NTT counterpart sections

Steel-manning NTT

Before proceeding we will rephrase the argument in the following way, to capture its essential reasoning

(P1) All sentient humans have moral value

(P2) If an individual is a sentient nonhuman animal, then there is no trait absent the individual, which is such that, if the trait were absent in a counterpart to a sentient human, then the counterpart would not have moral value

Therefore, (C) All sentient non-human animals have moral value

Here we have introduced the notion of the counterpart to a sentient human, to mean a replica of a sentient human from which any trait can be removed, and that can be or fail to be us/ourselves. In not requiring the counterpart to be us/ourselves we are straying from the original argument, however this is necessary so we can consider the case in which the traits that removed can include one's essential properties (those which make us 'us/ourselves'), which matches the recent explanations of NTT given by Ask Yourself.

Summary of Issues

As we will see below, the reason NTT is logically invalid because its logical form allows for a counterexample in which sentient humans have moral value, there exists a sentient non-human animal that lacks moral value, and for all traits that sentient non-human animals lack, human counterparts possess moral value with the traits removed. The reason we can pose such a counterexample is because there is nothing in the logical form of the argument that says moral value must be based on a trait (as is the case for the correction) nor is there anything to say counterparts lacking certain traits are animals (as is the case for the alternative interpretation).

Ask Yourself has argued that if one were to lose all of the traits that distinguish one from a non-human animal, then the resulting entity would be identical to that non-human animal, so P1 and P2 so interpreted [logically] entail C. The problem with this claim is that it presupposes something like the identity of indiscernibles: that if two entities have all of the same properties, then they must be the self-same object. This is a substantive metaphysical thesis, to which certain philosophers have objected (for discussion see [1]), so it must be added as an additional premise for the argument to be logically valid.

But as we will see below, the much greater problem with the argument so interpreted is that it has essentially no rational force. P2 so interpreted is essentially just asserting that beings with all and only the traits of sentient non-human animals have moral value. The argument thus offers little if any reason to change the mind of someone who does not already find this view plausible, and the defense of its premises offers no guidance on how to persuade such an individual, see how this version of the argument begs the question.

Proof of Invalidity in First Order Logic

In order to show the logical form of NTT, we will use what is known as First order logic. The sort of logic used to display the logical form of the argument above '(P1) If U then V; (P2) U; therefore, (C) V' is known as propositional logic, since it replaces full propositions flanking logical connectives like 'if...then', 'and' and 'or' with abstract symbols. First order logic goes inside the logical structure of propositions, and replaces predicates like 'is a sentient human' and 'is a sentient non-human animal' with abstract symbols. First order logic also considers variables for things to which those predicates are ascribed, and quantification over them (e.g. 'for all x, if x is a sentient human, then x has moral value'). It may be helpful to read the entry on logical connectives before reading the following sections.

Displaying the Logical Form of NTT in FOL

Symbols

• ∀ (for all)
• ⇒ (if, then; e.g. A ⇒ B means 'if A then B')
• ⇔ (if and only if; e.g. A ⇔ B means 'if A then B and if B then A')
• ¬ (negation i.e. not)
• ∃ (there exists)
• ∧ (and)
• ∨ (or)

Definitions

H(x) means 'x is a sentient human'

A(x) means 'x is a sentient non-human animal'

M(x) means 'x has moral value'

CP(x) means 'x is a counterpart of a sentient human'

T(x) means 'x is a trait'

P(x,y) means 'x has y; e.g. if x is a sentient human and t is a trait, P(x,t) means 'human x has trait t', or 'a property of x is t'

RNE(x) means 'we a morally required to not exploit x'

Note we will also use the shorthand Hx to represent H(x), Px,y or Pxy to represent P(x,y) etc.

The Logical Form of Part 1 of NTT

(P1) ∀x ( Hx ⇒ Mx )

(P2) ∀x ( Ax ⇒ ¬∃t ( Tt ∧ ¬Pxt ∧ ∀y ( CPy ⇒ ( ¬Pyt ⇒ ¬My ) ) ) )

Therefore, (C) ∀x ( Ax ⇒ Mx )

In English

(P1) for all x, if x is a sentient human, then x has moral value

(P2) for all x, if x is a sentient nonhuman animal, then there does not exist t, such that t is a trait, and x lacks t, and for all y, if y is the counterpart to a human, then if y lacks t, then y does not have moral value.

