If the statement If it is snowing then it is cold outside is assumed to be true is its reverse If it is cold outside then it is snowing also always true?

Answer 1

It is an inverse statement, which is not necessarily true. It is one of the laws of logic and logical reasoning.

What makes it illogical is:

• It may be cold but not cold enough.
• It can be too cold to snow.
  
• The air may be dry; without moisture, it will not snow.
• It may have snowed before and it may snow later, but it is not snowing now.
• It may be cold without snow falling because the temp is warming.
• It may be cold without snow falling because it is raining.
• It may be cold without snow falling because sleet is falling instead.

Answer 2

From a logic point of view, then YES, the second statement you give is always true (regardless of the temperature outside); this is the weird logic of "implies" (→)

A B →

T T T

T F F

F T T

F F T

If the first statement A (it is snowing) is true, then the overall truth of the sentence (if it is snowing then it is cold outside) is based on the truth of the second statement B (it is cold outside).

However, if the first statement A is false (ie it is not snowing) then the statement "if it is snowing then it is cold outside) is true regardless of whether it is cold outside or not.

All you are told is the state of the temperature if it is snowing; if it is not snowing then no determination of the state of the temperature can be made.

It is the logic of implies that is the downfall of many mathematical proofs: if ANY step is false, then you can prove anything you like. For example you can prove that 1 = 2 by employing the false step of dividing two different things by zero (though it is often hidden within algebra).

The sentence you give as its negation is not the logical inverse. The logical inverse is "If it is not cold outside then it is not snowing" - as you know that it is true that it is cold if it is snowing, then if it is not cold then it cannot be snowing must be also be true (otherwise if it was not cold and it was snowing, then is it not true that if it is snowing then it is cold).

This sentence also agrees with the inability to infer temperature if it is not slowing as if it is cold (statement A being false) then whether or not it is snowing (statement B), the sentence is true - you have no ability to infer the falling of snow if it is cold:

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No, it's not always true because it could be below freezing with no snow falling. It rarely snows in Antarctica, it just appears to be because of all the snow being blown around, but it is still very cold there.

Answer 3

no it is NOT true, it can be too cold to snow.