User:IssaRice/Strength of a mathematical statement: Difference between revisions

Revision as of 19:07, 1 October 2018

Negation

Negating a strong statement produces a weak statement, and negating a weak statement produces a strong statement. If a statement has strong and weak components, then the flip occurs at each stage. For example, in $\forall x W (x)$ with $W (x)$ a weak statement, negating it produces $\exists x \neg W (x)$ , where the strong $\forall x$ has become the weak $\exists x$ , and the weak $W (x)$ has become a strong $\neg W (x)$ . See Gowers's posts for more discussion on this.

Strong vs subset

A puzzle: why do we say P is stronger than Q if P is a subset of Q, but we also say that a theorem is stronger if it is more general (so bigger)?

One reply/intuition uses something like possible world semantics, e.g. see Wei Dai's post on Aumann's agreement theorem. There is just one possible world (a single $ω \in Ω$ ), but our information state is the set of all possible worlds that we cannot distinguish, so the less we know, the more possible worlds we think we could be in.
One visualization is to use a Venn diagram. The stronger the statement, the more our movement is restricted, as we are forced to be in more and more sets.
When we say a strong statement like $\forall x P (x)$ , we are saying $P (x_{1}) \land P (x_{2}) \land \dots \land P (x_{n})$ . When we say a weak statement like $\exists x P (x)$ , we are saying $P (x_{1}) \lor P (x_{2}) \lor \dots \lor P (x_{n})$ . It seems like in both cases we are accumulating more and more things.
In causal inference, I think $X ⊥ ⊥ Y \cup W$ is stronger than $(X ⊥ ⊥ Y) \lor (X ⊥ ⊥ W)$ , even though both seem to use a single "or"-type operation.

External links

https://gowers.wordpress.com/2008/12/28/how-can-one-equivalent-statement-be-stronger-than-another/ (haven't read this yet)
https://gowers.wordpress.com/2011/09/26/basic-logic-connectives-not/ (search strong)
https://gowers.wordpress.com/2011/10/02/basic-logic-relationships-between-statements-negation/ (search "strong")

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+==Negation==
+Negating a strong statement produces a weak statement, and negating a weak statement produces a strong statement. If a statement has strong and weak components, then the flip occurs at each stage. For example, in <math display="inline">\forall x W(x)</math> with <math display="inline">W(x)</math> a weak statement, negating it produces <math display="inline">\exists x \neg W(x)</math>, where the strong <math display="inline">\forall x</math> has become the weak <math display="inline">\exists x</math>, and the weak <math display="inline">W(x)</math> has become a strong <math display="inline">\neg W(x)</math>. See Gowers's posts for more discussion on this.
 ==Strong vs subset==
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 * One visualization is to use a Venn diagram. The stronger the statement, the more our movement is restricted, as we are forced to be in more and more sets.
 * When we say a strong statement like <math>\forall x P(x)</math>, we are saying <math>P(x_1) \wedge P(x_2) \wedge \cdots \wedge P(x_n)</math>. When we say a weak statement like <math>\exists x P(x)</math>, we are saying <math>P(x_1) \vee P(x_2) \vee \cdots \vee P(x_n)</math>. It seems like in both cases we are accumulating more and more things.
+* In causal inference, I think <math display="inline">X \perp\!\!\!\perp Y\cup W</math> is stronger than <math display="inline">(X \perp\!\!\!\perp Y) \vee (X \perp\!\!\!\perp W)</math>, even though both seem to use a single "or"-type operation.
 ==External links==