Alpha diversity

In modal logic, an accessibility relation is a binary relation, written as $R$ between possible worlds.

Description of Terms

A 'statement' in logic refers to a sentence (with a subject, predicate, and verb) that can be true or false. So, 'The room is cold' is a statement because it contains a subject, predicate and verb, and it can be true that 'the room is cold' or false that 'the room is cold.'

Generally, commands, beliefs and sentences about probabilities aren't judged as true or false. 'Inhale and exhale' is therefore not a statement in logic because it is a command and cannot be true or false, although a person can obey or refuse that command. 'I believe I can fly or I can't fly' isn't taken as a statement of truth or falsity, because beliefs don't say anything about the truth or falsity of the parts of the entire 'and' or 'or' statement and therefore the entire 'and' or 'or' statement.

A 'possible world' is any possible situation. In every case, a 'possible world' is contrasted with an actual situation. Earth one minute from now is a 'possible world.' The earth as it actually is also a 'possible world.' Hence the oddity of and controversy in contrasting a 'possible' world with an 'actual world' (earth is necessarily possible). In logic, 'worlds' are described as a non-empty set, where the set could consist of anything, depending on what the statement says.

'Modal Logic' is a description of the reasoning in making statements about 'possibility' or 'necessity.' 'It is possible that it rains tomorrow' is a statement in modal logic, because it is a statement about possibility. 'It is necessary that it rains tomorrow' also counts as a statement in modal logic, because it is a statement about 'necessity.' There are at least six logical axioms or principles that show what people mean whenever they make statements about 'necessity' or 'possibility' (described below). For a detailed explanation on modal logic, see here.

As described in greater detail below:

Necessarily $p$ means that $p$ is true at every 'possible world' $w$ such that $R(w^{*},w).$

Possibly $p$ means that $p$ is true at some possible world $w$ such that $R(w^{*},w)$.

'Truth-Value' is whether a statement is true or false. Whether or not a statement is true, in turn, depends on the meanings of words, laws of logic, or experience (observation, hearing, etc.).

'Formal Semantics' refers to the meaning of statements written in symbols. The sentence $(◻p\vee ◻q)\to ◻(p\vee q)$, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by the symbol $R$.

The 'accessibility relation' is a relationship between two 'possible worlds.' More precisely, the 'accessibility relation' is the idea that modal statements, like 'it's possible that it rains tomorrow,' may not take the same truth-value in all 'possible worlds.' On earth, the statement could be true or false. By contrast, in a planet where water is non-existent, this statement will always be false.

Due to the difficulty in judging if a modal statement is true in every 'possible world,' logicians have derived certain axioms or principles that show on what basis any statement is true in any 'possible world.' These axioms describing the relationship between 'possible worlds' is the 'accessibility relation' in detail.

Put another way, these modal axioms describe in detail the 'accessibility relation,' $R$ between two 'worlds.' That relation, $R$ symbolizes that from any given 'possible world' some other 'possible worlds' may be accessible, and others may not be.

The 'accessibility relation' has important uses in both the formal/theoretical aspects of modal logic (theories about 'modal logic'). It also has applications to a variety of disciplines including epistemology (theories about how people know something is true or false), metaphysics (theories about reality), value theory (theories about morality and ethics), and computer science (theories about programmatic manipulation of data).

Basic Review of (Propositional) Modal Logic

The reasoning behind the 'accessibility relation' uses the basics of 'propositional modal logic' (see modal logic for a detailed discussion). 'Propositional modal logic' is traditional propositional logic with the addition of two key unary operators:

$◻$ symbolizes the phrase 'It is necessary that...'

$◊$ symbolizes the phrase 'It is possible that...'

These operators can be attached to a single sentence to form a new compound sentence.

For example, $◻$ can be attached to a sentence such as 'I walk outside.' The new sentence would look like: $◻$'I walk outside.' The entire new sentence would therefore read: 'It is necessary that I walk outside.'

But the symbol $A$ can be used to stand for any sentence instead of writing out entire sentences. So any sentence such as 'I walk outside' or 'I walk outside and I look around' are symbolized by $A$.

