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Recently I come across the following question :

What can be said about the truth value of $Q$ when: $P$ is true and $P\Leftrightarrow Q$ is false?

The answer to this question is "clearly" that Q is false.

But for some time using only my intuition without using the true/false truth values table, I was convinced that we can't decide whether $Q$ is true or false. I was articulating in my mind that $P \Leftrightarrow Q$ is wrong. Hence the relation between P and Q can be anything but equivalence, Then even knowing that P is true we can't decide anything about Q.

I want to identify this maths logical/intuition fallacy. Is this type of "limitations" imposed by the vocabulary of the formal logic we study early in our mathematic courses, and which does not always reflect the human language, is a part of the study in advanced mathematics logic?

Update :

Most of the answers went in the direction of giving the definitions for the equivalence operator Propositions and open statements. I am aware of the mistake concerning these definitions. It is my fault for not stating my background. I was preparing for an instruction session given to some first-year bachelor students. Having an analysis background, I never give this kind of sloppy intuition/ logic fallacies a deep thought. My question can be roughly formulated: " are the study of these sloppy intuition/ logic fallacies part of mathematics"

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    $\begingroup$ 0. I guess that the reason for this fallacy is that theorems of the form $P\iff Q$ are rare in practice; see Henning Makholm’s answer. I would say they are unnatural. Hence we subconsciously “correct” this formula. 1. I am not aware of a branch of mathematics or a textbook that studies logical fallacies. This phenomenon belongs to psychology. Textbooks on philosophy contain some material on fallacies of another, more social kind. $\endgroup$ – beroal Oct 4 '17 at 18:14
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I think your confusion goes back to the Paradox of the Material Implication. Or rather, in this case, the paradox of material equivalence.

The short of it is: our intuitions regarding an English 'if and only if' statement (which, after all, is how we are told to read the $\leftrightarrow$) do not match the mathematically defined truth-functional operator $\leftrightarrow$. Following your example, if I am told that it is false that snow is white if and only if bananas are yellow (a statement most people would indeed hold to be false), and I am also told that snow is white, then I am not going to infer that bananas are not yellow. So yes, there is a mismatch.

But, given that the question involves the mathematically defined operator, you should answer it accordingly and, in a case like this, ignore your intuitions.

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The statement $P\Leftrightarrow Q$ does not say that the meanings of $P$ and $Q$ need to have anything in particular do to with each other. It asserts that $P$ and $Q$ have the same truth value -- nothing more, nothing less.

If $P$ is true, and it is not the case that $P$ and $Q$ have the same truth value, then they must have different truth values, and the only value that is different from "true" (and therefore possible for $Q$ to have) is "false".


One source of confusion is if $P$ and $Q$ contains variables. For example, we could let $P$ be $x=2$ and $Q$ be $x\ne 5$. Then the claim $$ x = 2 \iff x \ne 5 $$ is certainly not a general truth about numbers. However, now the truth values of everything depends on what $x$ is. If $x$ happens to be $5$ then $x=2$ and $x\ne 5$ do happen to have the same truth value (namely false), and in that case the formula $x=2 \Leftrightarrow x\ne 5$ is true.

In ordinary "sloppy" mathematics, we will often say something like

a) $x=2 \Leftrightarrow x\ne 5$ is false.

But what we actually mean by that is

b) $x=4 \Leftrightarrow x \ne 5$ is not always true.

or, with more symbols

c) $\forall x ( x=4 \Leftrightarrow x\ne 5)$ is false.

In the latter case, the quantifier $\forall x$ produces a statement that does not depend on selecting a particular value for $x$ in advance, and therefore it is now fully correct to declare the statement to be false, period.

One of the points of exercises such as you quote is to help you become aware of the unsaid parts of "sloppy" phrasings such as (a) above.

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  • $\begingroup$ The universal quantifier. I have been thinking of this same way of formalizing “$P\iff Q$ is wrong”. How often in practice we actually state this formula without implying a universal quantifier at the root of the formula? $\endgroup$ – beroal Oct 4 '17 at 18:08
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The equivalence $P\iff Q$ is false. What that means is that $P$ and $Q$ have opposite truth values. And, since $P$ is true, $Q$ is false.

Asserting that a statement is false does not mean that there's something wrong with it, whatever that means. You are being lead astray by a false equivalence between true and right (and therefore between false and wrong).

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That the equivalence is false means that it can be given that $P$ is true and false $Q$, or that $P$ is false and $Q$ is true. Since we also admit that $P$ is true, we conclude that $Q$ is false.

Formal language can be understood and managed by a machine. The human, more and more, but introducing statistical elements. But I do believe and hope that it will never come to control all the psychological and cultural variables.

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