On extending mathematical attitudes to natural languages

Kris Brown - Inaugural Tutorial Day 2023

(press s for speaker notes)



Artificial languages (including mathematical languages) have a different flavor from natural languages.

We tend to like some artificial languages (like mathematics).

  • Deductive reasoning
  • Compositionality (\(\phi\)’s meaning just depends on its syntactic structure and the meanings of its simple parts.)
  • Upfront, explicit definitions + premises
  • Monotonicity (once we are justified in saying something, we will forever continue to be justified in saying it)

Like Plato, we see them as an ideal to which other languages are to be held. Other discourse is inadequate insofar as it does not have these properties. Why? It would be nice if …

  • reasoning could be a mechanical, thoughtless process.
  • context had no impact on our utterances’ validity.
  • meaning were transparent / we had absolute certainty about some things.
  • one need not worry about justification of established facts / we had a feeling of progress towards something Eternal.


Formal language Natural language
Logical inference Material inference
Monotonic logic Nonmonotonic logic
Logical empiricism Pragmatism
Atomism about meaning Holism about meaning
Language/theory distinction The ‘vocabulary’ vocabulary
Truth Justification
Semantics Pragmatics
Representationalism Inferentialism

Material inference

  • If I strike this match, it will light.
  • If it’s red, then it’s colored
  • If the streets are wet, it recently rained.
  • If NH is to the East of CA, then CA is to the West of NH.
Material inference Logically-valid inference
Def Can be changed from a good inference into a bad one by altering some nonlogical vocabulary. True no matter what you plug in for the variables or substitute for the non-logical vocabulary.
Slogan Descriptive terms appear essentially Descriptive terms appear vacuously
Form True, but not because of its logical form True in virtue of its logical form
  1. There are some inferences that are good, not in virtue of their logical form.
  2. Rather than thinking the descriptive terms (“match”, “red”) have some essence which has the suppressed premises, we can understand the content of these descriptive terms in terms of the materially good inferences they appear in (as premises or conclusions).

Material proprieties of inference are more fundamental / conceptually prior.

Need the notion of a good inference in order to understand what a logically good inference is.

Nonmonotonic logic

  1. “If I rub this match along the striking surface, then it will ignite.” (\(p\rightarrow q\))
  2. “If \(p\), but the match is inside a strong electromagnetic field, then it will not ignite.” (\(p\land r\rightarrow \neg q\))
  3. “If \(p\) and \(r\), but the match is in a Faraday cage, then it will ignite.” (\(p\land r\land s\rightarrow q\))
  4. “If \(p\) and \(r\) and \(s\), but there is no oxygen in the room, then the match will not ignite.” (\(p\land r\land s\land t\rightarrow \neg q\))

How these new ideas might be in the ‘spirit’ of category theory:

  • Understanding the conceptual content of “bat” as defined by its interconnections in networks of material inference (rather than those being the proper inferences to make in virtue of the nature/Platonic idea of “bat”)
    • This is evocative of ‘morphisms-first’ approach to conceptual content.
  • Modeling the openness to change endogenously rather than as a meta-level attitude
    • This is very in spirit with open systems vs closed systems.

Logical empiricism

How might someone with an affinity for formal logic approach the problem of natural languages? Consider a tradition in philosophy called logical empiricism:

  • We have freedom to pick our language (e.g. “bachelor := unmarried man”)

principle of tolerance: we are not in the business of setting up prohibitions but of arriving at conventions… In logic there are no morals. Everyone is welcome to set up his logic, i.e., his form of language, as he pleases. If he wants to discuss it with us, though, he needs to give syntactical specifications rather than philosophical debates.

— Carnap, Logical Syntax of Language §17

  • Once we do this, the world settles what our terms refer to and which theory (i.e. set of true sentences) is correct.

This language/theory distinction is characteristic of artificial languages. Confusion stems from applying this model to natural languages, where the analogies (e.g. deduction, premise) don’t exactly work the same way.