Therefore, (C) For all x, if x is a sentient non-human animal, then x has moral value

The Logical Form of Part 2 of NTT

(P1) ∀x ( Ax ⇒ Mx )

(P2) ∀x ( Ax ⇒ ¬∃t ( Tt ∧ ¬Pxt ∧ ∀y ( CPy ∧ My ⇒ ( ¬Pyt ⇒ ¬RNEy ) ) ) )

Therefore, (C) ∀x ( (Ax ∧ Mx) ⇒ RNEx )

In English:

(P1) For all x, if x is a sentient nonhuman animal, then x has moral value.

(P2) For all x, if x is a sentient nonhuman animal, then there does not exist t, such that t is a trait, and x lacks t, and for all y, if y is the counterpart to a human, and y has moral value, then if y lacks t, then we are not morally required to not exploit y.

Therefore (C) for all x, if x is a sentient non-human animal, and x has moral value, then we are morally required not to exploit x.

Showing NTT is Logically Invalid

The standard way of showing that an argument is invalid is to construct a counterexample, or a model, which is allowed by the logical form of the premises and the conclusion, in which the premises are true and the conclusion is false. For part 1 of NTT, we will show the argument is invalid by constructing a case, allowed by its logical form as displayed above, in which P1 & P2 are true but C is not.

Part 1 Counterexample

In FOL

(E1) ∃x ( Hx ) ∧ ∀x ( Hx ⇒ ( Mx ∧ ¬Ax ∧ ¬Tx ∧ ¬Cx) )

(E2) ∃x ( Ax ) ∧ ∀x ( Ax ⇒ ( ¬Mx ∧ ¬Hx ∧ ¬Tx ∧ ¬Cx ∧ ( ∃y ( Cy ∧ ∀t (Tt ∧ ( ¬Pxt ⇒ ¬Pyt ) ) ) ) )

(E3) ∀x ( Cx ⇒ ( Mx ∧ ¬Ax ∧ ¬Tx ∧ ¬Hx ) )

(E4) ∀x ( Tx ⇒ ( ¬Ax ∧ ¬Hx ∧ ¬Cx ∧ ¬ Mx) )

In English

(E1) there is a sentient human, and all sentient humans have moral value, are not sentient non-human animals, are not traits, and are not counterparts / counterfactual alternative versions of sentient humans

(E2) there is a sentient non-human animal, and all sentient non-human animals do not have moral value, are not sentient humans, are not traits, and are not counterparts of sentient humans, and are such that there is a counterpart of a sentient human that lacks all traits that lacks all the same traits as the non-human animal animal

(E3) all counterparts of sentient humans have moral value, are not nnon-human animals, are not traits, and are not humans as they actually are

(E4) all traits are not sentient non-human animals, are not sentient humans, are not counterparts of sentient humans, and do not have moral value

Proving the Counterexample renders NTT invalid

If NTT is invalid, then the formula

(E1 ∧ E2 ∧ E3 ∧ E4) ⇒ ( P1 ∧ P2 ∧ ¬C )

should be valid. That is to say, if NTT is invalid, and we have chosen our counterexample correctly, then if the counterexample is true, then the premises of NTT should be true and its conclusion false. We can check this using a logical proof generator, with the input

((\existsx (Hx) \land \forallx(Hx\to(\negAx \land\negTx\land \negCx\landMx))) \land(\existsx(Ax) \land \forallx (Ax\to (\negHx \land \negTx\land \negCx\land\negMx \land(\existsy(Cy\land \forallt(Tt\land( \negPxt\to \negPyt))))))) \land(\forallx (Cx\to(\negAx \land\negTx \land\negHx\landMx)))\land (Tx\to( \negAx\land\negHx \land\negCx \land\negMx))) \to( \forallx ( Hx \to Mx ) \land \forallx ( Ax \to \neg \existst ( Tt \land \neg Pxt \land ( \forally (Cy \to ( \neg Pyt \to \neg My ) ) ) ) ) \to\neg\forallx (Ax \to Mx))

which gives us valid. Hence we have demonstrated that NTT is invalid.

