Accessibility relation
Encyclopedia
In modal logic
, an accessibility relation is a binary relation, written as
between possible worlds
.
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 ánd' or ór' 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 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).
As described in greater detail below:
Necessarily
means that 
is true at every 'possible world' 
such that 
Possibly
means that 
is true at some possible world 
such that 
.
'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
, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by the symbol 
.
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,'
between two 'worlds.' That relation, 
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 things like 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).
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
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 
.
Thus for any sentence
(simple or compound), the compound sentences 
and 
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: 
.
However, the symbols
, 
can also be used to stand for any statement of our language. For example, 
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: 
. The sentence 'It is possible that I walk outside' would look like: 
.
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 logic
s, and the last holds in all normal modal logic
s.
1st Modal Axiom:
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
symbol in the sentence '
' stand for 'it is not the case, or 'not.'
The
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,
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:
stands for 'The apple is red.'
So using the symbols and their meaning described above, the first modal axiom,
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:
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.
stands for 'Spring has not arrived.'
Using the symbols and their meaning described above, the second modal axiom,
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
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:
*
*
*
(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:
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
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
that holds between 'possible worlds' 
and 
only when 
is accessible from 
.
Additionally, to make things more specific, any formula, axiom like
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.
, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by the symbol 
, where 
takes the form of the 'necessity relation' described below.
So, the 'accessibility relation,'
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:
The four types of
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 
are:
'The Reflexive' or *Axiom (T):
This says that if every world is accessible to itself, then any world in which
is true will be a world from which there is an accessible world in which 
is true. Notice this is a variation, more detailed description of the 'necessity' definition above.
'The Transitive' or *Axiom (4) above:
This says that
is transitive, 
is true at a world 
only when 
is true at every world 
accessible from 
. Hence, 
is true at a world 
only when 
is true at every world accessible from every world accessible from 
. Notice that this is a variation, more detailed description of the 'possibility' definition above.
'The Euclidean' or *Axiom (5) above:
This says that
is euclidean. So, 
is true at a world 
if and only if 
is true at some world accessible from 
. 
is true at a world 
if and only if, for every world 
accessible from 
, there is a world 
accessible from 
at which 
is true.
The euclidean
property guarantees the truth of this. If
is true at a world accessible from 
, then if that world is accessible from every other world accessible from 
, it will be true that for every world accessible from 
there is an accessible world in which 
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
is symmetric. If 
is true in a world 
, then in every world 
accessible from 
, there is a world accessible from 
in which 
is true. Since 
is true in 
, this is guaranteed to be true provided that 
is accessible from it, which is precisely what symmetry says.
symmetric?" (David Lewis
, 1996)
In other words, the oddity of 'necessarily possible' can be remedied by simply asking if two statements, one about 'necessity' and one about 'possibility,' are symmetric logically. However, this doesn't really solve the issue at all. Any property characterizing
whether 'transitivity,' 'symmetry,' and so on, is dependent upon the very concepts of 'possibility' and 'necessity.' But in turn, this dependency means that all properties of modal logic depend on 'possibility' and 'necessity,' and not fundamentally 'symmetry,' 'transitivity' and so on. For instance, the first principle says that if a modal statement is classified under this principle, then whenever a statement about 'necessity' is uttered, it is the same as another statement about 'possibility.' To say that 'It's necessary that the apple is red' is to say that 'It's not possible that it is not the case that the apple is red.' In other words, to say that it's 'necessary' that something is to say that it is not 'possible' that something. In fact, in the example, I essentially said that it's necessarily possible that 'the apple is red' because the way I explain what I mean by 'necessity' is through a more detailed sentence about 'possibility,' albeit the statement about 'possibility' just happens to be one framed negatively, where the 'possibility' is that it is not 'possible' that it's not the case that something. Pointing this out helps to show that although the sentence about 'necessity' is characterized by duality, that duality is dependent on the concepts of 'necessity' and 'possibility,' and as long as these concepts are used, they inevitably give rise to the strange paradox about how something can be both 'necessary' and 'possible.' So talking of 'duality' or any other logical property like 'symmetry' or 'transitivity' doesn't clear up this paradox that the language of 'possibility' and 'necessity' give rise to.
Kripke may argue that this objection is a failure to understand what
is supposed to do. 
only uses the concepts of 'necessity' and 'possibility' to show that the seeming contradiction of a 'necessary possibility' is in fact not a contradiction but actually a relation between two propositions characterized in one of the four types, 'duality,' 'symmetry' and so on. In other words, he might say that it's not 'necessary' and 'possible' propositions that are primary, but that they only appear to be until the relation 
comes to light.
