[FOM] substitutional quantification, counterfactuals and ontological commitment
Paul Hollander
paul at personalit.net
Wed Apr 26 14:55:42 EDT 2017
[NOTE: This includes typographical corrections.]
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 natural
deduction by adding six rule schemas to natural deduction for
first-order quantified logic with identity.
Here, the sub-identity arrow 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 classical 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 the inheritance relation for monadic
properties while id- is the inheritance relation for negations of
monadic properties, like ~F.
With these rules, we stay within 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
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))', 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
it 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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