First Order Logic – Why Are the Quantifier Rules of Inference Reasonable?

first-order-logiclogicquantifiers

This picture is from the book "Mathematical Logic, 2nd edition, Christopher C. Leary, Lars Kristensen" :

quantifier rules of inference

I have two questions :

  1. Why are the quantifier rules of inference reasonable as they write in the start of the text ?

  2. Consider these variations :

$$(\{\psi\rightarrow \phi\}, \psi \rightarrow (\exists x \phi)\})$$

$$(\{\phi\rightarrow \psi\}, (\forall x \phi) \rightarrow \psi\})$$

Why is it not reasonable to have these quantifier rules of inference ?

Best Answer

Let us say , we have used matrix/vector/linear algebra with a list of axioms to show (Proof P1) that $0 \cdot x=0$

Now , we must have $0 \cdot 1 = 0$ , $0 \cdot 0 = 0$ , etc
We can not have some $x$ where that fails.
In case there is some $x$ where that fails , then Proof P1 is false : it is not working for that $x$ , whereas we claimed that P1 works & we claimed that P1 is true.
Hence there can be no $x$ where P1 fails.
Hence P1 works for all $x$ :
$[ P1 \implies 0 \cdot x=0 ] \equiv [ P1 \implies \forall x : 0 \cdot x=0 ]$

It is indeed true that we can change $\forall$ to $\exists$ :
$[ P1 \implies 0 \cdot x=0 ] \implies [ P1 \implies \exists x : 0 \cdot x=0 ]$
That is weaker claim. It is not equivalent claim.
That might be a new rule of inference like this :
$[ \forall x P ] \implies [ \exists y P ]$
Example :
"for all rational x , 2x is rational" versus "there exists some rational x , 2x is rational"
"all humans are made of carbon" versus "there is some human made of carbon"

The other rule is similarly justified.

It will roughly go like this :
Without Details on what $x$ is , we can show (Proof P2) that "when $\sin x$ is rational , then we must have that $\pi$ is irrational" : Proof P2 is not the Issue here.
We must then show that $\sin x$ rational does occur. Automatically , we can show $\pi$ is irrational via P2.
In case there is no way to make $\sin x$ rational , then we can make no claim on $\pi$ irrationality.

In other words : (A) When there does not exist a single $x$ where $\sin x$ is rational , then we can not infer the conclusion. (B) When there does exist at least a single $x$ where $\sin x$ is rational , then we can infer the conclusion using P2. (C) When for all $x$ we have $\sin x$ is rational , then we can still infer the conclusion using P2 , though that is unnecessary overwork.

OP variations are valid , though one is weaker & the other is overkill.

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