[FOM] substitutional quantification, counterfactuals and ontological commitment

[Paul Hollander]

[NOTE:  This is my final corrected submission -- my apologies for the 
previous drafts.]

Category theory uses the arrow, →, to express order-preserving relations 
weaker than identity.

The fragment of category theory pertaining to identity, "baby category 
theory," or "sub-identity theory," can be formalized for quantified 
logic by adding six natural deduction rule schemas.

Here, the sub-identity morphism from a to b is expressed by 'id+(a → 
b)'. Its inverse is 'id-(b → a)'.  They interact with identity and 
monadic properties like this:


(= Introduction)  ⊦ id+(a → b) and ⊦ id-(b → a) together imply ⊦ a = b,


(= Elimination)  ⊦ a = b implies ⊦ id+(a → b) and ⊦ id-(b → a),


(id+ Introduction)  ⊦ id+(a → a),


(id+ Elimination)  ⊦ Fa and ⊦ id+(a → b) together imply ⊦ Fb,


(id- Introduction)  ⊦ id-(a → a),


(id- Elimination)  ⊦ ~Fa and ⊦ id-(a → b) together imply ⊦ ~Fb.


These rules replace the introduction and elimination rules for = in 
natural deduction for quantified logic with identity.

The id+ and id- introduction rules are no-premiss rules analogous to = 
Introduction.

The id+ and id- elimination rules are classical rules of detachment. 
They differ in that id+ is an inheritance relation for monadic 
properties while id- is an inheritance relation for negations of monadic 
properties, like ~F.

With these rules, we stay close to the bounds of classical first-order 
quantified logic with identity.

However, every non-trivial identity 'a = b' ultimately entails four 
sentences, 'id+(a → b)', 'id-(a → b)', 'id+(b → a)' and 'id-(b → a)', 
each of which has a unique inferential role.a

For the substitutionalist, instantiating a variable 'x' requires not one 
but two names 'a' and 'b'.  Doing so provides the substitutionalist with 
a non-trivial counterfactual claim, like '~(a = b)', as I pointed out in 
my original submission.

But in this context, assuming this fragment of category theory, the 
non-trivial claim ⊦ ~(a = b) spawns four non-trivial claims:  ⊦ ~(id+(a 
→ b)), ⊦ ~(id-(a → b)), ⊦ ~(id+(b → a)) and ⊦ ~(id-(b → a)).  Each claim 
is amenable to its own non-trivial counterfactual evaluation because 
none requires assuming an "impossible possible world."

Additionally, because 'id+(a → b)' and 'id-(b → a)' are logically 
equivalent, as are 'id+(b → a)' and 'id-(a → b)', the counterfactuals 
spawned by '~(a = b)' involve a structured set of possible worlds.  The 
closest such worlds involve the non-trivial counterfactuals '~(id+(a → 
b) <--> id-(b → a))' and '~(id+(b → a) <--> id-(a → b))', where '<-->' 
is the double conditional, while the furthest such worlds involve the 
negation of each sentence separately.

Therefore, substitutionalist existential quantification provides six 
structured non-trivial counterfactual scenarios, given this fragment of 
category theory.  But objectualist quantification provides none, because 
the objectualist assumes each instantiated quantifier implies reference.

Again, I'd appreciate any feedback from FOM.

Cheers,

Paul Hollander



On 4/17/17 13:21, paul at personalit.net wrote:
> I was recently talking with a colleague, who teaches logic and 
> philosophy of science using Copi-style rules only, about an advantage 
> of substitutionalist quantification over objectualist quantification.
>
> An objectualist accepts the inference from |- (Ex)Fx to |- Fa by 
> assuming that 'a' refers. Not so for substitutional quantification, 
> for which this inference requires two names 'a' and 'b'.  This is 
> advantageous when 'Fa' is not equivalent to 'a=a' because it requires 
> application of a rule of detachment, like =-elimination, to infer |- 
> Fa from |- (Ex)Fx.  And applying =-elimination in this case requires 
> assuming the non-trivial identity 'a=b'.
>
> Therefore, substitutional quantification is advantageous because the 
> negation of 'a=b', '~(a=b)' is amenable to counterfactual reasoning, 
> while the negation of 'a=a', '~(a=a)', is not, except in the case of 
> an "impossible possible world."
>
> To me, this expresses a distinction between objectualist and 
> substitutionalist interpretations of quantification, while remaining 
> agnostic about ontological commitment, the question of whether 'a' and 
> 'b' refer.  The substitutionalist supports non-trivial counterfactual 
> reasoning, but not the objectualist.
>
> I'm curious as to any feedback from FOM.
>
> Cheers,
>
> Paul Hollander
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