Accessibility Relation - Basic Review of (Propositional) Modal Logic

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 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 logics, and the last holds in all normal modal logics.

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:

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.

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