Agrippan / Münchhausen trilemma

Assume a transitive structure of justification on set of beliefs.

Then, any knowledge requires a choice:

  1. Infinitely regressive argument: the chain of justification goes on ad infinitum.
  2. Circular / coherentist argument: justification is not acyclic
  3. Foundationalist / dogmatic argument: there exist self-justifying beliefs

\[...\implies p_{i-1} \implies p_i \implies p_{i+1} \implies ... \]

Challenge for logical empiricism

Belief Inference
Regress stopper Sense data Analytic / logical inference

But for something to be a regress stopper, it must be justified independent of any other collateral commitments.

Quine’s Two Dogmas of Empiricism

Analytic truths Synthetic truths
Statements which are ‘true by definition’. Require more than definitions to be true.
Doctors who work on eyes are doctors.
People who run move.
Doctors who work on eyes are rich.
People who run damage their bodies.

Quine looks at various characterizations of analyticity and finds them all inadequate (ultimately circular notions: self-contradictory, synonomy, necessity).

Each statement is not made (atomistically) true or false by the world. The meaning of \(\phi\) is dependent on our context \(\Gamma\) of background assumptions - only the whole constellation faces the tribunal of sense experience.

  • When we make a scientific measurement to falsify some theory, the interpretation of that measurement is made in some other theory. The world doesn’t falsify a theory, rather we have preferences of which beliefs we want to keep fixed and which we are willing to update.
  • We are free to choose our logic!

[Carnap’s] pragmatism leaves off at the imagined boundary between the analytic and the synthetic. In repudiating such a boundary I espouse a more thorough pragmatism.

Sellars’ Myth of the Given

The narrative being attacked is a process like this:

  1. Physical objects -> 2. Sense data -> 3. Noninferential beliefs -> 4. Inferential beliefs

Reductivist empiricism seeks a foundation for knowledge and takes sense-impressions as the unjustified justifiers. Against this, Sellars argues that:

  • For a sensory experience to play an evidential role (to serve as the justification of some other claim), it has to be conceptually articulated.
  • You can’t just have this concept in isolation, because having the concept is understanding the concept’s inferential relations (holism about conceptual content).

In characterizing an episode or a state as knowing, we are not giving an empirical description of it. We are placing it in the logical space of reasons, of justifying and being able to justify what one says. — Empiricism and the Philosophy of Mind §36

Attacking logical empiricism

Sellars Quine
Myth Myth of the Given Myth of the Museum
Relevant work Empiricism and the Philosophy of Mind (1956) Two Dogmas of Empiricism (1951)
Concept challenged Sense-givenness Meaning-givenness
Against atomism in… Epistemology Semantics
Kantian representation Sense impression Meaning-relation
Regress stopped premises inferences

The simplest way to describe the common features of Quine’s and Sellars’s attacks on logical empiricism is to say that both raise behaviorist questions about the epistemic privilege which logical empiricism claims for certain assertions, qua reports of privileged representations. — Rorty, Philosophy and the Mirror of Nature, Chapter 4

Quine (resp. Sellars) asks “What do you have to be able to do?” in semantics (resp. epistemology) in order to argue for his holism.

Alternative to the language-theory distinction

The ‘vocabulary’ vocabulary. Meanings and facts come together at the same time.

Rather than search for a fixed foundational vocabulary to deduce truth from, we ought acknowledge our power of redescription - to change our vocabulary to suit our interests.

  • E.g. Convincing someone who has defined a product of sets to redescribe this as the product in Set. It’s not an argument.
  • Redescription is rational but not logical: you can rationally persuade (not prove) someone who thinks 1 ought be (defined to be) a prime number.
  • No real difference between learning phlogiston (oxygen) has very different properties than we thought it had vs declaring phlogiston doesn’t (objectively) exist.

