Developing the Alternate version in FOL

Proposed Alternate version

(P1) ∀x(Hx∧Sx ⇒ Mx)
(P2) ¬∃t( Tt ∧ ∀x(Ax∧Sx ⇒ ¬Px,t) ∧ ∀y(Sy∧¬Py,t ⇒ ¬ My) )
(P3) ∀x ∀y ∀t ( (Tt ∧ Sx ∧ Ay∧Sy ∧ ( Px,t ⇔ Py,t ) ) ⇒ x = y)
(C) ∀x(Hx∧Sx ⇒ Mx)

Most Charitable Interpretation

(P1) ∀x(Hx∧Sx ⇒ Mx)
(P2) ¬∃t( Tt ∧ ∀x(Ax∧Sx ⇒ ¬Px,t) ∧ ∀y(Sy∧∃z(Hz∧Sz∧AVz,y)∧¬Py,t ⇒ ¬ My) )
(C) ∀x(Hx∧Sx ⇒ Mx)

Comment on Forum

On the topic of clarifying the logical form of NTT so as to show that the alternate version is, if made valid, basically question begging, how about this:

Hx = x is a human
Ax = x is a non-human animal
Sx = x is sentient
Mx = x has moral value
Tx = x is a trait
Px,y = x has y (or a property of x is y) AVx,y = an alternate version of x is y (i.e. y is a way that x could be / could have been and still have been the same individual in the ethically relevant sense; e.g. an alternate version of me is someone who had my exact history but who left the house 5 minutes earlier than I did today, while an individual who originated from different gametes would not be an alternate version of me)

(I’m leaving parentheses out of predicates and relations because I’m worried that things are getting too cluttered. To further cut down on parentheses I’m going to be using the standard grouping conventions that conjunction binds stronger than disjunction and conditionals; I think it should all be clear enough)

The logical form of NTT (part 1) on the natural reading is:
(P1) ∀x(Hx∧Sx ⇒ Mx)
(P2) ¬∃t( Tt ∧ ∀x(Ax∧Sx ⇒ ¬Px,t) ∧ ∀y(Sy∧∃z(Hz∧Sz∧AVz,y)∧¬Py,t ⇒ ¬ My) )
(C) ∀x(Hx∧Sx ⇒ Mx)

Or
(P1) For all x, if x is human and x is sentient, then x has moral value
(P2) There does not exist any thing t, such that t is a trait, and if x is a non-human animal and x is sentient then x does not have t, and for all y, if y is sentient, and there exists a z such that z is human and z is sentient and an alternate version of z is y, and y does not have t, then it is not the case that z has moral value
(C) For all x, if x is a non-human animal and x is sentient, then x has moral value

Now as we all know this isn’t valid, and one can coherently accept P1 and accept P2 by being a value narcissist who holds that if one lost the non-essential properties that one has and non-human animals lack (like rationality, or moral agency, or whatever – but here we’re blocked from considering originating from human gametes, since that’s an essential property) one would retain one’s moral value, but no non-human animals have moral value because they lack the essential properties which one has.
So now we want to move to the alternate version of NTT, on which P2 talks about “alterations” (or really just differences) from oneself that can change out one’s essential properties.

The logical form of NTT (part 1) on this alternative reading is:
(P1) ∀x(Hx∧Sx ⇒ Mx)
(P2) ¬∃t( Tt ∧ ∀x(Ax∧Sx ⇒ ¬Px,t) ∧ ∀y(Sy∧¬Py,t ⇒ ¬ My) )
(C) ∀x(Hx∧Sx ⇒ Mx)

Or (P1) For all x, if x is human and x is sentient, then x has moral value
(P2) There does not exist any thing t, such that t is a trait, and if x is a non-human animal and x is sentient then x does not have t, and for all y, if y is sentient, and y does not have t, then it is not the case that z has moral value
(C) For all x, if x is a non-human animal and x is sentient, then x has moral value
(The difference between this and the natural version is that in P2 we just drop the conjunct about y being an alternate version of some sentient human)

So now we know this isn’t valid, but what’s supposed to make the inference from P1 and P2 to C a good one (according to Isaac) is the identity of indisernibles, which, restricted to this context, is:

(P3) ∀x ∀y ∀t ( (Tt ∧ Sx ∧ Ay∧Sy ∧ ( Px,t ⇔ Py,t ) ) ⇒ x = y)

Or
(P3) for all x, all y, and all t, if t is a trait, and x is sentient, and y is a non-human animal and y is sentient, and x has t if and only if y has t, then x is the self-same entity as y

OK, so now my first question is: is the alternate version with P3 added:
(P1) ∀x(Hx∧Sx ⇒ Mx)
(P2) ¬∃t( Tt ∧ ∀x(Ax∧Sx ⇒ ¬Px,t) ∧ ∀y(Sy∧¬Py,t ⇒ ¬ My) )
(P3) ∀x ∀y ∀t ( (Tt ∧ Sx ∧ Ay∧Sy ∧ ( Px,t ⇔ Py,t ) ) ⇒ x = y)
(C) ∀x(Hx∧Sx ⇒ Mx)

Valid?

