Notice how you're trying to defend the premise by treating it as a syllogism, saying that the consequent follows from the antecedent? As I've pointed out before, it's an invalid syllogism. D can't be deduced from A, B, and C — Michael

No, I said "d follows from the transitive property", which is not the same as saying "d follows from the antecedent". d has a unique feature which makes it a consequent. The defense of d specifically and not the entire premise is the transitive property applied.

No, I said "d follows from the transitive property" — Soylent

But it doesn't. Nowhere in "¬Pa (= O¬a)", "¬a = b", or "c" does "d" appear.

But it doesn't. Nowhere in "¬Pa (= O¬a)", "¬a = b", or "c" does "d" appear. — Michael

d is a term and not a predicate statement. The predicate statement contained in d follows from the transitive property. The content of d is contained in "¬Pa (= O¬a)", "¬a = b", or "c" wherein the application of the transitive property on those terms is the term unique term d (i.e., the consequent).

The content of d is contained in "¬Pa (= O¬a)", "¬a = b", or "c" wherein the application of the transitive property on those terms is the term unique term d (i.e., the consequent). — Soylent

Still not making sense. I understand the transitive property as saying if 2 > 1 and if 3 > 2 then 3 > 1, or as saying if a = b and if b = c then a = c. I don't see how your example relates to this in any way.

Let's repurpose your example of transitivity in action.

Let:
a = 2>1
b = 3>2
c = 3>1

If a and b, then c

What justifies c?

Either modus ponens (the affirmation of the antecedent terms).
Or the transitivity of the content of the terms.

The second. But neither the first nor the second applies to P15. P15 is nothing like that example.

P15 is of the form:

If ¬Pa (= O¬a)
and ¬a = b
and c
then, d

¬Pa is true by definition
b is true by empirical evidence
c is true by supportive argument and empirical evidence

d follows from the transitive property applied to O¬a wherein ¬a = b and some b is c.

@Soylent

P6 remains unchanged, as therefore does my criticism, which you have yet to address.

Also, given your stipulative definition, P6, P7, and C3 are redundant. If, by definition, gratuitous suffering is preventable, then all you'd need to do is show that something causes gratuitous suffering, and it logically follows that it's preventable.

On second thought, given your 'if and only if' in P6, it must be false, since gratuitous suffering is preventable simply by virtue of being gratuitous suffering, not conditionally.

Other parts of your argument might be similarly effected. For example, P9, in which you talk about "the means of making gratuitous suffering... preventable", which wouldn't make sense.

I've addressed your criticism of P6 elsewhere and since P6 remains unchanged my response elsewhere is still applicable. If you feel your objection to P6 is different than the one already presented in this thread, I would gladly address your objection specifically and would appreciate it you could explain how yours is different.

I agree that there is an appearance of redundancy or begging the question by use of the term "gratuitous suffering" early in the argument, but the use is only to make the distinction between gratuitous suffering and other forms of suffering not obliged by the success of the argument. P6, P7 and C3 are useful to show that food production practices are preventable and fit into the larger argument to show that food production practices constitute gratuitous suffering as per the definition.

Considering your comment that one only needs to show that something causes gratuitous suffering and it logically follows that it's preventable, but to show that something causes gratuitous suffering one must show that the suffering is preventable, is there any way to avoid the redundancy?

Considering your comment that one only needs to show that something causes gratuitous suffering and it logically follows that it's preventable, but to show that something causes gratuitous suffering one must show that the suffering is preventable, is there any way to avoid the redundancy? — Soylent

Well, the converse is invalid, i.e. if something causes preventable suffering, it doesn't following that it's gratuitous suffering, because gratuitous suffering, as per your definition, requires more than that. It doesn't work both ways. So, showing that something causes preventable suffering wouldn't be redundant; it would satisfy one condition, but not all conditions, for something to count as gratuitous suffering.

The strongest criticism of your argument in it's current form is that P6 is necessarily false, and your argument is therefore necessarily unsound:

1. Gratuitous suffering, by definition, is preventable by virtue of being gratuitous suffering (and therefore not conditionally).
2. P6 contradicts (1.).
C. P6 is false.

You must either change the definition or change P6.

1. Gratuitous suffering, by definition, is preventable simply by virtue of being gratuitous suffering (and therefore not conditionally)

The bolded is false. An instance of gratuitous suffering is conditionally true (i.e., the instance satisfies the conditions).

Consider:

P1: a bachelor is, by definition, an unmarried man simply by virtue of being a bachelor (and therefore not conditionally).
P2 Mike is a bachelor on the condition that he is an unmarried man.
P3 P2 contradicts P1
C1 P2 is false

Well there is of course one condition, the condition which is implicit in the definition, and which Mike is a bachelor in virtue of.

This doesn't overcome my criticism. Analogously, your argument incurs the following contradiction:

1. If a man is unmarried, then he is a bachelor.
2. A man is a bachelor if and only if he drinks wine.

If there's one condition, then "therefore not conditionally" is false.

I would say it's more akin to:

1. If one is a bachelor, then one is an unmarried man.
2. One is unmarried if and only if one never marries.

1. If one is a bachelor, then one is an unmarried man.
2. One is unmarried if and only if one never marries. — Soylent

I noticed that I inverted the antecedent and the consequent. I apologize. The correct version is below:

1. If one is an unmarried man, then one is a bachelor.
2. One is unmarried if and only if one never marries.

If there's one condition, then "therefore not conditionally" is false. — Soylent

Yes, well done, I concede, but that is not what I meant, and is beside the point.

