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I am trying to get better at writing math ideas rigorously and symbolically. Consider the proposition: "The sum of two odd integers is an even integer."

Formally (I think), this can be expressed as: (1) "For all integers $x$ and $y$ such that $x=2k+1$ and $y=2q+1$ where $k$ and $q$ are integers, $x+y=2r$ where $r$ is an integer."

Would it be more formal to write: (2) "For all integers $x$ and $y$, if there exist integers $k$ and $q$ such that $x=2k+1$ and $y=2q+1$, then $x+y=2r$ where $r$ is an integer."?

Does statement(1) have the same logical structure as statement(2); that is, is there an implicit "if...then..." in statement(1)?

Is the following symbolic expression correct? $$\forall x,y\in\mathbb{Z}: (\exists k,q\in\mathbb{Z}: x=2k+1 \land y=2q+1 \implies \exists r\in\mathbb{Z}: x+y=2r)$$

**Every time I introduce a new variable, do I need a logical quantifier?

**In general, given proposition $P$, is $\forall x,y\in \mathbb{Z}: P(x)$ logically equivalent to $x,y\in\mathbb{Z} \implies P(x)$? If so, it seems that a statement in Predicate Logic has been replaced with a statement using Propositional Logic and a notion of sets?

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2 Answers 2

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It would be much simpler to put it as follows:$$(\forall m,n\in\Bbb Z)(\exists r\in\Bbb Z):(2m+1)+(2n+1)=2r.$$There is no need to introduce all those other symbols. But, yes, what you wrote is correct, except that instead of $2r+1$ you should simply have $2r$.

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  • $\begingroup$ Thanks, I replaced $2r+1$ with $2r$ in the question. $\endgroup$
    – Mithrandir
    Sep 14, 2020 at 18:27
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Every time I introduce a new variable, do I need a logical quantifier?

Yes, absolutely. Sometimes universal quantifiers are suppressed (that is, we write down a formula with some free variables but think in terms of that formula's universal closure) but in my opinion this is bad practice especially when first learning formal logic.

In general, given proposition $P$, is $\forall x,y\in \mathbb{Z}: P(x)$ logically equivalent to $x,y\in\mathbb{Z} \implies P(x)$?

Not really, since the latter hasn't introduced $x$ and $y$ via a quantifier. The former is a sentence, the latter merely a formula; the former is however (modulo some mild rewriting) the universal closure of the latter.

If so, it seems that a statement in Predicate Logic has been replaced with a statement using Propositional Logic and a notion of sets?

Implicitly taking universal closures does let us strip away an "outermost" universal quantifier. However, it doesn't get rid of all quantifiers in general. Consider for example the $\epsilon$-$\delta$ definition of continuity, which has the form $$\forall \epsilon\in\mathbb{R}_{>0}\exists \delta\in\mathbb{R}_{>0}\forall x\in\mathbb{R}[\mbox{stuff}].$$ We can "elide" the "$\forall\epsilon\in\mathbb{R}_{>0}$" by looking instead at a formula whose universal closure is that sentence, but we can't similarly erase the remaining quantifiers. So very little "propositionalization" is actually possible.

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