Why is the class of basic statements permitted by "p" a subclass of the class of basic statement permitted by "q"?

Question

Excerpt from section 37 "Logical Ranges. Notes on the Theory of Measurement" of "The Logic of Scientific Discovery":

If a statement "p" is more easy to falsify than a statement "q", because is of a higher level of universality or precision, then the class of the basic statements permitted by "p" is a proper subclass of the class of the basic statements permitted by "q".

I don't quite get how Popper makes this assertion. Take the following two statements:

1. Plane geometry shapes are made of lines
2. Squares are made of lines

Clearly, #1 is more universal than #2. But doesn't #1 permit more basic statements than #2?

From #1, I can deduce:

• Triangles are made of lines
• Rectangles are made of lines
• Polygons are made of lines
• ...

How can the class of basic statements permitted by #1 be a subclass of the class of basic statements permitted by #2?

Answer 1

You have to consider falsifiable statements.

Consider e.g. the prediction "All ravens are black". It is more general [it has a higher level of universality] that "All ravens in my backyard are black" because it applies to a wider population.

Thus, it is more easily falsifiable: if I met a white raven in the street this fact will falsify the first prediction but not the second one.

The same for statements with a "higher level of precision": a physical law predicting that we can measure magnitude A with an approximation of +/- 1% will be more easily falsifiable than a law predicting that we can measure magnitude A with an approximation of +/- 10%.

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Thus, more general (more universal) will mean more easily falsifiable, and thus the number of cases falsifying it will include the class of falsifiers of the less general: the white raven in the street will falsify the general law but not the more restricted one ("All ravens in my backyard are black").

This means that the class of falsifiers of the more restricted law is a subclass of the class of falsifiers of the more general one.

If we define the class of "permitted cases" as the complement of the class of falsifiers, from elementary laws of sets we have that:

the class of permitted cases corresponding to a more general law will be a subclass of the class of permitted cases corresponding to a more restricted law.