David Corfield Brandom and material inference

Non-enthemematic reasoning

Why does Brandom, following Sellars, fall back so quickly on the idea that we must reason non-enthemematically? Presumably the idea is that it’s implausible to imagine we have countless premises ready for use in our head. But is this just because of the kind of logic they consider to be the only candidate for this enthemematic reasoning?

So Brandom opts for ‘material inference’ as it’s implausible to think we have ready: X is red all over implies X is not green all over; X is red all over implies X is not blue all over;… and this for every pair of colours. And worse: X is red all over implies X is not blue and green; X is red all over implies X is not red and green. Then, changing topic, we’d need ‘if today is Wednesday, tomorrow is Thursday’ etc. for every day, and so on over countless topics.

But, in the colour case, what if we have implicitly,

• Colours_of: Concrete objects $\to$ Sets of colours

And the type of colours has a structure of a set, a collection of elements, for which two given elements are either the same or different, being aware that some terms are used interchangeably. [It’s probably more subtle with terms for subshades.]

Then for any particular concrete object sent to a singleton colour, say, {red}, we’d know it couldn’t be sent to any different colour, {green}, {blue}, or even set of colours {blue, green}, {red, green}.

The nature of Colours_of as a function to a subset of Colours, a set, integrates all the exclusion clauses for free. We also have that red implies coloured, since

coloured(x) is true by definition when Colours_of(x) is inhabited (that is, nonempty).

So Red:Colours_of(x) entails coloured(x).

It’s reminiscent of Wittgenstein after the Tractatus when confronting ‘X is red all over’ and ‘X is green all over’ as not atomic, and trying out wavelengths, and this not working.

So ‘Wavelength of light(x) = 650 nm’ and ‘Wavelength of light(x) = 450 nm’. Again we need the structure of the type of numbers to entail that these predicates are incompatible.

Perhaps the key is a different logic.

González de Prado Salas, J., de Donato Rodríguez, X. & Zamora Bonilla, J. 2017. Inferentialism, degrees of commitment, and ampliative reasoning. Synthese. https://doi.org/10.1007/s11229-017-1579-5

“if I am committed to endorsing the proposition ‘The house is empty,’ I will also be committed to endorsing ‘Jane is not in the house.’”

“…if one wants to characterize inferentially the content associated with non-logical vocabulary, one has to consider inferences that are not formally good, but that are made good by the content of the (non-logical) concepts they involve—that is, materially good inferences (like the inference about the empty house mentioned above).”

A type of house, we have singled one out, and can refer to it now as ‘the house’.

Occupants: House to P(People)

Occupants(the house) = 0, The house is empty.

jane is not in the house

Or a syllogism:

• No person is in the house
• Jane is a person
• So, Jane is not in the house.

A distinction is made within philosophy between formal and material inference. The first of these operates purely through the logical form of the relevant propositions, where the second relies on conceptual content within them.

A classic example of a formal inference is $A \& B$, therefore $A$. Substitute any propositions for $A$ and $B$ and the inference goes through. The conjunction $\&$ is a piece of logical vocabulary. By contrast, the thinking goes, that $C$ is west of $D$ implies that $D$ is east of $C$ is a piece of material inference, relying on the relation between the non-logical concepts, east and west. Substitute ‘older’ for ‘east’ and ‘larger’ for ‘west’ and the inference fails.

The philosopher Wilfred Sellars famously asserted that in such cases we’re not merely employing a tacit proposition, i.e., here ‘If $X$ is west of $Y$ then $Y$ is east of $X$, instantiating and then employing modus ponens. For Sellars, and those following him, like Robert Brandom, material inference is primary; only for a limited portion of our inferential practices has humankind managed to extract formal inference schemas.

But then how to decide what is ‘logical’ and what ‘non-logical’? As an adherent of dependent type theory, can’t I hold any inference carried out in that system to be formal?

Say I have judged $\vdash j: P$ and $\vdash f: \neg (\sum_{x: P}H(x))$. Then defining for $x: H(j)$, $g(x):\equiv f(j, x)$, I can now judge $g: \neg H(j)$, using standard type-theoretic rules.

Rewriting this inference in something closer to English, we find the syllogism:

$j$ is a $P$, No $P$s are $H$, therefore $j$ is not $H$.

As a valid syllogism, any substitution should do, so let’s choose a type, $P$, say $Person$. $j$ is an element of $P$, so let’s say Jane. $H$ is a property of people, let’s say ‘being in this house’.

Then we have the inference

Jane is a person. No people are in the house. Therefore Jane is not in the house.

OK, why this example? Because it appears in the literature:

• González de Prado Salas, J., de Donato Rodríguez, X. & Zamora Bonilla, J. 2017. Inferentialism, degrees of commitment, and ampliative reasoning. Synthese. (paper)

if I am committed to endorsing the proposition ‘The house is empty,’ I will also be committed to endorsing ‘Jane is not in the house.’

