How to contradict a general statement?

Question

Consider the statement, "All the geometrical shapes on the screen have radius equal to 6".

Suppose on the screen we have 6 circles with radius equal to 6 and also 1 triangle. In order to contradict the above statement, we need to show that there is at least 1 shape with no radius equal to 6. But is it correct if we say, "there is one object with no radius equal to 6 and that is the triangle" as radius is a property defined for circles?

Can someone clear up if we should consider a sentence like, "A triangle has radius equal to 6" a statement or meaningless? Because the truth value of "A triangle has radius equal to 6" will determine the truth value of the statement, "All the geometrical shapes on the screen have radius equal to 6".

Answer 1

Regarding the example, we can apply Russell's analysis of definite descriptions to the statement "the Radius of Triangle has lenght 6" (of the general form "the F is G") :

exists x ((Rad(x,T) and for all y (Rad(y,T) → x=y)) and x=6).

Thus, mimicking Russell's example regarding "the current Emperor of Kentucky is gray", if we assume that "the Radius of Triangle T" is not a referring expression, we have that:

according to Russell's theory, if there are no entities x with property F, the proposition "x has property G" is false for all values of x.

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Regarding the title-question:

"How to contradict a general statement?"

the answer is: with a counter-example showing a specific instance that contradicts it.

E.g. if we have the general statement "every x is P", a counterexample will be an object c that is not-P.

Answer 2

It comes down to how you interpret the meaning of the sentence "x has radius y". (As has been pointed out in the comments, one can meaningfully define the radius of a triangle, but for the sake of argument we take it to be defined only for circles.) Consider the following two interpretations:

(1) x is a circle that has radius y.

(2) x is a circle that has radius y, or x is not a circle.

Under (1), the general statement "All the geometrical shapes on the screen have radius equal to 6" would be false in your case, while under (2) it would be true.

One could also simply say that the statement "All the geometrical shapes on the screen have radius equal to 6" is only valid when all objects on the screen have a radius, i.e. are circles, and in any other case makes a category error. I think this is how people generally react to more extreme examples. For instance, the assertion "Everything depicted on the screen is diesel-powered" makes sense if the screen depicts only locomotives, ships or other objects that are powered in some way, is false if an electric car is shown, but becomes invalid if the screen depicts a stone. The reason the OP's question seems confusing is because the concepts of radius and geometrical object are closely related in our minds, while those of diesel power and stone are not.

Answer 3

Mathematics or some other means of proof to suggest otherwise is generally the definitive means of negating a generalizing statement. As it is to any claim or statement. If it is presented as a factual statement, evidence that goes against this would render the statement false. If it’s a generalization (ie. loosely using ‘all’ in place of ‘most’- such as “all these trees are squirming with squirrels’ as opposed to ‘Exactly 82% of these trees are teeming with squirrels’) then the standards of a -generalization- are less strictly based than that of a presentedly factual statement. Ergo, a generalization isn’t always regarded as an evidence-based fact as opposed to the latter. I’m order to negate something's factual nature, or must first be presented as a fact. Otherwise to negate a -generalization- you must merely provide enough evidence to suggest a contrary or opposite generalization, albeit by less strictly defined guidelines. All have a radius of 6 presented as a generalization wouldn’t be negated in its entirety despite one or two shapes being 6.2 or 6.5, whereas is it was presented specifically as a factual statement, the existence of these one or two anomalies would negate it by definition, were it presented as a fact to begin with.