eolas/zk/Logical_possibility_and_necessity.md

---
tags: [propositional-logic]
---

# Logical possibility and necessity

## Logical possibility

In distinguishing the properties of
[logical consistency](Logical_consistency.md) and
[validity](Validity_and_entailment.md) we make tacit use
of the notion of **possibility**. This is because when we consider the validity
of an argument we are assessing truth-conditions and this consists in asking
ourselves what could or could not be the case: were it such that _P_, then it
would be the case that _Q_. It is important to understand what possibility means
in the context of logic and how it differs from what we might mean ordinarily
when we use the term.

It is evident from the case of arguments that are valid but not sound that logic
operates with a specialised notion of possibility. For example it has to be the
case that the proposition _Every woman can levitate_ is logically possible since
the following argument is valid:

```
1.  P: Janice is a woman.
2.  P: Every woman can levitate.
3.  C: Janice can levitate.
```

But we know of course that women cannot levitate. When we assert that this is
impossible we are relying on a stronger notion of possibility than logical
possibility. It follows that the concept of possibility can have different
degrees. The scope of the concept of possibility has been the concern of
logicians and philosophers since at least the time of Plato and numerous
different formulations exist. The notion that we mostly work with unreflectively
in everyday life is nomological possibility. This means ‘governed by the
application of laws’ where these laws pertain to our current understanding of
the natural world as determined by physics. Levitation is therefore
nomologically impossible but logically possible.

If logical possibility is not constrained by the laws of physics does it place
any restrictions on what is possible? Logic applies a single restriction, the
law of non-contradiction: a proposition cannot both be true and false at once.
The following propositions are examples of a contradictory propositions.

Some examples of contradictions:

- There is a dog that is not a dog
- Today is Tuesday and today is not Tuesday
- The cat that is dead is alive

From this we can derive the following property of logical possibility:

> A proposition is logically possible just if it does not imply a contradiction.

## Logical necessity

A proposition is _logically necessary_ if it is true in every logically possible
circumstance which is to say: true on every possible truth functional
assignment. Necessity and
[logical truth](Logical_truth_and_falsity.md#logical-truth)
are therefore synonyms: anything that is logically true (a tautology) is true by
necessity (could not be otherwise.)

Further, every logical truth is logically possible but not everything that is
logically possible is logically true. It is possible that it is raining but this
is not logically necessary - it could be otherwise, i.e not raining. However it
is not possible that it could be both raining and not raining.