My textbook says "If P then Q" and "Q only if P" are logically equivalent, but consider this:

"If it's green, it's poisonous."

"It's poisonous only if it's green."

Now say there's a purple poisonous frog. The first statement is true and the second statement is false. Thus, they're not equivalent.

What am I doing wrong?


In mathematical logic P → Q must be read :

"if P, then Q", as well as : "P only if Q".

Thus "Q only if P" is Q → P, which is not equivalent to P → Q.

You can see also this post as well as this one.

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Here is another way of looking at this.

The statement X if and only if Y expresses the logical equivalence of X and Y and is written X ⇔ Y.

This is the conjuction of the two conditionals X ⇒ Y and Y ⇒ X.

The "if" conjunct corresponds to Y ⇒ X and the "only if" conjuct corresponds to X ⇒ Y.

It should be obvious then, that the statement "if P then Q" is not equivalent to "Q only if P".

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There are implications in your 1st statement that make it false! The only way to make it true, would be if all and only green frogs are poisonous. Then the second statement, poisonous only if green, would be true. An example of this would be if you put green poisonous frogs in a room, together with non-poisonous purple, orange, yellow, etc. frogs. Under these conditions, both statements: if green, then poisonous and poisonous only if green, would be true!

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No; they are not. Please see my intuitive explanation located at Math SE here, which I cannot reproduce here because Philosophy SE lacks MathJax formatting, as at Dec 2015.

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Since your question is "what am I doing wrong", you are making one of two possible mistakes:

One is that you are committing an "appeal to authority" fallacy. You believe the statement because you found it in your textbook. However, the statement is false: "If P then Q" and "Q only if P" are not equivalent.

The other and frankly more likely mistake is that you have misread what your textbook says. Maybe there was a section in the textbook titled "show which of the following statements are true and which are false".

A simple example where the statements are not equivalent: "If a person is a man, then that person is a human being" vs. "A person is only a human being if that person is a man".

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