Thus for any sentence $A$ (simple or compound), the compound sentences $◻A$ and $◊A$ can be formed. Sentences such as 'It is necessary that I walk outside' or 'It is possible that I walk outside' would therefore look like: $◻$$A$ $◊A$.

However, the symbols $p$, $q$ can also be used to stand for any statement of our language. For example, $p$ can stand for 'I walk outside' or 'I walk outside and I look around.' The sentence 'It is necessary that I walk outside' would look like: $◻$$q$. The sentence 'It is possible that I walk outside' would look like: $◊$$q$.

Six Basic Axioms of Modal Logic:

There are at least six basic axioms or principles of almost all modal logics or steps in thinking/reasoning. The first two hold in all regular modal logics, and the last holds in all normal modal logics.

1st Modal Axiom:

• $◻p\leftrightarrow \mathrm{\neg }◊\mathrm{\neg }p$ (Duality)

The double arrow stands symbolizes 'if and only if,' 'necessary and sufficient' conditions. A 'necessary' condition is something that must be the case for something else. Being literate, for instance, is a 'necessary' condition for reading about the 'accessibility relation.' A 'sufficient condition' a condition that is good enough for something else. Being literate, for instance, is a 'sufficient' condition for learning about the accessibility relation.' In other words, it's good enough to be literate in order to learn about the 'accessibility relation,' however it may not be 'necessary' because the relation could be learned in different ways (like through speech). Aside from 'necessary and sufficient,' the double arrow represents equivalence between the meaning of two statements, the statement to the left and the statement to the right of the double arrow.

The half square symbols before the diamond and $p$ symbol in the sentence '$◻p\leftrightarrow \mathrm{\neg }◊\mathrm{\neg }p$' stand for 'it is not the case, or 'not.'

The $p$ symbol stands for any statement such as 'I walk outside.' Therefore it could also stand for 'The apple is Red.'

Example 1:

The first principle says that any statement involving 'necessity' on the left side of the double arrow is equivalent to the statement about the negation of 'possibility' on the right.

So using the symbols and their meaning, the first modal axiom, $◻p\leftrightarrow \mathrm{\neg }◊\mathrm{\neg }p$ could stand for: 'It's necessary that I walk outside if and only if it's not possible that it is not the case that I walk outside.'

And when I say that 'It's necessary that I walk outside,' this is the same as saying that 'It's not possible that it is not the case that I walk outside.' Furthermore, when I say that 'It's not possible that it is not the case that I walk outside,' this is the same as saying that 'It's necessary that I walk outside.'

Example 2:

$p$ stands for 'The apple is red.'

So using the symbols and their meaning described above, the first modal axiom, $◻p\leftrightarrow \mathrm{\neg }◊\mathrm{\neg }p$ could stand for: 'It's necessary that the apple is red if and only if it's not possible that it is not the case that the apple is red.'

And when I say that 'It's necessary that the apple is red,' this is the same as saying that 'It's not possible that it is not the case that the apple is red.' Furthermore, when I say that 'It's not possible that it is not the case that the apple is red,' this is the same as saying that 'It's necessary that the apple is red.'

Second Modal Axiom:

• $◊p\leftrightarrow \mathrm{\neg }◻\mathrm{\neg }p$ (Duality)

Example 1:

The second principle says that any statement involving 'possibility' on the left side of the double arrow is the same as the statement about the negation of 'necessity' on the right.

$p$ stands for 'Spring has not arrived.'

Using the symbols and their meaning described above, the second modal axiom, $◊p\leftrightarrow \mathrm{\neg }◻\mathrm{\neg }p$ could stand for: 'It's possible that Spring has not arrived if and only if it is not the case that it is necessary that it is not the case that Spring has not arrived.'

Essentially, the second axiom says that any statement about possibility called 'X' is the same as a negation or denial of a different statement about necessity 'Y.' The statement about necessity shows the denial of the same original statement 'X.'

The other axioms can be read and interpreted in the same way, by substituting letters $p$ for any statement and following the reasoning. Brackets in a symbolized sentence mean that anything inside the brackets counts as a whole sentence. Any symbol before the brackets therefore applies to the sentence as a whole, not just the letters or an individual sentence.

