1 |
(i)
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Valid.
(Often called "modus tollens", or "modus tollendo
tollens".)
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(ii) |
Not valid.
(This fallacious form of argument is sometimes called "affirming
the consequent".)
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(iii) |
Valid.
(Sometimes called "modus tollendo ponens".)
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(iv)
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Valid.
(Called, as you might have guessed, "modus ponendo tollens".
How does one remember which modus is which. Well, modus ponendo
tollens is the mode (of inference) which denies (tollens) by positing
(ponendo). That is, the conclusion is negative (¬y)
and the first premise is positive (j).
And so on.)
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(v) |
Not valid.
If j is true, both [j®y] and [j®¬y] will be true.
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2 |
(i) |
Yes.
(Though "beat" in the original sentence is ambiguous. If
it is referring to the past, the translation should omit "will".)
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(ii) |
No.
The first sentence says something general, but the "it"
in the second sentence must refer to some particular thing. Suppose
it refers to West Ham, and they do not win. Then the second sentence
will be false, because some team will win. But the first sentence
is, presumably, true.
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(iii) |
No.
Beating Valencia may be necessary for winning the cup without being
sufficient.
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(iv) |
Yes.
We could equally translate it as "[¬Manchester United will win
the cup « Valencia will win
the cup]"; or as "¬[Manchester United will win the cup « Valencia will win the cup]". (Check
that these really are equivalent.)
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(v) |
No.
If Mary comes and John doesn't, the second sentence will be false,
but the first could still be true. The suggested translation would
have been correct if the first sentence had been, "John will
not come unless Mary comes, in which case he will".
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3 |
(i) |
¬[JÙ¬M]
would do. [J®M] is false
just when J is true and M is false. So it is false just when [JÙ¬M]
is true. So it is true just when ¬[JÙ¬M]
is true. ¬[¬MÙJ] would do
instead. If you have a different answer, check whether it is right
by means of a truth-table.
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(ii) |
[¬JÚM]
would do. [J®M] is true whenever
J is false – i.e. whenever ¬J is true. It is also true whenever M
is true. Otherwise it is false. So it is true just when one or the
other (or both) of these is true – i.e. when [¬JÚM]
is true.
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(iii) |
[JÙ¬M]
would do. (Or, of course, ¬¬[JÙ¬M].) [JÙ¬M]
is true just when [J®M] is
false.
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(iv) |
[¬JÚ¬M],
for instance. [JÙM] is false
just when either J or M or both are false.
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(v) |
[¬JÙ¬M].
[JÚM] is false just when
both J and M are false.
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(vi) |
[¬[JÙ¬M]Ù¬[MÙ¬J]],
for instance. One way of arriving at this is by remembering that [j«y]
is equivalent to [[j®y] Ù[y®j]]. |