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I'm reading a logic book by Tarski and it states:

"Rule of substitution: If a universal sentence, which has already been accepted as true, contains sentential variables, and if these variables are replaced by other sentential variables or by sentential functions or by sentences—always substituting the same expression for a given variable throughout—, then the sentence obtained in this way may also be recognized as true."

"When we want to apply the rule of substitution, we omit the quantifier and substitute for the variables which were previously bound by this quantifier other variables or related compound expressions any other bound variables which may occur in the sentential function have to remain unaltered, and in the substituted expressions we cannot admit any variables having the same form as the bound ones; finally, if necessary, a universal quantifier is set in front of the expression which is obtained in this way, in order to turn it into a sentence."

then he proceeds with an example:

"applying the rule of substitution to the sentence: for any number $x$ there is a number $y$ such that $x+y=5$ the following sentence can be obtained: for any number $z$ there is a number $y$ such that $z²+y=5$"

how is this possible? isn't "$z²$" wrong since he says "always substituting the same expression for a given variable throughout"?

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  • $\begingroup$ Looks like it may be a typo, unless I'm misreading it. (Which I did when I wrote my earlier, now deleted, comment!) $\endgroup$ – Dave L. Renfro Jan 30 at 10:54
  • $\begingroup$ another thing I dont get is: "Every object is equal to itself: x = x" the proof is: "Use the rule of substitution, and substitute in Leibniz's law "x" for "y"" where Leibniz's law is: "x = y if, and only if, x has every property which y has, and y has every property which x has". how can you do this? isn't y a bound variable? He said in the rule of substitution: "any other bound variables which may occur in the sentential function have to remain unaltered, and in the substituted expressions we cannot admit any variables having the same form as the bound ones" $\endgroup$ – Robert Jan 30 at 11:01
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    $\begingroup$ In the example of the post, Tarski at first omits the quantifier (for any number $x$) and then substitutes $z^2$ for $x$ in the expression $x + y = 5$. Finally, he introduces a universal quantifier (for all $z$) to obtain a sentence. $\endgroup$ – frabala Jan 30 at 11:16
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    $\begingroup$ There may be a formal language and metalanguage conflation here (not sure, however), which wouldn't be all that surprising because the book is likely a translation from Polish (or the non-native English speaker Tarski originally wrote it in English) and because I've found that authors often tend to be careless about these issues, as I suspect the authors, being experts steeped in the subject, can easily overlook difficulties beginners might have in correctly assigning meaning to their prose. $\endgroup$ – Dave L. Renfro Jan 30 at 11:16
  • $\begingroup$ @frabala oooh I got it! I was substituting "x" of the quantifier with "z" smh thank you so much $\endgroup$ – Robert Jan 30 at 11:37
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It's possible, because of the implicit universal quantification of variables. In other words, because every permissible value for the variables maintains the truth of the sentence. Thus, any substitution for the variables maintains/preserves the truth of the sentence.

In your example, (x + y) = 5, holds true for all values of x, and y. z² has the same value as some value of x. So, ((z²) + y) = 5 also holds true.

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