If a sentence, especially one with two or more negations, does not yield up its secrets on first reading, try contraposition.
In syllogistic logic contraposition is an operation where the subject and predicate terms are changed around and replaced by their term complements (read “negation”). For example:
All even numbers are divisible by 2. (True)
All numbers not divisible by 2 are non-even. (True)
Some solids are non-hexahedra. (True)
Some hexahedra are non-solids. (False)
Here, contraposition does not always preserve the truth of the original statement (as can be seen). In propositional and predicate logic though, the contrapositive is equivalent to the original statement, and that statement is a conditional. The mechanism is rather similar: we negate the antecedent and the consequent and swap them round.
If the accountant sees the figures, he will quit.
If the accountant hasn’t quit, then he hasn’t seen the figures yet.
Statements such as this where a pronoun in the consequent is anaphoric on the antecedent of the conditional can be easily turned into statements with relative clauses.
The accountant who has seen the figures has quit.
This is but a step away from complex sentences with relative clauses and multiple negations, such as lawyers are fond of. The additional complication is that parts of the sentence need not be in an order which makes it immediately obvious what is what. Here is an example from a limitations of liability clause:
"Nothing herein shall be taken to exclude or limit the Council’s liability for: (iii) any liability which cannot be excluded or limited by applicable law."
We work backwards.
First, restate the sentence identifying the relative clause:
The Council’s liability which cannot be excluded or limited by applicable law shall not be taken to be excluded or limited in this agreement (‘herein’).
Then, turn this into a conditional:
If the Council’s liability is not excluded or limited by applicable law, then it is not taken to be excluded or limited in this agreement.
Restate by contraposition and simplify:
If this agreement excludes or limits the Council’s liability, then so does the applicable law.
The second part of the sentence (the consequent) is in fact a kind of supposition. What it says is that if the agreement in question exludes or limits the Council's liability, then the liability will have been exluded or limited by the applicable law too, but students often find it hard to get their heads around the future perfect tense.
In syllogistic logic contraposition is an operation where the subject and predicate terms are changed around and replaced by their term complements (read “negation”). For example:
All even numbers are divisible by 2. (True)
All numbers not divisible by 2 are non-even. (True)
Some solids are non-hexahedra. (True)
Some hexahedra are non-solids. (False)
Here, contraposition does not always preserve the truth of the original statement (as can be seen). In propositional and predicate logic though, the contrapositive is equivalent to the original statement, and that statement is a conditional. The mechanism is rather similar: we negate the antecedent and the consequent and swap them round.
If the accountant sees the figures, he will quit.
If the accountant hasn’t quit, then he hasn’t seen the figures yet.
Statements such as this where a pronoun in the consequent is anaphoric on the antecedent of the conditional can be easily turned into statements with relative clauses.
The accountant who has seen the figures has quit.
This is but a step away from complex sentences with relative clauses and multiple negations, such as lawyers are fond of. The additional complication is that parts of the sentence need not be in an order which makes it immediately obvious what is what. Here is an example from a limitations of liability clause:
"Nothing herein shall be taken to exclude or limit the Council’s liability for: (iii) any liability which cannot be excluded or limited by applicable law."
We work backwards.
First, restate the sentence identifying the relative clause:
The Council’s liability which cannot be excluded or limited by applicable law shall not be taken to be excluded or limited in this agreement (‘herein’).
Then, turn this into a conditional:
If the Council’s liability is not excluded or limited by applicable law, then it is not taken to be excluded or limited in this agreement.
Restate by contraposition and simplify:
If this agreement excludes or limits the Council’s liability, then so does the applicable law.
The second part of the sentence (the consequent) is in fact a kind of supposition. What it says is that if the agreement in question exludes or limits the Council's liability, then the liability will have been exluded or limited by the applicable law too, but students often find it hard to get their heads around the future perfect tense.
The material we work with can vary in complexity, and the patterns may not be apparent at first sight. Sometimes we may find it convenient to contrapose by implicit negations, as in this example:
If an accountant makes a mistake, the calculations are out.
If the calculations are true and correct, the accountants have been very scrupulous.
One way or another, contraposition tends to produce reliable and satisfying outcomes.
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