Checking the Validity Directly

We can also use the logical proof generator to directly show the argument is invalid by checking the formula

P1 ∧ P2 ⇒ C

We can do this with the input (for part 1)

(P1) \forallx (Hx \to Mx)

(P2) \forallx ( Ax \to \neg \exists t (Tt \land \neg Pxt \land ( \forally (Cy \to ( \neg Pyt \to \neg My ) ) ) ) )

(C) \forallx ( Ax \to Mx )

Or all together (P1 ∧ P2 ⇒ C)

\forallx ( Hx \to Mx ) \land \forallx ( Ax \to \neg \existst ( Tt \land \neg Pxt \land ( \forally (Cy \to ( \neg Pyt \to \neg My ) ) ) ) ) \to\forallx (Ax \to Mx)

which yields invalid. This is unlike the correction which gives valid.

Remark on Invalidity

The reason why we can construct such counterexamples is because the truth of P2 has no bearing on whether or not animals have moral value, as there is nothing in the logical form of the argument that says moral value must be based on a trait (as is the case with the correction) nor is there anything to say counterparts lacking certain traits are animals (as is the case with the alternative interpretation).

The Alternative Interpretation of NTT

Ask Yourself explains his interpretation of P2 with an animation

In the video to the right Ask Yourself explains his interpretation of NTT and why he believes it to be valid. The claim made is that, paraphrasing, all objects are constellations of traits, and that if the traits of the two objects are equalized, then the objects become the same object, and furthermore, to resist this is to deny the very first law of logic, which is the law of identity. This assumes a number of substantive metaphysical premises, which are outlined below:

Bundle Theory

The first claim, that objects are constellations of traits is not built into logic, it is a form of bundle theory which states :

Substances are in fact no more than bundles of properties conceived of as universals.[2]

This is a substantive metaphysical premise that needs to be stated if it's to be used as part of an argument. It's worth noting that the identity of indiscernibles would follow from such a premise, as correctly identified by Ask Yourself. This is one of the reasons why many philosophers have raised objections to such a premise.

Identity of Indescernibles

The claim that two objects with the same traits are the same object is a principle similar to the the identity of indiscernibles, which says: if for every property P, object x has P if and only if object y has P, then x is identical to y. Or in the notation of symbolic logic:

(∀P)(Px ⇔ Py) ⇒ x=y.

In the notation we have been using, this could be expressed as

∀x ∀y ∀t ( Px,t ⇔ Py,t ) ⇒ x = y)

(for all things x, y, and t, if x has t if and only if y has t, then x and y are the self-same thing. )

Or in that case that we are only considering traits, that may or may not refer to all properties, the principle would be expressed as

∀x ∀y ∀t ( Tt ∧ ( Px,t ⇔ Py,t ) ) ⇒ x = y)

i.e. for all things x, y, and t, if t is a trait, and x has t if and only if y has t, then x and y are the self-same thing.

The identity of indiscernibles is not simply a fact about the logical form of propositions or their components - it is instead a metaphysical thesis about the conditions under which entities are identical. Thus, an argument the conclusion of which follows only from its premises together with the identity of identity of indiscernibles in virtue of their logical form is not logically valid. To be logically valid, such an argument must include the statement of the identity of indiscernibles among its premises. In contrast, the law of identity, understood simply as 'x is identical to x for each x' or ∀x (x = x), is a fact about the logical form of the identity relation (so it does not need to be stated among the premises of an argument that involves propositions that employ the identity relation).

