Are you using the turnstile as used here, e.g. where "⊢ A" means "I know that A"?

I'm suggesting we play with it how we please. You can define clearer rules if you like?

Srap Tasmaner I was using the turnstile as a shorthand for Frege's judgement stroke, so read "⊢⊢the cat is on the mat" as "I think that I think..." or "I think that I judge..." or whatever. Not as "...is derivable from..." — Banno

"My thought of judging that things are so is a different act of the mind from my judging that they are so." - J

Just to be clear, that's Rödl, not me, though I think he's right. And @Banno's paraphrase is also right if we agree that both "I judge" and "my thought of judging" can be captured by the judgment stroke, in the later case by simple recursion. Complications can ensue about exactly how to understand "my thought of judging", if it isn't understood as a type of assertion.

Right. I think (judge!) that distinguishing a thought from a judgement is an unnecessary complication. Hence my liking for @Banno's framing.

A sentence is already an assertion sign. (I assert.) How does it end up needing reinforcement? — bongo fury

:smile: You know the answer to that. But for the record: I seriously doubt if your use of the sentence "The cat is on the mat", above, genuinely asserted anything. And when T. S. Shmeliot uses the sentence in a poem, it's even less likely to be an assertion. And when I scream it in a crowded theater . . . it's art.

That said, the "proposition solution" is somewhat enigmatic to many of us.

Good. That sets out the issue quite clearly and simply. Some of us around here, following Kimhi and Rödl, want to make "thought" more complicated, but it's by no means obvious that this must be correct.

You know the answer to that. — J

I have the vaguest inkling (as yet) of it being due to the inscrutability of reference.

What answer should I have known?

has the same post in two places, so some of the context was lost. My reply:

There's a difference between understanding what it would take for "the cat is on the mat" to be true, and asserting that the cat is on the mat.

Between "p" and ⊢p".

One might, somewhat redundantly, further assert that one asserts that the cat is on the mat. If the need arose. — Banno

What answer should I have known? — bongo fury

Oh, sorry, from your post I thought you acknowledged that "all sentences assert" can't be quite right. I didn't have anything more esoteric in mind.

Haha, no, I do (unironically) think a sentence is an assertion sign. Alright... a naming sign.

"⊢the cat is on the mat"

and

"the cat is on the mat"

... A sentence is already an assertion sign. (I assert.) How does it end up needing reinforcement? — bongo fury

This is tricky because you say we need not worry about Frege, but then you immediately introduce Frege's notation (which Banno was using incorrectly in the relevant examples).

The answer is that Frege's judgment-stroke does not reinforce a sentence, but rather judges some content (i.e. a proposition). The judgment-stroke is not a referential symbol in the object language, and so it is not added to "the cat is on the mat," but is rather a vertical stroke added to the horizontal in:

"—the cat is on the mat"

Frege was inherently opposed to meta-analysis using the object language (and even simpliciter), if I recall.

-

So I think that in order to get away from Frege we need to avoid his judgment-stroke (and any symbols that attempt to represent it).

The prefix, however we phrase it - "I hereby assert that...", "I think that...", "I judge that..." etc - does seem to iterate naturally. — bongo fury

Apparently we are asking about how the following relate:

• The cat is on the mat.
• I judge that the cat is on the mat.
• I judge that I judge that the cat is on the mat.
• I judge that I judge that I judge that the cat is on the mat.
• ... (and so on)

Hence the recursivity of judgment.

One might, somewhat redundantly, further assert that one asserts that the cat is on the mat. If the need arose. — Banno

Does it seem less redundant if it read, "I assert that I've made the judgment that the cat is on the mat"? This formulation tries to bring in "making a judgment" as a 1st-person activity, not just a semantical stance. And you couldn't just lop off "I assert" because "I've made the judgment" doesn't quite say the same thing as "I assert". Unless you think Frege would just replace that phrase as well with a judgment stroke?

Perhaps.

I'm not seeing a problem here. Seems @bongo fury is stirring the possum, which is fair enough. Something might come of it.

I agree that it (the solution) must be about recognising the interplay of object- and meta-language.

Google, for those from foreign parts:

In Australian slang, "stirring the possum" means to provoke, instigate, or cause a disturbance, often by raising controversial topics or engaging in heated debate. It implies deliberately riling things up or causing a reaction, much like disturbing a sleeping possum would likely result in a negative response. The phrase can also be used to describe someone who is a "stirrer," someone who enjoys causing a bit of trouble or debate.

A noble activity.

Haha, no, for me it's actually philosophy big deal number one. Lately.

I agree that it (the solution) must be about recognising the interplay of object- and meta-language. — bongo fury

Okay, and what are the questions that are at stake? I assume there must be quite a few different questions.