Alternative to the language-theory distinction

For certain aims, we want a fixed vocabulary, for reasoning to be monotonic. In mature sciences, a lot of work is taken to allow for discourse to proceed as if the vocabulary were fixed:

  • Discursive “clean rooms”, maintained through heroic social disciplinary measures.
  • For mature sciences: e.g. in the history of temperature, each time a new way of measuring temperature was discovered, the concept changed.

We may not want to take this attitude towards all forms of discourse, e.g.  moral discourse.

Even scientific discourse we want periodic redescription (Kuhnian paradigm shifts).


  • Material (logical) inference, (non) monotonic logic
  • Holism vs atomism about conceptual content
  • The vocabulary vocabulary vs the language / theory distinction
  • David paints a beautiful picture in “What are we tracking?” of good math as crystallizing a phenomenon.

    • ✅ Formalizing a human practice makes it explicit, exposing it to rational criticism.

    • ❌ Thinking of an abstraction as ‘right’ crystalizes it - making it immune to revision.

  • ✅ Compositionality / monotonicity are nice properties to aspire for.

    • ❌ But they are inherently atomistic, rather than holistic.
  • Determining who Topos works with

  • Attitude towards the subject matter experts we work with

Achilles and the Tortoise: regress of inferences

The Tortoise accepts a proposition \(p\) and \(p \implies q\).

  • The exact \(p\) and \(q\) aren’t important to the moral of the story, though
  • The Tortoise is acting like: I’ll do anything you tell me to do, so long as you make explicit the rule you’re asking me to follow.

Achilles tries to convince the Tortoise to accept \(q\) (logic obliges you to acknowledge \(q\)).

The Tortoise is willing to go along with this but demands that this rule be made explicit: so Achilles adds an extra axiom: \(p \land (p \implies q) \implies q\).

Achilles says that, now, you really have to accept \(q\), given that you’re committed to:

  • \(p\)
  • \(p \implies q\)
  • \(p \land (p \implies q) \implies q\).

But the Tortoise notes that, if taking those three propositions and concluding \(q\) is really something logic obliges one to do, then it bears writing down: \[p \land (p \implies q) \land (p \land (p \implies q)) \implies q\]

This can go ad infinitum; the Tortoise wins.

Myth of the Given

The narrative being attacked is a process like this:

  1. Physical objects
  2. Sensing of sense contents
  3. Noninferential beliefs
  4. Inferential beliefs
  • (1->2) Because there is a red object with a triangular facing surface in front of me, I find myself with a sensing of red-triangular content.

The first inference is a causal notion (studied by neurophysiologists).

It is a matter-of-factual relation. It is describable in non-normative vocabulary.

Myth of the Given

The narrative being attacked is a process like this:

  1. Physical objects
  2. Sensing of sense contents
  3. Noninferential beliefs
  4. Inferential beliefs
  • (1->2) Because there is a red object with a triangular facing surface in front of me, I find myself with a sensing of red-triangular content.
  • (2->3) ?
  • (3->4) Because I have this belief (along with others), I am justified in believing there is a yield sign in front of me.

The third inference is epistemic, an inferential notion which relates sententially-structured beliefs/believables which are repeatable abstracta. This is a matter of reasons rather than causes. It is not a natural justificatory relation but rather a normative one; the logician rather than the scientist has the final say in adjudicating it.

Myth of the Given

The narrative being attacked is a process like this:

  1. Physical objects
  2. Sensing of sense contents
  3. Noninferential beliefs
  4. Inferential beliefs
  • (1->2) Because there is a red object with a triangular facing surface in front of me, I find myself with a sensing of red-triangular content.

  • (2->3) Because I have such as sense content that I acquire the noninferential belief that there is a red-triangular object in front of me.

  • (3->4) Because I have this belief (along with others), I am justified in believing there is a yield sign in front of me.

What of the second arrow? It is pointing from content in the causal order to the conceptual order. Sellars denies that sensings can ground the non-inferential beliefs (as reasons);

Two Dogmas pragmatism

The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man- made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field. Truth values have to be redistributed over some of our statements.