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 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 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).
Read more about this topic: Accessibility Relation
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