# Are the following premises and conclusions begging the question?

Argument 1

P1 Everything that is not X is Y

P2 Given that Z is not X

C Therefore Z is Y

Argument 2

P1 Everything that is X is not Y

P2 Given that Z is X

C Therefore Z is not Y

To be clear 'is' between emboldened terms stands for 'is a subset of' or ⊆.

If I say: "Everything that is not a fruit (not-X) is edible (Y). Everything that is a fruit (X) is not edible (not-Y). Bob (Z) is not a fruit (not-X). Therefore Bob (Z) is edible (Y)".

Does this make sense?

Similarly, if I say "Everything that is not a fruit (not-X) is edible (Y). Everything that is a fruit (X) is not edible (not-Y). This tomato (Z) is a fruit (X). Therefore this tomato (Z) is not edible (not-Y)".

Does this also make sense?

Thank you very much.

• Comments are not for extended discussion; this conversation has been moved to chat. (questioningthis: I hope you can see and participate in that room. Normally, one needs more reputation for that but it should work in this case. If not, please @ me and we'll find a solution. -- Also, welcome to Philosophy.SE!) – user2953 Apr 9 '18 at 15:16

The question is still ambiguous: "is" can refer either to individuals belonging to a set or to a subset included into a set.

Consider the example with Bob; we have:

P1) Everything not X is Y --- we read it as set inclusion : not-X ⊆ Y [where is the subset relation]

P2) Bob is not X --- B ∈ not-X [where is the elementhood relation]

C) Bob is Y --- B ∈ Y.

The deduction is correct, because the definition of subset is : A ⊆ M iff every element of A is an element of M.

Thus, Bob is an element of set not-X and not-X is subset of Y; therefore, Bob is element of Y.

"Everything that is not a fruit (not-X) is edible (Y). Bob is not a fruit (X). Therefore Bob is edible (Y)".

Similarly for :

"Everything that is a fruit (X) is inedible (not-Y). Tomatos (Z) are fruits (X). Therefore tomatos (Z) are not edible (not-Y)".

We have :

P1) Everything that is X is not Y --- X ⊆ not-Y

P2) Everything that is Z is X --- Z ⊆ X

C) Z is not Y --- Z ⊆ not-Y.

Also this deduction is correct, because the relation of inclusion ("being a subset of") is transitive.

• The question was edited while you were answering. – MichaelK Apr 9 '18 at 13:06
• @MichaelK - nothing changes. – Mauro ALLEGRANZA Apr 9 '18 at 13:10
• Does this mean I would need to clarify whether 'is' defines a set or a subset belonging to a set to be more precise? Furthermore, is this impossible to do? – questioningthis Apr 9 '18 at 13:15
• @MauroALLEGRANZA Eh, yes it does because P3 was added. – MichaelK Apr 9 '18 at 13:16
• @questioningthis - I've used the standard math symbols: for element-set and for subset-set. – Mauro ALLEGRANZA Apr 9 '18 at 13:21

Are the following premises and conclusions begging the question?

Both arguments are valid. Assuming the truth of the premises, I do not see a problem with the reasoning itself. Both arguments are in the Form AAA in the first figure, which is valid.

I do not see any question-begging.

To beg the question is to assume the truth of the conclusion of an argument in the premises in order for the conclusion to follow. It is a type of circular reasoning and an informal fallacy, in which an arguer makes an argument that requires the desired conclusion to be true. (Wikipedia, “Begging the question”.)

Barker demands more rigor when defining "to beg the question":

A speaker can prove a conclusion only if his argument contains premises…. (2) that are such that the speaker and his hearers can know them to be true without being aware of whether the conclusion is true….

If the premises are related to the conclusion in such an intimate way that the speaker and his hearers could not have less reason to doubt the premises than they have to doubt the conclusion, then the argument is worthless as a proof…. (Barker, The Elements of Logic (1965), pp 175-76.)

Here is how I interpret Barker. The argument begs the question when the premises are related to the conclusion in this way: The following becomes impossible: There is less reason to doubt the premises than there is to doubt the conclusion.

Thus, when there is less reason to doubt the premises than there is to doubt the conclusion, then the argument is valid. Otherwise, the listener must assume some part of the conclusion to balance the lingering doubts about the premises.