(I wrote a bit about the general topic in <this post>, which is another thread where it came up.)

More from elsewhere...

A double judgment-stroke would make no sense for Frege. It is precisely a syncategorematic expression, and therefore cannot be nested in that way.
— Leontiskos
Syncategorematic means it has no meaning in isolation, only in context (like logical connectives), but that doesn't automatically rule out meta-level use — i.e., a second-order application about a judgment. Your argument again does not arrive at your conclusion.

"~" is also syncategorematic; yet we can write ~(~(A)). Necessity is syncategorematic; yet we can write ☐(☐(A)).

In Grundgesetze, Frege does not propose nesting judgment-strokes, but he does engage in meta-logical reflection — talking about what is asserted, and about the act of asserting.

A nested judgment-stroke would not violate Frege’s logical vision; it simply belongs to a different level — a meta-logical one — where judgments themselves become the objects of analysis. — Banno

And since can't help himself, and also hasn't responded to the point in question:

A nested judgment-stroke would not violate Frege’s logical vision; — Banno

You simply do not know what you are talking about. For Frege the notation is unified and continuous. The horizontal represents the content and the vertical represents the judging or asserting of that content. There is no such thing as a double vertical or a nested judgment-stroke. Here is Kimhi:

Since the vertical does not belong to the functional composition of a proposition, it has no referential import. This distinguishes it, within the Begriffsschrift, as the sole syncategorematic expression. The whole symbol governed by a judgment-stroke, for example, “⊢p,” is itself a syncategorematic unit since it cannot be embedded as a functional or predicative component within a logically complex whole. (In particular, it cannot be either a subject or a predicate term in a proposition.) As such, it cannot be repeated in different logical contexts, but can only stand by itself.
— Irad Kimhi, Thinking and Being, 41-2 — Leontiskos

And again, your argument does not meet your conclusion.

Have fun.


Added: Just to be clear the Kimhi quote is against writing ⊢(⊢p → ⊢q), not ⊢⊢(p→q).

let's not worry about whether it fits Frege's vision. The prefix, however we phrase it - "I hereby assert that...", "I think that...", "I judge that..." etc - does seem to iterate naturally. — bongo fury

Okay, and what are the questions that are at stake? — Leontiskos

Yep, I'll be trying to contribute.

It is interesting, though, that Banno thinks Frege's judgment-stroke is a functional symbol that can simply be nested contextually. So his difference with Frege has to do with whether the judgment-stroke belongs to the object language.

I don't think this thread will ultimately get away from those sorts of puzzles, namely the puzzles of how and why the boundaries between the meta-language and object-language exist, and whether they ought to. So given that Frege has already come in, we might ask why he placed his strictures around the judgment-stroke and second-order predicates. That is the sort of question that is apropos.

↪J Haha, no, I do (unironically) think a sentence is an assertion sign. Alright... a naming sign

At least most commonly. I am not sure however, if other uses really get that far away from assertion. Of course, if "the cat is on the mat," is used as an example, it isn't being asserted, but if it's being used as an example, then it is going to be something we are asserting something about. So the "sentence" is really more of a name/variable that is having something predicated of it as an assertion.

I have always found the difference between declarative statements (i.e., the assertion of the self, "I," as an "agent of truth" who is responsible for the statement) and informational sentences (i.e., those that displace the agent/make them transparent) more interesting re assertion. For me, it's enough that "a man is standing" and "it is true that a man is standing," is the same thing in its most basic, common usage (just like "a man" or "a duck" is the same as "one man," and "one duck," etc.), and that it is implied that "exists" follows from "is true" or "one" or any basic assertion.

(I wrote a bit about the general topic in <this post>, which is another thread where it came up.) — Leontiskos

Great, I invite people to bring that kind of thing here, if it's off topic there (or wherever).

There are so many points people have made that I'm highly disposed to agreeing or disagreeing with at length. In the end, I prefer one move of chess (cooperative chess :wink: ) at a time, so I won't. But thanks for all contributions.

I don't think this thread will ultimately get away from those sorts of puzzles, namely the puzzles of how and why the boundaries between the meta-language and object-language exist, and whether they ought to. — Leontiskos

:100:

It is interesting, though, that Banno thinks Frege's judgment-stroke is a functional symbol that can simply be nested contextually. So his difference with Frege has to do with whether the judgment-stroke belongs to the object language. — Leontiskos

Does this help with the puzzles of how and why and whether they ought?

Of course, if "the cat is on the mat," is used as an example, it isn't being asserted, — Count Timothy von Icarus

Is it then not an assertion? Is a name not a name when it's an example?

