Reasoning with may and could is no less a challenge to a student of logic than to a student of English, and both would rather banish the modality outside the sentence than wait for it to come up in a mid-position and then discover that the sentence does not say what they meant it to say.
For example, ‘Gordon may have apologized for the gaffe,’ will come out as: ‘Maybe Gordon has apologized for the gaffe,’ through the mouth of someone unsure about the force of ‘may’, or ‘It is possible that Gordon has apologized for the gaffe,’ for the purposes of deduction with modal logic.
The adverb ‘maybe’ at the beginning of a sentence may seem just the trick to avoid the uncertainty of ‘may’ inside it, but true to the theory that no exact synonyms exist in natural languages, the overall effect may be different. The sentence-initial ‘maybe’ may be begging a response from the listener, something the more matter-of-fact ‘may’ in a mid-position does not do.
Possibility can be expressed in terms of necessity, and modal logic has developed a guide, of sorts, to paraphrasing into non-modal language. In so far as the modal verb expresses possibility, the guide is reliable. Where other concepts (ability, permission, criticism, regret, etc) are thrown into the mix, this translation will not suffice.
It is necessary that P = It is not possible that not P
It is not necessary that P = It is possible that not P
It is possible that P = It is not necessary that not P
It is not possible that P = It is necessary that not P
Thus, ‘Gordon may have apologized for the gaffe,’ is ‘It is possible that he did,’ or ‘It is not necessary that he didn’t.’ And, ‘Gordon must have apologized for the gaffe,’ is ‘It is not possible that he didn’t,’ or ‘It is necessary that he did.’
For negations: ‘Gordon couldn’t have apologized for the gaffe’, comes out as ‘It is not possible that he did,’ or ‘It is necessary that he didn’t’, while ‘Gordon may not have apologized for the gaffe,’ is rendered as ‘It is possible that he didn’t,’ and ‘It is not necessary that he did.’
As can be seen, the position of the negation in our paraphrases changes the degree of possibility.
From the logical point of view, the implications of the particular statements are worked out somewhat differently. Thus, if we know that Gordon necessarily apologized for the gaffe, we can infer that he possibly apologized for the gaffe, but not the other way round. If we know that Gordon necessarily did not apologize for the gaffe, then he possibly did not do so, but not the other way round. The inference is always from more to less, or stronger to weaker.
For example, ‘Gordon may have apologized for the gaffe,’ will come out as: ‘Maybe Gordon has apologized for the gaffe,’ through the mouth of someone unsure about the force of ‘may’, or ‘It is possible that Gordon has apologized for the gaffe,’ for the purposes of deduction with modal logic.
The adverb ‘maybe’ at the beginning of a sentence may seem just the trick to avoid the uncertainty of ‘may’ inside it, but true to the theory that no exact synonyms exist in natural languages, the overall effect may be different. The sentence-initial ‘maybe’ may be begging a response from the listener, something the more matter-of-fact ‘may’ in a mid-position does not do.
Possibility can be expressed in terms of necessity, and modal logic has developed a guide, of sorts, to paraphrasing into non-modal language. In so far as the modal verb expresses possibility, the guide is reliable. Where other concepts (ability, permission, criticism, regret, etc) are thrown into the mix, this translation will not suffice.
It is necessary that P = It is not possible that not P
It is not necessary that P = It is possible that not P
It is possible that P = It is not necessary that not P
It is not possible that P = It is necessary that not P
Thus, ‘Gordon may have apologized for the gaffe,’ is ‘It is possible that he did,’ or ‘It is not necessary that he didn’t.’ And, ‘Gordon must have apologized for the gaffe,’ is ‘It is not possible that he didn’t,’ or ‘It is necessary that he did.’
For negations: ‘Gordon couldn’t have apologized for the gaffe’, comes out as ‘It is not possible that he did,’ or ‘It is necessary that he didn’t’, while ‘Gordon may not have apologized for the gaffe,’ is rendered as ‘It is possible that he didn’t,’ and ‘It is not necessary that he did.’
As can be seen, the position of the negation in our paraphrases changes the degree of possibility.
From the logical point of view, the implications of the particular statements are worked out somewhat differently. Thus, if we know that Gordon necessarily apologized for the gaffe, we can infer that he possibly apologized for the gaffe, but not the other way round. If we know that Gordon necessarily did not apologize for the gaffe, then he possibly did not do so, but not the other way round. The inference is always from more to less, or stronger to weaker.
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