But notice that if this were true, the concepts of 'duality' and so on would lack substance. As such, they wouldn't be useful in some instances, like when examining the idea of time as linear, or plotting out different courses of action. In fact,
is useful only when it is articulated as a characterization between propositions of 'necessity' and 'possibility.' So although the property of 'transitivity' can be understood readily (as for instance, a bushel of straw gradually decreasing in size from a handful to two few blades), its application is only meaningful when looking at the specifics of say, the application of moral principles or laws from one society to another or others. At this point, 'necessity' and 'possibility' will certainly come into play since 'transitivity' involves at what point something is 'possible' and something isn't, or 'necessary' for that matter. Put this way, deducing from general concepts of 'transivity' and so on will inevitably lead to propositions about (and thus concepts of) 'necessity' and 'possibility.' 
cannot be understood without 'necessity' and 'possibility,' though 'possibility' and 'necessity' can be understood without the relation 
. So rather than understanding Kripke's contribution as getting rid of the seemingly messiness of 'necessary' and 'possible' propositions, it's better to understand his relation as helping to clarify what's happening when using principles in modal logic, or modal logic in general since, according to logicians, all modal propositions can be characterized in as at least one type of modal axiom. Still, this doesn't clear up the oddity of the very concepts of 'necessity' and 'possibility.'
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
can be necessarily 
without 
being necessarily necessarily 
. 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
in the world 
is interpreted as "it is possible to transform 
into a world in which 
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
is epistemically accessible from 
for an individual 
in 
if and only if 
does not know something which would rule out the hypothesis that 
. We can ask whether the relation is transitive. If 
knows nothing that rules out the possibility that 
and knows nothing that rules the possibility that 
, it does not follow that 
knows nothing which rules out the hypothesis that 
. 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.
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
, an accessibility relation is a binary relation, written as
between possible worldsPossible Worlds
Possible Worlds may refer to:* Possible worlds, a concept in philosophy* Possible Worlds , by John Mighton** Possible Worlds , by Robert Lepage, based on the Mighton play* Possible Worlds , by Peter Porter...
.
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 ánd' or ór' 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 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).
As described in greater detail below:
Necessarily
means that is true at every 'possible world' such that Possibly
means that is true at some possible world such that .'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
, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by the symbol .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,'
between two 'worlds.' That relation, 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
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
(theories about 'modal logic'). It also has applications to things like epistemology (theories about how people know something is true or false), metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
(theories about reality), value theory
Value theory
Value theory encompasses a range of approaches to understanding how, why and to what degree people should value things; whether the thing is a person, idea, object, or anything else. This investigation began in ancient philosophy, where it is called axiology or ethics. Early philosophical...
(theories about morality and ethics), and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
(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 logicModal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
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
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 .Thus for any sentence
(simple or compound), the compound sentences and 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: .However, the symbols
, can also be used to stand for any statement of our language. For example, 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: . The sentence 'It is possible that I walk outside' would look like: .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 logic
Regular modal logic
In modal logic, a regular modal logic L is a modal logic closed under the duality of the modal operators:\Diamond A \equiv \lnot\Box\lnot Aand the rule\to C \vdash \to\Box C....
s, and the last holds in all normal modal logic
Normal modal logic
In logic, a normal modal logic is a set L of modal formulas such that L contains:* All propositional tautologies;* All instances of the Kripke schema: \Box\toand it is closed under:...
s.
1st Modal Axiom:
-
(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
symbol in the sentence '' stand for 'it is not the case, or 'not.'The
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,
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:
stands for 'The apple is red.'So using the symbols and their meaning described above, the first modal axiom,
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:
-
(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.
stands for 'Spring has not arrived.'Using the symbols and their meaning described above, the second modal axiom,
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
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:
*
*
*
(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)
- (T)
-
- (4)
- (4)
-
- (5)
- (5)
-
- (B)
- (B)
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
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
in terms of quantification over possible worlds, subject to the relation of accessibility. A sentence like
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
that holds between 'possible worlds' and only when is accessible from .Additionally, to make things more specific, any formula, axiom like
can be translated into a formula of first-order logicFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
through standard translation
Standard translation
In modal logic, standard translation is a way of transforming formulas of modal logic into formulas of first-order logic which capture the meaning of the modal formulas. Standard translation is defined inductively on the structure of the formula. In short, atomic formulas are mapped onto unary...
. 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, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by the symbol , where takes the form of the 'necessity relation' described below. So, the 'accessibility relation,'
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
means that 
is true at every 'possible world' 
such that 
.
- Possibly
means that 
is true at some possible world 
such that 
.
The four types of
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 are:'The Reflexive' or *Axiom (T):
This says that if every world is accessible to itself, then any world in which
is true will be a world from which there is an accessible world in which is true. Notice this is a variation, more detailed description of the 'necessity' definition above.'The Transitive' or *Axiom (4) above:
This says that
is transitive, is true at a world only when is true at every world accessible from . Hence, is true at a world only when is true at every world accessible from every world accessible from . Notice that this is a variation, more detailed description of the 'possibility' definition above.'The Euclidean' or *Axiom (5) above:
This says that
is euclidean. So, is true at a world if and only if is true at some world accessible from . is true at a world if and only if, for every world accessible from , there is a world accessible from at which is true.The euclidean
Euclidean relation
In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's "Common Notion 1" in The Elements: things which equal the same thing also equal one another.-Definition:...
