Experiments on Semantic Intuitions

A few more comments on why I think the setup of Weinberg, Nichols and Stich's experiments on intuitions is unfortunate. The problem seems particularly obvious in the experiments on semantic intuitions reported by Machery, Mallon, Nichols and Stich, but I think it carries over to many (though perhaps not all) of the experiments of Weinberg, Nichals and Stich. Here is one of the questions Machery, Mallon, Nichols and Stich asked:

Suppose that John has learned in college that GÃ¶del is the man who proved an important mathematical theorem, called the incompleteness of arithmetic. John is quite good at mathematics and he can give an accurate statement of the incompleteness theorem, which he attributes to GÃ¶del as the discoverer. But this is the only thing that he has heard about GÃ¶del. Now suppose that GÃ¶del was not the author of this theorem. A man called "Schmidt" whose body was found in Vienna under mysterious circumstances many years ago, actually did the work in question. His friend GÃ¶del somehow got hold of the manuscript and claimed credit for the work, which was thereafter attributed to GÃ¶del. Thus he has been known as the man who proved the incompleteness of arithmetic. Most people who have heard the name "GÃ¶del" are like John; the claim that GÃ¶del discovered the incompleteness theorem is the only thing they have ever heard about GÃ¶del. When John uses the name "GÃ¶del," is he talking about:
(A) the person who really discovered the incompleteness of arithmetic?
or
(B) the person who got hold of the manuscript and claimed credit for the work?

What would you answer?

What would you answer if instead of options (A) and (B) there was a text field in which you could enter your answer?

I would write something like "in one sense, John talks about the man who proved the incompleteness theorem. But there is also a sense in which he talked about the man really called 'GÃ¶del'." Or maybe I would answer "it depends." I really don't have any strong intuitions about the case, in particular as the question is put in terms of "talking about". Even a proponent of the causal theory of reference could admit that there is a sense in which John, claiming that "Hilbert's programme was shattered when GÃ¶del proved the incompleteness of arithmetic", talks about the man who proved the incompleteness of arithmetic.

If many people are like me, and would choose (A) or (B) more or less at random because they believe the answer isn't clear, the result of the experiment will look like there are important differences in intuition even if there are no such differences at all.

I suspect that if there was a free answer field, or more options like "both answers are ok" and "it depends", the results would be much less interesting.

PS: Notice that the presented text contains sentences like "Now suppose that GÃ¶del was not the author of this theorem" which presuppose that "GÃ¶del" doesn't denote the author of the incompleteness theorems. These sentence are taken more or less directly from Kripke. Now I agree with Lewis and Jackson that the intension of names is far less determinate than often assumed, so that if someone's utterances presuppose one resolution of the indeterminacy, we usually follow that resolution. If that is right, then philosophers generally agree with the judgments of Kripke (and Putnam) only because Kripke (and Putnam) never put their thought experiments in neutral terms.

This would be worth testing, and it suggests an even better setup than directly asking people about reference or aboutness. The idea would be to present subjects at random with one of two texts, presupposing different determinations of reference:

1) Read the following text carefully once, then answer the questions below it without looking back at the text.
In 1930, Kurt GÃ¶del proved two important mathematical results, known as GÃ¶del's incompleteness theorems. The first incompleteness theorem says that some truths of mathematics are not mathematically provable. According to the second theorem, one cannot prove mathematically that "0 = 1" is not mathematically provable. These results came as a big shock to many logicians and mathematicians, and their proof is so ingenious that GÃ¶del has often been called the greatest logician since Aristotle. About himself very little is known, though. He lived in Vienna in the 1920s, where he was murdered by Soviet agents in 1930, shortly after finishing his manuscripts. His real name wasn't actually "GÃ¶del", but "Schmidt". It is not entirely clear why his manuscripts were published under the name "GÃ¶del". Recent evidence suggests that there was another logician in Vienna at the time called "GÃ¶del", who published GÃ¶del's theorems under his own name to help his careeer. However, the plan didn't work out: this other logician was himself murdered by fascists shortly afterwards in 1932.
Question 1: What does GÃ¶del's second incompleteness theorem show?
Question 2: Kurt GÃ¶del has been called the greatest logician since -- whom?
Question 3: Who murdered GÃ¶del?
2) Read the following text carefully once, then answer the questions below it without looking back at the text.
Two of the most important theorems in modern logic are known as "GÃ¶del's incompleteness theorems". The first incompleteness theorem says that some truths of mathematics are not mathematically provable. According to the second theorem, one cannot mathematically prove that "0 = 1" is not mathematically provable. These results came as a big shock to many logicians and mathematicians. Recent evidence shows that they weren't really proven by Kurt GÃ¶del, an Austrian logician, but by one of his colleagues, Karl Schmidt, who was murdered by Russian agents in 1930, shortly after finishing the manuscripts. Apparently Kurt GÃ¶del got hold of the manuscripts and published them under his own name to help his career. So Schmidt's results came to be known as "GÃ¶del's incompleteness theorems", and GÃ¶del, who wasn't such a good logician at all, was praised as the greatest logician since Aristotle. He didn't live long to enjoy his fame, though: he himself was murdered by fascists in 1932.
Question 1: What does the second incompleteness theorem show?
Question 2: Kurt GÃ¶del has been called the greatest logician since -- whom?
Question 3: Who murdered GÃ¶del?