An arrow stands for "then" where the left statement before the arrow is the "if" of the entire sentence.

Other Modal Axioms:

* $◻(p\wedge q)\leftrightarrow (◻p\wedge ◻q)$

* $(◻p\vee ◻q)\to ◻(p\vee q)$

* $◻(p\to q)\to (◻p\to ◻q)$ (Kripke property)

Most of the other axioms concerning the modal operators are controversial and not widely agreed upon. Here are the most commonly used and discussed of these:

(T) $◻p\to p$

(4) $◻p\to ◻◻p$

(5) $◊p\to ◻◊p$

(B) $p\to ◻◊p$

Here, "(T)","(4)","(5)", and "(B)" represent the traditional names of these axioms (or principles).

According to the traditional 'possible worlds' semantics of modal logic, the compound sentences that are formed out of the modal operators are to be interpreted in terms of quantification over possible worlds, subject to the relation of accessibility. A sentence like $(◻p\vee ◻q)\to ◻(p\vee q)$ is to be interpreted as true or false in all or some 'possible worlds.' In turn, the grounds on which the sentence is true (symmetry, transitive property, etc.) in all 'possible worlds' is the 'accessibility relation.'

The relation of accessibility can now be defined as an (uninterpreted) relation $R(w_1,w_2)$ that holds between 'possible worlds' $w_1$ and $w_2$ only when $w_2$ is accessible from $w_1$.

Additionally, to make things more specific, any formula, axiom like $(◻p\vee ◻q)\to ◻(p\vee q)$ can be translated into a formula of first-order logic through standard translation. That first-order logic formula or sentence makes the meaning of the boxes and diamonds in modal logic explicit.

The Four Types of the 'Accessibility Relation' in Formal Semantics

'Formal semantics' studies the meaning of statements written in symbols. The sentence $(◻p\vee ◻q)\to ◻(p\vee q)$, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by the symbol $R$, where $R$ takes the form of the 'necessity relation' described below.

So, the 'accessibility relation,' $R$ can take on at least four forms, that is, the 'accessibility relation' can be described in at least four ways.

Each type is either about 'possibility' or 'necessity' where 'possibility' and 'necessity' is defined as:

• (TS) Necessarily $p$ means that $p$ is true at every 'possible world' $w$ such that $R(w^{*},w)$.

• Possibly $p$ means that $p$ is true at some possible world $w$ such that $R(w^{*},w)$.

The four types of $R$ will be a variation of these two general types. They will specify on what conditions a statement is true either in every possible world, or some possible. The four specific types of $R$ are:

'The Reflexive' or *Axiom (T):

This says that if every world is accessible to itself, then any world in which $A$ is true will be a world from which there is an accessible world in which $A$ is true. Notice this is a variation, more detailed description of the 'necessity' definition above.

'The Transitive' or *Axiom (4) above:

This says that $R$ is transitive, $◻A$ is true at a world $w$ only when $A$ is true at every world $w^{\prime }$ accessible from $w$. Hence, $◻◻A$ is true at a world $w$ only when $A$ is true at every world accessible from every world accessible from $w$. Notice that this is a variation, more detailed description of the 'possibility' definition above.

'The Euclidean' or *Axiom (5) above:

This says that $R$ is euclidean. So, $◊A$ is true at a world $w$ if and only if $A$ is true at some world accessible from $w$. $◻◊A$ is true at a world $w$ if and only if, for every world $w^{\prime }$ accessible from $w$, there is a world $w^{″}$ accessible from $w^{\prime }$ at which $A$ is true.

The euclidean property guarantees the truth of this. If $A$ is true at a world accessible from $w$, then if that world is accessible from every other world accessible from $w$, it will be true that for every world accessible from $w$ there is an accessible world in which $A$ is true. Notice that this is a variation, more detailed description of the 'necessity' definition above.

'The Symmetric' or *Axiom (B) above:

This says that $R$ is symmetric. If $A$ is true in a world $w$, then in every world $w^{\prime }$ accessible from $w$, there is a world accessible from $w^{\prime }$ in which $A$ is true. Since $A$ is true in $w$, this is guaranteed to be true provided that $w$ is accessible from it, which is precisely what symmetry says.

