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I'm learning predicate logic and in the textbook I'm using it's written that "whether or not an assignment σ satisfies a formula α solely depends on the values that σ assigns to the free variables in α". I can't understand precisely why this is so. Can anyone provide a clear explanation?

Thank you very much

user405159
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2 Answers2

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Consider a formula without free variables like $\forall x \,. \exists y \,. y < x$. This formula is either true or false, depending on what structure it is applied to, but not on the values of $x$ and $y$. For instance, if $<$ stands for the usual strict ordering relation, it is false if applied to the natural numbers and true if applied to the integers.

Now consider $\exists y \,. y < x$. Here $x$ is a free variable and the formula's truth depends on its value. In fact, you can use this formula to define a set: for the natural numbers, it would be all numbers except $0$.

I suspect the context of that comment in your book is the inductive definition of truth of a formula. If that's the case, read on. When we define truth of a formula, we need values for the free variables. We can't decide whether $x > 0$ is true or false without knowing the value of $x$.

So, in the definition of truth of a formula we introduce an environment, which assigns values to all variables, both free and bound. How do we choose the values of the bound variables? Fortunately, it doesn't matter, precisely because of the statement in your book. Hence we can make arbitrary choices for them and keep the definition simple.

Why doesn't it matter? The definition of truth of a formula that starts with a quantifier tells you to take the provided environment and derive other environments that have given values for the variable bound by that quantifier. Hence, the value assigned to it by the provided environment is ignored.

Fabio Somenzi
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A free variable in a formula acts as a pronoun : "it is red" is true if with "it" I mean my shirt, while it is false if with "it" I mean my car.

Thus, in order to evaluate the truth value of the formula :

$x=0$,

we have to see what value (in this case : a number) the assignment $\sigma$ assocaites to the variable $x$.

If $\sigma(x)=1$, then $(x=0)[\sigma]$ is false : $\sigma \nvDash (x=0)$, while if $\sigma(x)=0$, then $(x=0)[\sigma]$ is true : $\sigma \vDash (x=0)$.

For more details, you have to see the semantical specifications on your textbook...

Mauro ALLEGRANZA
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