In modal logic, an accessibility relation is a binary relation, written as between possible worlds.
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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 a possible situation. In every case, a possible world is contrasted with an actual situation. Earth one minute from now is a "possible world". Earth as it is 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
Possibly
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
The 'accessibility relation' is a relationship between two 'possible worlds.' More preciselyplease clarify definition, 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,'
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:
These operators can be attached to a single sentence to form a new compound sentence.
For example,
But the symbol
Thus for any sentence
However, the symbols
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:
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 is enough to be literate in order to learn about the 'accessibility relation,' but may not be 'necessary' because the relation could be learned in different ways (such as 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
The
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,
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:
So using the symbols and their meaning described above, the first modal axiom,
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.
Using the symbols and their meaning described above, the second modal axiom,
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
An arrow stands for "then" where the left statement before the arrow is the "if" of the entire sentence.
Other Modal Axioms:
*
*
*
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
The relation of accessibility can now be defined as an (uninterpreted) relation
Additionally, to make things more specific, any formula, axiom like
The Four Types of the 'Accessibility Relation' in Formal Semantics
'Formal semantics' studies the meaning of statements written in symbols. The sentence
So, the 'accessibility relation,'
Each type is either about 'possibility' or 'necessity' where 'possibility' and 'necessity' is defined as:
The four types of
Reflexive, or *Axiom (T) above:
If
Transitive, or *Axiom (4) above:
If
Euclidean or *Axiom (5) above:
If
Symmetric or *Axiom (B) above:
If
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:
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
Another interpretation of the 'accessibility relation' with a physical meaning was given in Gerla 1987 where the claim “is possible
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
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.