Alternative Version in FOL

With the addition of the identity of indiscernibles we can present a logically valid alternative version of NTT, as shown below:

(P1) ∀x ( Hx ⇒ Mx )

(P2) ∀x ( Ax ⇒ ( ∃y ( CPy ∧ My ∧ ∀t ( Tt ⇒ ( Px,t ⇔ Py,t ) ) ) ) )

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

Therefore (C) ∀x ( Ax ⇒ Mx )

In English

In the following take human to mean sentient human & animal to mean sentient nonhuman animal

(P1) humans are of moral value
(P2) for all animals there exists the counterpart to a human that has moral value and has all the same traits as the animal
(P3) All things with the same traits are the same thing.
Therefore (C) Animals are of moral value

This Alternative Version of NTT Begs the Question

With the addition of the identity of indiscernibles as P3, this alternative version of NTT becomes logically valid. However, the validity of this version of the argument comes at a very high price. The "counterparts" to sentient humans spoken of P2 are allowed to lack the essential properties of humans - like originating from human gametes - and to have the essential properties of non-human animals - like originating from bovine gametes. Hence, these "counterparts" need not be human, or us, at all. The only requirement of these entities is that they (i) have moral value, and (ii) have all and only the properties of sentient non-human animals. So in asserting that, for each sentient non-human animal, there is a "counterpart" that has moral value and has all and only the properties of the non-human animal, P2 is essentially just asserting that sentient beings with the properties of non-human animals have moral value. In doing so P2 assumes the substance of what the argument is supposed to prove, namely that sentient beings like non-human animals have moral value, thus rendering the argument question-begging.

To note how unconvincing this version of NTT really is, note that P1, about human moral value, really plays no role at all in deriving the conclusion of this version of NTT at all. The conclusion follows from simply P2, which asserts that for each sentient non-human animal, there is a being that has moral value and has all and only the same traits as the non-human animal, and P3, which entails that if the being with moral value has all and only the same traits as the non-human animal, then the being is identical to the non-human animal. But clearly, the argument that:

(P2) For each non-human animal, there is an entity with all and only the same traits that has moral value

(P3) If two entities have all and only the same traits, then they are the self-same entity

Therefore, (C) All non-human animals have moral value

is not a very compelling argument for the moral value of non-human animals at all. Anyone not inclined to accept the conclusion will have no more inclination at all to accept P2. Moreover, the structure of this version of the argument offers no guidance as to how to convince someone not inclined to accept the conclusion of the truth of the argument's premises, since it seems just as difficult to convince someone of the truth of P2 as to convince them of the truth of C directly.

Excluding Essential Properties

Based on the written version, P2 seems clearly to speak of traits that do not include essential properties and include only accidental properties. The distinction is that

An essential property of an object is a property that it must have [in order to count as the same object], while an accidental property of an object is one that it happens to have but that it could lack [and still count as the same object]. [3]

If the traits spoken of in P2 include only accidental properties, we are envisioning humans like oneself losing accidental properties that one has and non-human animals lack (like abstract reasoning ability, moral agency, etc.) and still retaining one’s status as oneself. The original written version of NTT seems to require this interpretation, since it says

"there are no traits absent in animals which if absent in humans would cause us to deem ourselves valueless,"

so the entities losing the traits are clearly supposed still to be us after losing them.

This interpretation of P2 as speaking only of accidental traits is not question begging. But it is is clearly invalid, as one could hold the view that we might call

Value Narcissism: one’s essential properties (or the essential properties of being human) are what give one moral value – or are such that, if one lost them, one would lose moral value.

A value narcissist could hold that although humans like oneself (or just oneself) have moral value, and one would retain this value if one lost the accidental properties that one has and sentient non-humans lack, non-human animals still lack moral value. They could thus accept P1 and P2 but reject C.

Some individuals faced with arguments for veganism like NTT may well be tempted to embrace something like value narcissism. For instance, in Ask Yourself's debates / discussions with the Warskis, Friend Ed, and Patty Politics, Ask Yourself's conversants seemed tempted to the view that being human (i.e. possessing the essential properties of a human) is necessary to have moral value - at least independent of other considerations like membership in a human community (the trait named by FriendEd). It is certainly true that Ask Yourself argued substantively against this view, although in doing so he unfortunately seemed to make invalid generalizations about the ways in which species membership might matter morally. But it seems quite possible that addition confusion in debates may stem from the fact that, on the natural interpretation of P2 as restricted to accidental traits, the argument simply invalidly overlooks the option of value narcissism, and makes it unclear how a substantive attack on the plausibility of value narcissism is needed for the argument to establish its conclusion.