68 lines
3 KiB
Markdown
68 lines
3 KiB
Markdown
---
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tags: [propositional-logic]
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---
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# Logical possibility and necessity
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## Logical possibility
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In distinguishing the properties of
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[logical consistency](Logical_consistency.md) and
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[validity](Validity_and_entailment.md) we make tacit use
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of the notion of **possibility**. This is because when we consider the validity
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of an argument we are assessing truth-conditions and this consists in asking
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ourselves what could or could not be the case: were it such that _P_, then it
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would be the case that _Q_. It is important to understand what possibility means
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in the context of logic and how it differs from what we might mean ordinarily
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when we use the term.
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It is evident from the case of arguments that are valid but not sound that logic
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operates with a specialised notion of possibility. For example it has to be the
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case that the proposition _Every woman can levitate_ is logically possible since
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the following argument is valid:
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```
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1. P: Janice is a woman.
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2. P: Every woman can levitate.
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3. C: Janice can levitate.
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```
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But we know of course that women cannot levitate. When we assert that this is
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impossible we are relying on a stronger notion of possibility than logical
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possibility. It follows that the concept of possibility can have different
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degrees. The scope of the concept of possibility has been the concern of
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logicians and philosophers since at least the time of Plato and numerous
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different formulations exist. The notion that we mostly work with unreflectively
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in everyday life is nomological possibility. This means ‘governed by the
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application of laws’ where these laws pertain to our current understanding of
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the natural world as determined by physics. Levitation is therefore
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nomologically impossible but logically possible.
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If logical possibility is not constrained by the laws of physics does it place
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any restrictions on what is possible? Logic applies a single restriction, the
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law of non-contradiction: a proposition cannot both be true and false at once.
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The following propositions are examples of a contradictory propositions.
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Some examples of contradictions:
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- There is a dog that is not a dog
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- Today is Tuesday and today is not Tuesday
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- The cat that is dead is alive
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From this we can derive the following property of logical possibility:
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> A proposition is logically possible just if it does not imply a contradiction.
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## Logical necessity
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A proposition is _logically necessary_ if it is true in every logically possible
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circumstance which is to say: true on every possible truth functional
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assignment. Necessity and
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[logical truth](Logical_truth_and_falsity.md#logical-truth)
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are therefore synonyms: anything that is logically true (a tautology) is true by
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necessity (could not be otherwise.)
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Further, every logical truth is logically possible but not everything that is
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logically possible is logically true. It is possible that it is raining but this
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is not logically necessary - it could be otherwise, i.e not raining. However it
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is not possible that it could be both raining and not raining.
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