As I said, it’s been awhile since I taught this stuff. Please tell me that someone else on this forum has more recently taught or taken a symbolic logic course, and can thus give a proof of this thing’s validity (or show a counterexample, and then maybe give a fixed up version that is valid, because I think this should be pretty close). I’m already spending too much time on this and I’d really hate to have to dig up and go over my materials on this.

Second, it seems to me that the alternate version of P2,

¬∃t( Tt ∧ ∀x(Ax∧Sx ⇒ ¬Px,t) ∧ ∀y(Sy∧¬Py,t ⇒ ¬ My) )

Is dangerously close to just question-beggingly asserting the moral value of all sentient beings. To see this it may help to simplify it to the logically equivalent:
∀t ( ¬Tt v ∃x(Ax∧Sx∧Px,t) v ∃y(Sy∧¬Py,t∧ My) )
Or
For all t, either t is not a trait, or there is an x such that x is a non-human animal and x is sentient and x has t, or there exists a y such that y is sentient and y does not have t and y has moral value

Can anyone help bring out how this seems very very close to saying that sentient beings have moral value?

Now I think that I can perhaps see Dr. Singer’s point about the relevant part of P2 just saying that there exists a sentient being with moral value. If we had wanted it to be able to say that all sentient beings have moral value, then perhaps the premise should initially be:

(P2*) ¬∃t( Tt ∧ ∀x(Ax∧Sx ⇒ ¬Px,t) ∧ ∃y(Sy∧¬Py,t∧¬ My) )

This simplifies to
∀t ( ¬Tt v ∃x(Ax∧Sx∧Px,t) v ∀y(¬SyvPy,t)vMy) )
Or
For all t, either t is not a trait, or there exists an x such that x is a non-human animal and x is sentient and x has t, or for all y, either y is not sentient, or y has t, or y has moral value

Does this version of P2 help make the argument in the alternate version valid? Does this version of P2 make it clear how P2 seems to be just asserting the value of all sentient beings, and thus make the argument in the alternate version circular?

Comments

[Margaret's new comment: I actually think that I am encountering this issue in dealing with the alternate version of NTT. Isn't the way we can correct this (and remember, WE'RE the ones putting it into FOL, so it's OUR job to do this in a way that corrects things so as to be as charitable as possible to the English gist of the argument) just to paraphrase P2 in essence as saying (roughly, as I may be proposing a more fine-grained LF now):

(P2) ¬∃t ( T(t) ∧ ∀x ( A(x) ⇒ ¬P(x,t) ) ∧ ∃y ( ( CP(y) ∧ ¬P(y,t) ) ∧ ¬ M(y) ) )

or

There does not exist a thing t such that t is a trait and for all x if x is a sentient non-human animal then x does not have t, and there exists a y such that y is a counterpart of a sentient human and y does not have t and y does not have moral value

Instead of the old way we had it which was

(P2) ¬∃t ( T(t) ∧ ∀x ( A(x) ⇒ ¬P(x,t) ) ∧ ∀y ( ( CP(y) ∧ ¬P(y,t) ) ⇒ ¬ M(y) ) )

or

There does not exist a thing t such that t is a trait and for all x if x is a sentient non-human animal then x does not have t, and for all y such that if y is a counterpart of a sentient human and y does not have t then y does not have moral value

[Margaret: this looks pretty nit-pickey / uncharitable to me. The brute-force way to fill in the missing premise is just to say that if P1 and P2 are true, then C has to be true, i.e.:

(P3) ( ∀x ( H(x) ⇒ M(x) ) ∧ (¬∃t ( T(t) ∧ ∀x ( A(x) ⇒ ¬P(x,t) ) ∧ ∀y ( ( CP(y) ∧ ¬P(y,t) ) ⇒ ¬ M(y) ) ) ⇒ ∀x ( A(x) ⇒ M(x) )

So if adding something like that (or maybe trying to pretend that it's a fact about logical form that doesn't need to be added?) is how we imagine the argument succeeding in showing that the conclusion follows from the premises simply in virtue of logical form, I just don't see the point in imagining that it first establishes "there is no moral value giving trait absent in all animals" and then fails to entail "all animals have the moral value giving trait"]

[DrSinger: I think it's important and often overlooked/dismissed, I'll post on the forum about it]

[Margaret: If it is easy to correct (and I'm still not sure what the point of this 'even if did establish..' thing is), then why quibble about it? Why not just move it to / leave it in the 'getting to the most charitable version' section? Given the extreme danger of this all just coming across as pointless, butt-hurt nit-picking, I would think that nothing like this should be included unless it has some important relevance to actual, important confusions people have about the argument when interpreted in as charitable a way as possible...]