“A false thought is not a thought that has no being….It must be possible to negate a false thought, and for this I need the thought; I cannot negate what is not there.” (tr. Geach)

“[I]s negation of a thought to be regarded as dissolution of the thought into its component parts?….such as happens, eg, if a sentence written on paper is cut up with scissors, so that on each scrap of paper there stands the expression for part of a thought. These scraps can then be shuffled at will or carried away by the wind; the connexion is dissolved, the original order can no longer be recognized. Is this what happens when we negate a thought? No! The thought would undoubtedly survive even this execution of it in effigy.”

“People speak of affirmative and negative judgments; even Kant does so. Translated into my terminology, this would be a distinction between affirmative and negative thoughts…. I know of no logical principle whose verbal expression makes it necessary, or even preferable, to use these terms…. Accordingly I am in favour of dropping the distinction between negative and affirmative judgments or thoughts…”

“Are there two different ways of judging, of which one is used for the affirmative, and the other for the negative, answer to a question? Or is judging the same act in both cases?”

“These inferences proceed on the same principle, which is in good agreement with the view that judging is the same act whether the answer to a question is affirmative or negative.”

• Acts of judging don’t embed into larger constructions like conditionals (questions, negations)
• Negations do embed
• So negations aren’t acts of judging

“A thought may have just the same content whether you assent to its truth or not; a proposition may occur in discourse now asserted, now unasserted, and yet be recognizably the same proposition.”

Geach is not particularly concerned with negation, but gives a clean shape to Frege’s argument and generalises it.

Geach takes the Frege point to undermine (all?) attempts to use actions to give meanings to words:

“[I]t is natural to cast ‘it is true that…’ for the role of assertion sign. But this will not do, for this expression may come in an unasserted clause without change of meaning; nor is there any equivocation in an argument ‘it is true that \(p\); and if it is true that \(p\), then \(q\); ergo \(q\)’….[T]he identification of the assertoric force with the meaning of ‘it is true that…’ is just a mistake.”

Or, in opposition to a kind of expressivism according to which calling something ‘bad’ is to condemn (rather than describe) it:

• If doing something is bad, you shouldn’t do it.
• Mismanaging public funds is bad.
• Therefore, you shouldn’t mismanage public funds.

As Geach notes, someone who endorses the first premise here isn’t condemning doing something.

Or, in opposition to a view where saying that something “looks red” is to tentatively assert that it is red (rather than describing how it looks):

• If that looks red, then the lights are too low.
• That looks red.
• Therefore, the lights are too low.

Someone who endorses the first premise isn’t asserting (tentatively or otherwise) that anything is red.

The expressivists haven’t gone anywhere. But it’s commonly recognised that they owe Geach an answer.

Similarly, any attempted analysis of “not” as indicating an act of some sort needs to answer this argument.

It seems like I can assert “If the accused was not in Berlin, they didn’t do it” without rejecting that the accused was in Berlin. So what is that “not” doing?

“Bilateralism” is a name for a broad family of views that take linguistic meaning to stem from two acts or states or processes, one positive and one negative.

For example:

• assertion and denial
• acceptance and rejection
• proving and refuting
• inferring to and inferring from

Bilateralism is a family of views about linguistic meaning in general, so it certainly applies to negation.

It’s common (but not required) for bilateralists to treat negation as flip-flopping between the two sides.

An example:

• an assertion of \(\neg A\) is in order whenever a denial of \(A\) is in order, and
• a denial of \(\neg A\) is in order whenever an assertion of \(A\) is in order.

A bilateralist owes semantics for embedding environments in the same way.

eg (inclusive) disjunction:

• an assertion of \(A \vee B\) is in order whenever an assertion of \(A\) is in order or an assertion of \(B\) is, and
• a denial of \(A \vee B\) is in order whenever a joint denial of \(A\) and \(B\) is in order.

Since everything is in terms of the same two sides, these semantics are compositional. For example, we can conclude that an assertion of \(A \vee \neg B\) is in order whenever an assertion of \(A\) is in order or a denial of \(B\) is in order.

This answers Geach.

Someone who asserts “Either Carl is a good dog or Melbourne isn’t in Victoria” has not thereby denied that Melbourne is in Victoria.

And yet the meaning of their “not” (like all other meaning) is given by its tie to the act of denial.

Negation, on this kind of view, is not a marker of denial in the following sense: it’s not true that whenever a negation occurs, something is denied.

But we can still maintain that negation gets its meaning from its tie to denial.

This is one way to answer Frege/Geach: by positing a looser tie between vocabulary and act, so that the vocabulary can meaningfully occur without the act actually being performed.

“[Negation] leads to processing difficulties across a broad range of sentence types, experimental tasks, and paradigms” (Kaup & Dudschig, “Understanding negation” (2020))

Sentences with negations are slower and more error-prone to process in being produced, being understood, being remembered, and being obeyed.

This is a huge question, and I’m not attempting to answer it here, or even to fully evaluate a possible answer.

