### Formal and Material Inference

#### Posted by David Corfield

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, whereas the second relies on conceptual content occurring 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 it and then using *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 $j: P$ and $f: \neg (\sum_{x: P}H(x))$, where $H$ is a predicate on $P$. 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, or something very like it, appears already 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’s defined to mean that they’re aren’t people in the house. I must learn that the presence of doors within doesn’t count. And I’d better know that Jane is a person. I wouldn’t have concluded from the house’s emptiness that just anything is not in the house.

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. To be able to use such language, to know that there is a type of colours, that red is a colour, that some things have a colour, when all this is written type-theoretically, I don’t see why we can’t take it too as a piece of formal inference.

## Re: Formal and Material Inference

The red ball case is, I claim, an instance of a general inference scheme which says that if we have two types, $A$ and $B$, which are sets, and a relation between them, $R$, then given $a:A$, $b:B$ and that $R(a,b)$ is the case, then $\sum_{x: B}R(a,x)$ is inhabited, and so the propositional truncation $||\sum_{x: B}R(a,x)||$ is true.

So, for some choices of the components of this set-up we have special words to employ for the latter proposition: the object $a$ is colored; the person $a$ is a parent; the item $a$ is used; the house $a$ is occupied; the number $a$ is composite, and so on.

To insist that such instances of the inference scheme are to be counted as material will require a difficult line to be drawn as to acceptable rewriting. Presumably all allow the rewriting that enables us to consider the inference from ‘Jill is happy, and Jack is too’ to ‘Jack is happy’ as formal.