How is this argument deductively invalid?

Question

I am reading the book An Introduction to Formal Logic by Peter Smith. I was checking the following argument (Exercise-1 Question no-15) :

Miracles cannot happen. Why? Because, by definition, a miracle is an event incompatible with the laws of nature. And everything that happens is always consistent with the laws of nature.

Now, the author has written the following in the answer sheet:

Invalid. The premisses tell us that in fact no miracle ever happens. But they don’t tell us that miracles cannot happen. Compare:

Crimes cannot happen. Since, by definition, a crime is an act incompatible with the criminal law. And everything that happens is always consistent with the criminal law.

In a law-abiding community the premisses may be true: it doesn’t follow that criminal activity is impossible!

The author and some people seem to stress that 'And everything that happens is always consistent with laws of nature' means everything that has happened or is happening is consistent with laws of nature. It doesn't eliminate the possibility that anything can happen in future which is incompatible with the laws of Nature.

Similarly, they want to stress that 'No miracle ever happens' doesn't eliminate the possibility that it can happen in future. However, 'Miracle cannot happen' is the statement that eliminates that possibility. The same thing can be said about the second argument.

Now, Is the author making the right point? I mean we can certainly have counter-examples:

1. Two plus two equals four.
2. Zero cannot divide any number.

Now, does the former statement mean that up to now two plus two has been equal to four but it doesn't need to remain the same way in future? Does the latter statement mean that zero never divided, is not dividing and will not divide any number? I mean is it conceivable that the former statement can be false but still the latter statement is always true?

The way these simple present tenses is interpreted by the author seems wrong. That's why I have cross-posted it on English Language and Literature Stack Exchange. So, shouldn't this argument be deductively valid?

Answer 1

I agree with you that the example can be misleading.

IMO, the key point is the usage of the modal operator expressing possibility, that is not present in the premises of the argument.

If we formalize the premises in predicate logic, what we get is the conclusion:

$\lnot \exists x (\text {Miracle}(x) \land \text {Happen}(x))$,

that reads: "No miracle happens."

This (following the Author's comment) asserts that miracles do not happen, not that it is not possible that... This one need modalities to be expressed:

$\lnot \diamond \exists x (\text {Miracle}(x) \land \text {Happen}(x))$.

Answer 2

Suppose we have a chess tournament. After the tournament ends, we analyze the results of all of the games. We come to the conclusion that:

"Every move that gets made does not involve a pawn promoting into a knight."

In other words we might say that:

"Knight promotion does not happen."

However, in the rules of chess, it's still possible for a pawn to promote into a knight. In other words, knight promotion can happen.

Thus, the argument "knight promotion does not happen implies that knight promotion cannot happen" is an invalid argument.

Perhaps as a better example there may exist some positions in chess or other games where a position of pieces on the board can happen in accordance with the ordinary rules of chess, but has never, in fact, happened in a game that people have played.

Also, to say that something cannot happen in the real world implies that such couldn't happen at any future events. Everything that happens being consistent with the laws of nature applies to past and present events. The quote doesn't say that everything that will happen will be consistent with the laws of nature. Thus, the premise of the argument that everything that is happening and has happened doesn't apply to future events, while the claim that something cannot happen does apply to future events. So, by that account also, the argument is invalid... at least as stated.

Answer 3

I disagree with the author on both counts:

• By the principle of charity, the crime argument is valid, albeit unsound (its 2nd premise is false, whichever way one interprets “everything that happens is always consistent with the criminal law”).

• On the other hand, the miracles argument also is valid; however its 2nd premise (“everything that happens is always consistent with the laws of nature”) may or may not be true.

  The author rightly observes that the premises together say that no miracle ever happens, i.e., miracles never happen. However, whether “never happens” necessarily implies “cannot happen” really is a matter of semantics: the author says nay, while I say aye (so, when I state that speed is never negative, I mean it in the sense that speed can never be negative).

  (If the premises had instead been along the lines of “miracles do not happen”, then yes, reasonably, that wouldn't have necessarily implied that “miracles cannot happen”.)