…if one wants to characterize inferentially the content associated with non-logical vocabulary, one has to consider inferences that are not formally good, but that are made good by the content of the (non-logical) concepts they involve–that is, materially good inferences (like the inference about the empty house mentioned above).

Surely it can’t be that the translation between ‘No people are in the house’ and ‘The house is empty’ changes the inference from being formal to being material. When I learn to say of a house that it’s empty, I know that it is defined to mean that they’re aren’t people in the house. I must learn that the presence of doors within doesn’t count.

Another case of claimed material inference, one given by Sellars himself, is that from ‘This is red’ to ‘This is colored’. On p. 17 of Chapter 1 of my book, which OUP has now made available, I sketch a type-theoretic account.

It seems to me that far more of our inferences should be seen as formal than Sellars allowed.

Formal inference is a special case of material. One often expects the word for a more general concept to exclude a special case. I still remember the shock of being told that a square is a rectangle. I wanted ‘rectangle’ to imply adjacent sides of different lengths.

So first comes material correct inference, and a special subclass of this is formally logically correct.

(I’ll use ‘formally’ now for ‘logically formally’.)

But now come what I see as the problems. The rules of the game of telling when materially correct inference is formally correct are just too vague. Their operation seems to rely on an existing understanding of what logic is (or what a typical 21st century analytic philosopher takes logic to be).

Start with

This bat is green and that ball is red. So, that ball is red.

Surely this is formally correct. We’re supposed to know that ‘and’ is logical vocab, and any replacement of the other words (if it doesn’t produce nonsense) should produce something of the form P&Q where inference to Q is accepted.

But what about ‘is’? Is that logical? If not, am I allowed to change one instance?

This bat is green and that ball was red. So, that ball is red.

Presumably not, since this inference is incorrect. Is the rule then that when replacing, all instances of the same word must be replaced in the same way? So

This bat was green and that ball was red. So, that ball was red.

And that’s OK? Will I then have to know about homonyms

I can bear any bear This is a bear So I can bear this bear.

The test can’t be that I have to replace all instances of ‘bear’ the same way. That would limit the variety of replacements enormously, when a much broader range preserves correctness.

Now,

This ball is red and that ball is too. So, that ball is red.

Is ‘too’ logical? If not, then it’s easy to generate an incorrect inference (replace ‘too’ with ‘green’).

All men are mortal Socrates is a man So, Socrates is mortal.

Are ‘are’ and ‘is’ logical? Must I know that ‘men’ is the plural of ‘man’? If not, correctness is easily broken.

When the sun is out, it is hot The sun is out So, it is hot.

Is ‘when’ logical? Replacing it with ‘a year after’ breaks correctness (or am I only to replace single words - but that’s very specific to the choice of native tongue?).

If is/are are logical, then is it the case that

A and B are happy So A is happy

is formally correct?

But then this fails when only jointly can they carry it:

A and B are able to carry this box So A is able to carry this box

And so on. Either one is extraordinarily strict on what counts as formally correct, or else the rules seem very vague.

Perhaps one could have degrees of formality - how much massaging is needed to put a piece of inference into a logical form.

This ball is red and that ball is too. So, that ball is red.

This is very nearly formal under the rule of replacing ‘too’ with a copy of the previous predicate.

Other cases of being very nearly formal include the use of definitions.

A is a bachelor So, A is male.

At the other extreme, inference is far from formal if it requires some implicit premises

E.g.,

P So, Q

rewritten as

$P$ $P \to Q$ So, $Q$.

Any piece of inference can be made formal with this trick. No need for definitions, just blast it with modus ponens and an implicit premise.

My thought about the red/coloured case is that it’s not as far from formal as the latter. Here the kind of logic available makes a difference. ‘is red’ and ‘is coloured’ are not just any two predicates. They’re related. The inference belongs to a well-defined inference scheme.

$\vdash X: Type$ $x:X \vdash B(x): Type$ $\vdash a: X$ $\vdash b: B(a)$

---

$\vdash ||B(a)||$ is true

Instances of this when rendered in natural language, include

Jane lives in this house This house is occupied

Jane has a son Jane is a parent

I really don’t think this is anywhere near as great a reformulation as the modus ponens one. That applies to just any ‘P, so Q’. This one is much more specific, much more like the specificity of ‘P&Q, so Q’.

I rather suspect that the irredeemably non-formal inferences are then ampliative ones, and that all the non-ampliative ones are formal with a degree of massaging.

Much will depend on how terms are defined: the ‘east’/‘west’ inference is ampliative if ‘east/west’ means in the direction of the rising/setting sun. On the hand, ‘to the east/west of’ defined as opposite relations is closer to being formal, and shares a common form R(a,b), so R(b, a) for the opposite relation.

Macfarlane:

To see whether Brandom’s theory counts as “strongly” pragmatist, we need to look more closely at the notions of commitment and entitlement in terms of which his semantic primitives (commitment preservation, entitlement preservation, incompatibility) are defined. Brandom calls these “deontic statuses” and says that they “correspond to the traditional deontic primitives of obligation and permission” (MIE, p. 160). (p. 89)

Necessity and possibility.