Re: A implies I


William of Ockham wrote:
The following argument for the 'existential import' of the universal
proposition just came to mind.

1. Assume 'every car in the street is green' is true.

2. Then 'no car in the street' is green is false, for 'every A is B'
and 'no A is B' are contraries: they cannot be true at the same time.

3. But 'no car in the street is green' is equivalent to 'it is not the
case that some car in the street is green', i.e. is equivalent to the
negation of 'some car in the street is green'.

4. But if it is false, the negation of the negation of 'some car in
the street is green' is true.

5. So 'some car in the street is green' is true.

6. So (1, 5) 'every car in the street is green' implies 'some car in
the street is green'.


Obviously this is not true in predicate logic, if 'every car in the
street is green' is read as, not for some x, x is a car in the street
and x is not green, and if 'no car in the street is green 'not for some
x, x is a car in the street and x is green. For both of these can be
true (if there are no cars in the street). But it does seem odd to say
that every A is B, and no A is B, could both be true.

I am familiar with at least one system of natural deduction using
predicate logic that explicitly deals with nonemtpy sets. In this
case, classical rules apply, and a universal affirmation A does imply
an existential afformation I. But predicate logic is also used in
connection with empty sets, and in this case, this case, an
existential affirmation doesn't necessarily follow from a universal
one. Statements about the emtpy set do tend to appear odd or
nonsensical when they are reinterprted classically. If you aren't aware
of the differences that can occur when you disallow or allow the empty
set, it's (frightfully) easy to become confused.

.

Relevant Pages

  • The suppressed term of negation
    ... Rule 1) Negation moves an object or variable from one group to another group, such as moving a car from a group of red cars to a group of differently coloured cars. ... So in this case, if we want to retain the fact that the coloured car we have negated is red, then we must stipulate a group to which the term red belongs. ... For then the negation refers to car and not red - "a red non-car", rather than the original proposition "a non-red car". ...
    (sci.logic)
  • Re: The suppressed term of negation
    ... the car belongs, such as the class of machines or red objects. ... Empty = = Universe ... negation, for while a red car is one colour, its colour negation could ...
    (sci.logic)
  • Re: A implies I
    ... Then 'no car in the street' is green is false, ... negation of 'some car in the street is green'. ... Of course it's odd; just as it would be odd to say that every car on ... be true (by the modern interpretation) in the above case. ...
    (sci.logic)
  • A implies I
    ... proposition just came to mind. ... Then 'no car in the street' is green is false, ... negation of 'some car in the street is green'. ... But it does seem odd to say ...
    (sci.logic)