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The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case $x$ is free while in the second case $x$ is a bound variable. Now for these two assertions:

If $x>2$ then $x>3$.($x$ is understood to be a real number).

For every $n$ if $n>2$ then $n>3$.

Is the $x$ in the first assertion free while the $n$ in the second assertion bound ? The Handbook of Mathematical Discourse states that in the first assertion $x$ is actually universally quantified like the second assertion. Can someone elaborate on that ?

Also, when proving them, their proofs are exactly the same except in the second one we add "let n be arbitrary"(How to Prove it). So do the two assertions differ in their Logical structure ?

Nameless
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2 Answers2

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In translating from English to formal language, one has to be careful to translate the meaning of a phrase rather than merely translating the individual words. Usually, one states "If $x > 2$ then $x > 3$" because I mean to assert its truth for all $x$. Thus, when formalizing the statement, I need to include $\forall x$.

An even better translation than $\forall x: x > 2 \implies x > 3$ would be $x > 2 \models x > 3$.

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Formulas with free variables are said to be TRUE if they hold for all values of the variable (in the domain of the structure at hand). So the unquantified implication IS true in that sense of the word true.

However, in this same technical sense, the formula "If x < 3 then x < 1" is NOT FALSE

since it does hold for some values of x (e.g. x = 5) even though in informal mathematical English, we would all say that it is false as a shorthand for "it is not true." Technically, it is neither true nor false, just like the formula x > 2. But with implications, we do make the informal assumption that the Handbook of Math. Discourse mentions because there is no purpose, outside of formal logic and model theory, for saying that the formula "If x < 3 then x < 1" is neither true nor false.

Ned
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  • In The Handbook of Mathematical Discourse, – SixWingedSeraph Feb 25 '16 at 01:09
  • In the entry "Open Sentence", The Handbook of Mathematical Discourse says "In many circumstances such an assertion is taken as being true for all instantiates of its variables." If I read the sentence, "If $x\gt 2$ then $x \gt 3$, I would assume that it is a (false) claim that it is universally true, so that $x$ is bound. But you have to be careful; there are places where that is not intended. That is why the book says "in many circumstances". But Ned is right: when the statement is an implication an open sentence like that is usually read as a claim that it is true for all $x$. – SixWingedSeraph Feb 25 '16 at 01:18