Your argument results in contradiction, because if gratuitous suffering is preventable if and only if it is possible for some agents to adopt a vegan diet, then that rules out the possibility that gratuitous suffering is preventable simply by virtue of being gratuitous suffering.

You are committed to both:

1. X is gratuitous suffering, and is therefore preventable.
2. X is gratuitous suffering, and is preventable if and only if Y.

Can you not see the contradiction?

You are also special pleading if you're claiming that the gratuitous suffering resulting from food production practices is somehow an exception to (1.).

then that rules out the possibility that gratuitous suffering is preventable simply by virtue of being gratuitous suffering. — Sapientia

I see your point now and will take some time to give it proper consideration.

A preliminary thought is that 1 and 2 are functionally indistinguishable and by the principle of identity of indiscernibility I need not commit to the position that 2 precludes 1. In other words, P6 satisfies both 1 and 2 because they are identical.

A preliminary thought is that 1 and 2 are functionally indistinguishable and by the principle of identity of indiscernibility I need not commit to the position that 2 precludes 1. In other words, P6 satisfies both 1 and 2 because they are identical. — Soylent

I don't understand how you can think them identical. They are different and distinguishable, they are not logically equivalent, they do not have the same function, nor do they have the same meaning. P6 has the logical form of 2 (well, not exactly, but it could put that way), and contradicts 1 (which is implicit in your argument).

d follows from the transitive property applied to O¬a wherein ¬a = b and some b is c. — Soylent

No it doesn't.

Look:

One ought make gratuitous suffering preventable
Gratuitous suffering is preventable iff it is possible to adopt a vegan diet
It is possible for some to adopt a vegan diet
Therefore one ought adopt a vegan diet.

There's no modus ponens. There's no transitivity. As I've said before, the conclusion must be "one ought make it possible to adopt a vegan diet". However, now that you've introduced P3, even this conclusion doesn't work; instead, the conclusion "gratuitous suffering is preventable" is deducible from P2 and P3. And if gratuitous suffering is (already) preventable then there's nothing left for us to do. The condition of our obligation has already been satisfied.

As I have repeatedly said, if you want the conclusion to be "one ought adopt a vegan diet" then premise 2 must be "gratuitous suffering is preventable iff a vegan diet is adopted". Nothing else will work (with premise 1).

Considering your comment that one only needs to show that something causes gratuitous suffering and it logically follows that it's preventable, but to show that something causes gratuitous suffering one must show that the suffering is preventable, is there any way to avoid the redundancy?

You need to get rid of the word "gratuitous" and have:

One ought make known suffering preventable at a reasonable cost
One makes known suffering preventable at a reasonable cost iff one adopts a vegan diet
Therefore one ought adopt a vegan diet

I'm going to try a different approach since I think I partially see what you're saying.

This is what I want P15 to say, and it may or may not depending on how it's unpacked:

It is wrong to allow gratuitous suffering cause by food production practices

The only means to prevent gratuitous suffering caused by food production practices is to adopt a vegan diet (I think you deny the statement says this)

There are some for whom it is wrong to allow gratuitous suffering caused by food production practices.

Those people for whom it is wrong to allow gratuitous suffering caused by food production practices ought not allow gratuitous suffering caused by food procuction practices.

It is wrong to allow is logically equivalent to ought not allow.
If not allow gratuitous suffering is equivalent to adopt a vegan diet.
Then "those for whom ought not allow gratuitous suffering" is logically equivalent to "those for whom ought to adopt a vegan diet".

It is wrong to allow gratuitous suffering cause by food production practices

The only means to prevent gratuitous suffering caused by food production practices is to adopt a vegan diet (I think you deny the statement says this)

There are some for whom it is wrong to allow gratuitous suffering caused by food production practices. — Soylent

To clarify, the first premise is "one ought prevent gratuitous suffering"?

Those people for whom it is wrong to allow gratuitous suffering caused by food production practices ought not allow gratuitous suffering caused by food procuction practices.

That's not a conclusion. That's a tautology.

No it's not:

P15 If it is wrong to allow gratuitous suffering caused by food production practices and gratuitous suffering caused by food production practices is preventable if and only if it is possible to adopt a vegan diet, and there are those who are in a position to adopt a vegan diet, then a vegan diet ought to be adopted by all those who are in a position to adopt a vegan diet.

I'm trying to understand what you mean by "it is wrong to allow gratuitous suffering". Can you rewrite it in a "one ought X" format?

Usually I'd understand "it's wrong to allow so-and-so to steal" as "one ought prevent so-and-so from stealing", so I'd understand "it's wrong to allow gratuitous suffering" as "one ought prevent gratuitous suffering".

To make this very simple, this is what your argument needs to be:

P1. ∀x: C(P(x)) → O(P(x))
P2. ∀x: P(x) ↔ V(x)
C1. ∀x: C(V(x)) → O(V(x))

Which is to be read as:

For any person, if that person can P then that person ought P.
For any person, that person Ps if and only if that person adopts a vegan diet.
For any person, if that person can adopt a vegan diet then that person ought adopt a vegan diet.

So what sentence is "P"? "Prevents gratuitous suffering"? "Makes known suffering preventable at a reasonable cost"?

The logical equivalence must not introduce new terms.

"wrong to x" is only logically equivalent to "ought not x". The introduction of prevention is implied by the word "allow", but not logically contained.

1. X is a bachelor, and is therefore an unmarried man.
2. X is bachelor, and is unmarried if and only if X never marries.

By your contention, 2 is false because 1 also satisfies the conditions of what it means to be a bachelor and one must be committed to say that the mere name "bachelor" is sufficient to make one a bachelor, if even one has married or not.

*edit* 2 is false, please disregard this comment.