If Lewis and Jackson are right about this case, people might give different answers to question 3 depending on which version of the story they read, showing that they take "GÃ¶del" to refer to the man really called "Schmidt" in the first case, but to the man called "GÃ¶del" in the second.

# on 17 November 2004, 03:50

This comment is quite striking. For twenty years, textbooks have repeated that Kripke's thought experiments show that descriptivism is deeply wrong. These thought-experiments were supposed to elicit overwhelming intuitions (even descriptivists like Jackson endorsed these intutions). Notice that the first text is *directly* drawn from Kripke. But now we are told that these intuitions do *not* elicit strong intuitions. Hindsight bias?

Moreover, it is not the case that people choose randomly. The difference between the two cultural groups is significant. As a group, Westerners tend to have more Kripkean intuitions than Easterners. If people did choose entirely randomly, this would of course not be true.
Conclusion: people are not quite like the commentator.

# on 12 April 2005, 19:03

Macherie and al. have done a good job at popularizing the idea that as logically deep- and far-reaching intuitions as Kripke's rigidity thesis are in principle testable. In fact it is not easy to have a clear view between what could discriminate between the robust intuition of intensional indeterminacy (which led some prominent philosophers and semanticists to adopt a 2-Dimensional style semantics) and intuitions of rigidity which have been left undiscussed for a while and continue to show serious robustness when not directly confronted with the intuitive phenomenon of intensional indeterminacy. It's not the case though that Macherie and al. have directly tested this theoretical conflict between descriptivism and rigidity. They have reported on supposedly elicited naive semantic intuitions. But their data have some impact of course on the more theoretical side of the debate. One point in favor of the commentator is that he reminds us of the fact that the way a story or experiment is framed has a direct influence on the result, and that Putnam and Kripke's were framed in order to elicit intuitions of rigidity. Did Macherie and al. show that in spite of that framing semantic intuitions are variable, in particular culturally variable. Partly yes maybe but it will take some experimental refinements to answer more positively.
One point in case. It is partially indeterminate whether subjects performed one rather than the other of those two cognitive tasks while answering the question proposed by Macherie and al. : falsifying the sentence : "Godel is the discoverer of Godel's theorem" [truth-value attribution to some sentence]/ considering that the person named Godel is the person I continue to refer to by the name "Godel" even though I learnt new facts about him [preservation of usual reference]. We can well imagine a subject falsifying a sentence to the effect that Godel in the author of a theorem named after him while preserving her usual rigid reference to Godel. In fact in order to falsify that sentence she'd better keep the usual reference intact. So a further test could aim at a better unraveling of maybe two implicit tasks required to answer Macherie and al.'s test. This would in fact go in the direction of a Stichean view of the mind, modularity and fragmented reason, when semantic intuitions concerning truth-values of sentences and semantic intuitions concerning reference could be less than perfectly aligned one with the other.

# on 24 January 2020, 19:27

Actually, Gödel never proved the Second Incompleteness Theorem. He proved First Incompleteness Theorem. The first full proof of the Gödel's Second Incompleteness Theorem was published in the book: David Hilbert and Paul Bernays, "Grundlagen der Mathematik", vol. 2, in 1939 (please, see for example: Roman Murawski, "Recursive Functions and Metamathematics", Springer Dordrecht 1999).

In 2009 Dr Teodor J. Stepien and me delivered a talk at "2009 European Summer Meeting of Association for Symbolic Logic, Logic Colloquium 2009", in Sofia (Bulgaria). During this talk, a sketch of the proof of the consistency of the Arithmetic System was presented. This proof was done within this Arithmetic System.
The abstract of this talk: T. J. Stepien and L. T. Stepien, "On the consistency of Peano's Arithmetic System" , Bull. Symb. Logic 16, No. 1, 132 (2010); http://www.math.ucla.edu/~asl/bsl/1601-toc.htm

The full version of the proof (mentioned above), of the consistency of the Arithmetic System, was published in the paper: T. J. Stępień, Ł. T. Stępień, „On the Consistency of the Arithmetic System”, Journal of Mathematics and System Science, vol. 7, 43 (2017), arXiv:1803.11072 .