Instead, I’ll look at one candidate answer from the literature and ask what Frege’s argument can show us about it.

What is the relationship between understanding something and believing it?

Here two views have come to be called “Cartesian” and “Spinozan”, following Gilbert et al, “You can’t not believe everything you read” (1993).

Some terminology: let me use “belief” and “disbelief” to pick out whatever states we adopt towards propositions, and let me use “accept” and “reject” to pick out whatever processes result in those states. (My focus here is entirely on the processes.)

The Spinozan sees acceptance and rejection as not much like each other at all:

• acceptance is automatic, easy, involuntary, arational, while
• rejection is effortful, difficult, quasi-voluntary (like breathing/blinking?), quasi-rational

“Rejecting a proposition is more like thinking than seeing, while accepting is more like seeing than thinking” (Mandelbaum, “Thinking is believing” (2014))

Knowles & Condon, “Why people say ‘yes’” (1999)

• participants shown 100 personality-trait adjectives (from Big Five personality test)
• asked to judge “like me” or “not like me”
• divided into “yea-sayers”, “nay-sayers”, and “appropriate responders” based on proportion of answers
• yea-sayers answer quickest, then appropriate responders, then nay-sayers
  • for all, “not like me” takes about the same time;
  • the difference is in “like me” responses
• cognitive load during the task increases “like me” responses

Mandelbaum (ibid), advancing a Spinozan theory, ties negation to rejection and offers this as an explanation of negation’s processing difficulties.

Rejection is hard; negation is hard; they’re both negative—it would be nice to make something of this.

The details, though, are not clear: he talks of negations being “a subspecies of rejections”, but rejection is a process and negation is a lexical item, so this is hard to take at face value.

Pion et al, “Believe what we think!” (forthcoming) uses something like Frege’s argument to object to Mandelbaum, but without developing the argument in much detail.

In the rest of the talk, I claim that Frege’s argument does not refute Mandelbaum’s idea, although it may force him to clarify it in certain ways.

I hope that this isn’t just a move three steps down a dialectic, but also a more detailed development of a Spinozan theory of negation’s difficulty than has yet been offered.

This view might be wrong, but I think it’s at least Frege/Geach-proof. If that’s right, Pion et al’s objection fails against it; it must be wrong for some other reason.

As above, I don’t think negation can be a kind of rejection; it’s the wrong kind of thing. So what kind of connection can a Spinozan postulate here?

The key is to look at what’s hard: it’s hard to understand negations, and it’s hard to do rejection. So let’s suppose that understanding negation requires rejecting the thing negated.

This would give the Spinozan what they’re after: an explanation of negation’s difficulty that turns on rejection’s difficulty. Plus, it’s nicely simpleminded.

What about “Carl is a good dog or Melbourne isn’t in Victoria”?

Embeddings like this are key to Frege’s argument. What should the Spinozan say about them?

When we’re looking at assertion and denial, the situation is obvious: I can assert “Carl is a good dog or Melbourne isn’t in Victoria” without denying that Melbourne is in Victoria.

Here, I’ll do it: Carl is a good dog or Melbourne isn’t in Victoria.

It’s less obvious when it comes to thought. We can’t reliably introspect what we believe, and the Spinozan should maintain that we can’t reliably introspect the process of acceptance either.

The Spinozan holds that in understanding “Carl is a good dog or Melbourne isn’t in Victoria”, we accept the full disjunctive claim. But they also hold that we accept (since we understand) both “Carl is a good dog” and “Melbourne isn’t in Victoria”.

Since understanding the whole requires understanding the parts, it also requires accepting them.

This defuses Frege’s argument in a new way.

The Spinozan—unlike the bilateralist—doesn’t need to answer Frege; Frege’s argument doesn’t get off the ground here in the first place.

According to any Spinozan, understanding “If the accused was not in Berlin, they didn’t do it” involves accepting that the accused was not in Berlin.

Accepting the proposed link between negation and rejection, it also involves rejecting that the accused was in Berlin.

Now, you might think that it’s just implausible that understanding “If the accused was not in Berlin, they didn’t do it” involves rejecting that the accused was in Berlin.

Or you might thing that it’s implausible that understanding “Carl is a good dog or Melbourne isn’t in Victoria” involves rejecting that Melbourne is in Victoria.

But you should then think the Spinozan approach is implausible full stop—no special implausibility attaches to the negation/rejection link, as far as I can see.

(Fwiw, Spinozans tend to admit that their view is initially implausible—they just take it to be supported by evidence.)

• Frege has a number of targets in “Die Verneinung”; the key step for us is a focus on embeddings
• Geach develops this into a general argument against using acts to understand vocabulary
• Bilateralists have tried to answer Frege/Geach; it takes some care and turns on compositionality
• In some Spinozan theories, rejection is used to understand negation, and Frege’s argument has been used to try to rule this out
• But even a simpleminded version of this kind of Spinozan theory doesn’t need to worry about Frege/Geach
• If understanding involves acceptance, it can involve rejection too