"P" probably entails that I know P, just as it entails that I exist and I'm communicating and I'm speaking a language.

"P" is not identical to any of those, though, I don't think. Whether it's identical to "P is true." is another matter. I would say yes.

Does this help with the puzzles of how and why and whether they ought? — bongo fury

It's worth seeing how there is a way in which Frege and Kimhi are correct in seeing judgment as syncategorematic or unembeddable, and this can be seen by looking at one metaphysical aspect of judgment, namely its temporality.

First, Rombout points out that Frege's judgment-stroke is "performative language" (38). She says:

The vertical part of the symbol is not just signifying some act of judgment it seems, rather “the judgment stroke contains the act of assertion”(I3). This raises the question whether the judgment stroke is a sign signifying an act or whether writing it actually effects the assertion. — Frege, Russell and Wittgenstein on the Judgment Stroke, by Floor Rombout, 11

This underscores the fact that human judgment is an act. Similarly, the stroke that indicates this act is not therefore a mere semantic construction. Further, acts are temporal, and the act of judgment is no exception. This is why the act of judgment is by its very nature syncategorematic or unembeddable. To judge is to act in the present. Hence:

The issue here is that we reason discursively, and we do not (strictly speaking) ever simultaneously engage in more than one judgment. So when <I judge that I judge that a is F> there is at least a temporal distinction between the two instances of judgment, and in this case there is also a logical priority issue, i.e. one of the two judgments must be logically prior to the other.

So if Rodl wants to read that proposition as a non-temporal angelic intellection, it won't make any sense. That is, if we try to make both instances of 'judge' temporally and logically identical, it won't make any sense.

One way for Rodl to dispute true recursivity would be to say that the only way to interpret <I judge that I judge that a is F> in a non-vacuous way is to interpret it as <I judge that I have judged that a is F>. — Leontiskos

So we have 1 vs 2. Here is 1:

• I judge that a is F
• I judge that I judge that a is F
• I judge that I judge that I judge that a is F
• ...

And here is 2:

• I judge that a is F
• I judge that I have judged that a is F
• I judge that I have judged that I have judged that a is F
• ...

The equivocation of the judgment-terms in (1) can be more clearly seen once we realize that only one judgment is ever a judgment in the primary sense, namely a judgment in the present. This equivocation is remedied in (2) by explicitly allowing only the first judgment-term to be in the present tense. The others are in the past tense.

I think this sheds light on the recursivity of judgment, insofar as we can see that there is a judgment-hierarchy in any judgment about judgments. That hierarchy is most obvious when we think about tense, but we could also divert from tense and think about other forms of primary-ness and secondary-ness of judgments. For example, we might hold that (1) is acceptable, while each judgment-term is still not univocal in that primary sense.

There are two related issues. The first asks about the relation between a present-tense judgment and a past-tense judgment. There is a curious sense in which a past-tense judgment is not yet past, and is still "in effect"—at least insofar as we have not rescinded our assent in the meanwhile. (This entails that "I have judged" could mean at least two different things.) Still, this does not make a past-tense judgment active or present or primary in the same way that a present-tense judgment is active and present and primary. The second issue regards the question of whether and how a single judgment can encompass many judgments, i.e. how judgment can be mereologically complex. Philosophers are primarily concerned with this second question as it pertains to primary or present-tense judgments (e.g. "Dave, Sue, John, and Marie are all in attendance at the party"), but the question could also be extended to the various sentences of (1) or (2).


NB: I gave a number of relevant sources in the post <here>.

---

Added: Just to be clear the Kimhi quote is against writing ⊢(⊢p → ⊢q), not ⊢⊢(p→q). — Banno

No, when Kimhi says, "[The judgment-stroke] cannot be repeated in different logical contexts, but can only stand by itself," he is obviously excluding your strange embeddings. ⊢(⊢(p→q)) is obviously repeating the judgment-stroke within a logical context. Indeed, once one understands Frege's notation they realize that it is not even notationally possible to do this. Here is Rombout in the post where this was already explained to you:

Instead of writing the whole inference, consisting of the three assertions “ ⊢ p”, “ ⊢ (p ⊃ q)” and “ ⊢ q” , Russell and Whitehead propose an abbreviation containing the assertions of the two atomic propositions connected by an implication: “ ⊢ p ⊃⊢ q”. Frege would consider this a category mistake; in the Begriffsschrift it is not possible to have a judgment stroke within the scope of a conditional.

...

A reason why Russell and Whitehead consider this abbreviation acceptable can be found in their explanation of syllogisms... — Rombout, 44-5

(Rombout's whole paper examines why Wittgenstenians characteristically fail to understand this aspect of Frege, just as Wittgenstein did.)