
teaching modality	for some more ideas transferable to the analysis above
modality and aspect	which considers modal auxiliary verbs with perfect and progressive forms and considers some of the types of modality discussed here

Reference:
Palmer, F. R, 2001, Mood and Modality, 2nd ed., Cambridge: Cambridge University Press