property guarantees the truth of this. If
is true at a world accessible from , then if that world is accessible from every other world accessible from , it will be true that for every world accessible from there is an accessible world in which 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
is symmetric. If is true in a world , then in every world accessible from , there is a world accessible from in which is true. Since is true in , this is guaranteed to be true provided that is accessible from it, which is precisely what symmetry says.Comment about the 'accessibility relation'
Though useful in all kinds of philosophical applications, the key innovation is, as David Lewis states, that "old disputes give way to new. Instead of asking the baffling question whether whatever is actual is necessarily possible, we could simply try asking: is the relation symmetric?" (David LewisDavid Kellogg Lewis
David Kellogg Lewis was an American philosopher. Lewis taught briefly at UCLA and then at Princeton from 1970 until his death. He is also closely associated with Australia, whose philosophical community he visited almost annually for more than thirty years...
, 1996)
In other words, the oddity of 'necessarily possible' can be remedied by simply asking if two statements, one about 'necessity' and one about 'possibility,' are symmetric logically. However, this doesn't really solve the issue at all. Any property characterizing
whether 'transitivity,' 'symmetry,' and so on, is dependent upon the very concepts of 'possibility' and 'necessity.' But in turn, this dependency means that all properties of modal logic depend on 'possibility' and 'necessity,' and not fundamentally 'symmetry,' 'transitivity' and so on. For instance, the first principle says that if a modal statement is classified under this principle, then whenever a statement about 'necessity' is uttered, it is the same as another statement about 'possibility.' To say that 'It's necessary that the apple is red' is to say that 'It's not possible that it is not the case that the apple is red.' In other words, to say that it's 'necessary' that something is to say that it is not 'possible' that something. In fact, in the example, I essentially said that it's necessarily possible that 'the apple is red' because the way I explain what I mean by 'necessity' is through a more detailed sentence about 'possibility,' albeit the statement about 'possibility' just happens to be one framed negatively, where the 'possibility' is that it is not 'possible' that it's not the case that something. Pointing this out helps to show that although the sentence about 'necessity' is characterized by duality, that duality is dependent on the concepts of 'necessity' and 'possibility,' and as long as these concepts are used, they inevitably give rise to the strange paradox about how something can be both 'necessary' and 'possible.' So talking of 'duality' or any other logical property like 'symmetry' or 'transitivity' doesn't clear up this paradox that the language of 'possibility' and 'necessity' give rise to.Kripke may argue that this objection is a failure to understand what
is supposed to do. only uses the concepts of 'necessity' and 'possibility' to show that the seeming contradiction of a 'necessary possibility' is in fact not a contradiction but actually a relation between two propositions characterized in one of the four types, 'duality,' 'symmetry' and so on. In other words, he might say that it's not 'necessary' and 'possible' propositions that are primary, but that they only appear to be until the relation comes to light.But notice that if this were true, the concepts of 'duality' and so on would lack substance. As such, they wouldn't be useful in some instances, like when examining the idea of time as linear, or plotting out different courses of action. In fact,
is useful only when it is articulated as a characterization between propositions of 'necessity' and 'possibility.' So although the property of 'transitivity' can be understood readily (as for instance, a bushel of straw gradually decreasing in size from a handful to two few blades), its application is only meaningful when looking at the specifics of say, the application of moral principles or laws from one society to another or others. At this point, 'necessity' and 'possibility' will certainly come into play since 'transitivity' involves at what point something is 'possible' and something isn't, or 'necessary' for that matter. Put this way, deducing from general concepts of 'transivity' and so on will inevitably lead to propositions about (and thus concepts of) 'necessity' and 'possibility.' cannot be understood without 'necessity' and 'possibility,' though 'possibility' and 'necessity' can be understood without the relation . So rather than understanding Kripke's contribution as getting rid of the seemingly messiness of 'necessary' and 'possible' propositions, it's better to understand his relation as helping to clarify what's happening when using principles in modal logic, or modal logic in general since, according to logicians, all modal propositions can be characterized in as at least one type of modal axiom. Still, this doesn't clear up the oddity of the very concepts of 'necessity' and 'possibility.'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)
is nomologically necessary means that 
is true at all possible worlds that are nomologically accessible from the actual world. In other words, 
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
Counterpart theory
In philosophy, specifically in the area of modal metaphysics, counterpart theory is an alternative to standard possible-worlds semantics for interpreting quantified modal logic. Counterpart theory still presupposes possible worlds, but differs in certain important respects from the Kripkean view...
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
can be necessarily without being necessarily necessarily . 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
in the world is interpreted as "it is possible to transform into a world in which 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
is epistemically accessible from for an individual in if and only if does not know something which would rule out the hypothesis that . We can ask whether the relation is transitive. If knows nothing that rules out the possibility that and knows nothing that rules the possibility that , it does not follow that knows nothing which rules out the hypothesis that . 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
Deontic logic
Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts...
. 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.