Philosophical Applications

One of the applications of 'possible worlds' semantics and the 'accessibility relation' is to physics. Instead of just talking generically about 'necessity (or logical necessity),' the relation in physics deals with 'nomological necessity.' The fundamental translational schema (TS) described earlier can be exemplified as follows for physics:

• (TSN) $P$ is nomologically necessary means that $P$ is true at all possible worlds that are nomologically accessible from the actual world. In other words, $P$ is true at all possible worlds that obey the physical laws of the actual world.

The interesting thing to observe is that instead of having to ask, now, "Does nomological necessity satisfy the axiom (5)?", that is, "Is something that is nomologically possible nomologically necessarily possible?", we can ask instead: "Is the nomological accessibility relation euclidean?" And different theories of the nature of physical laws will result in different answers to this question. (Notice however that if the objection raised earlier is true, each different theory of the nature of physical laws would be 'possible' and 'necessary,' since the euclidean concept depends on the idea about 'possibility' and 'necessity'). The theory of Lewis, for example, is asymmetric. His counterpart theory also requires an intransitive relation of accessibility because it is based on the notion of similarity and similarity is generally intransitive. For example, a pile of straw with one less handful of straw may be similar to the whole pile but a pile with two (or more) less handfuls may not be. So $x$ can be necessarily $P$ without $x$ being necessarily necessarily $P$. On the other hand, Saul Kripke has an account of de re modality which is based on (metaphysical) identity across worlds and is therefore transitive.

Another interpretation of the 'accessibility relation' with a physical meaning was given in Gerla 1987 where the claim “is possible $P$ in the world $w^{″}$ is interpreted as "it is possible to transform $w$ into a world in which $P$ is true". So, the properties of the modal operators depend on the algebraic properties of the set of admissible transformations.

There are other applications of the 'accessibility relation' in philosophy. In epistemology, one can, instead of talking about nomological accessibility, talk about epistemic accessibility. A world $w^{\prime }$ is epistemically accessible from $w$ for an individual $I$ in $w$ if and only if $I$ does not know something which would rule out the hypothesis that $w^{\prime }=w$. We can ask whether the relation is transitive. If $I$ knows nothing that rules out the possibility that $w^{\prime }=w$ and knows nothing that rules the possibility that $w^{″}=w^{\prime }$, it does not follow that $I$ knows nothing which rules out the hypothesis that $w^{″}=w$. To return to our earlier example, one may not be able to distinguish a pile of sand from the same pile with one less handful and one may not be able to distinguish the pile with one less handful from the same pile with two less handfuls of sand, but one may still be able to distinguish the original pile from the pile with two less handfuls of sand.

Yet another example of the use of the 'accessibility relation' is in deontic logic. If we think of obligatoriness as truth in all morally perfect worlds, and permissibility as truth in some morally perfect world, then we will have to restrict out universe to include only morally perfect worlds. But, in that case, we will have left out the actual world. A better alternative would be to include all the metaphysically possible worlds but restrict the 'accessibility relation' to morally perfect worlds. Transitivity and the euclidean property will hold, but reflexivity and symmetry will not.

Computer Science Applications

In modeling a computation, a 'possible world' can be a possible computer state. Given the current computer state, you might define the accessible possible worlds to be all future possible computer states, or to be all possible immediate "next" computer states (assuming a discrete computer). Either choice defines a particular 'accessibility relation' giving rise to a particular modal logic suited specifically for theorems about the computation.

See also

• Modal logic
• Possible worlds
• Propositional attitude
• Modal depth

References

• Gerla, G.; Transformational semantics for first order logic, Logique et Analyse, No. 117–118, pp. 69–79, 1987.
• Fitelson, Brandon; Notes on "Accessibility" and Modality, 2003.
• Brown, Curtis; Propositional Modal Logic: A Few First Steps, 2002.
• Kripke, Saul; Naming and Necessity, Oxford, 1980.
• Lewis, David K.; Counterpart Theory and Quantified Modal Logic ^{(subscription required)}, The Journal of Philosophy, Vol. LXV, No. 5 (1968-03-07), pp. 113–126, 1968
• List of Logic Systems List of most of the more popular modal logics.