Theory-overarching theorems — and why they are typically no-go theorems

When consider influential theory-overarching theorems in quantum gravity, what we find is that most theorems on the list are no-go theorems. That is, these theorems such that, for rather generic conditions, a certain positive result is ruled out. To make things more concrete, I have compiled a list which, albeit not perfectly exhaustive, I would claim to be representative (although I cannot back up such a claim through a quantitative study or similar). Our interest in this section is to understand better why these general theorems are no-go theorems (rather than `go’ theorems).
\begin{center}- \Large \textbf{Key Theorems in (approach-)independent) Quantum Gravity} \end{center}
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\renewcommand{\arraystretch}{1.4} \setlength{\tabcolsep}{8pt} \begin{longtable}{>{\raggedright\arraybackslash}p{3.8cm} >{\raggedright\arraybackslash}p{4.5cm} >{\raggedright\arraybackslash}p{7cm}} \toprule \textbf{Theorem} & \textbf{Domain} & \textbf{Formal Statement} \\ \midrule \endfirsthead \toprule \textbf{Theorem} & \textbf{Domain} & \textbf{Formal Statement} \\ \midrule \endhead
\textbf{Hawking–Penrose Singularity Theorem} & Classical General Relativity & Under physically reasonable energy conditions, global hyperbolicity, and the existence of trapped surfaces, the spacetime must contain geodesic incompleteness (i.e., a singularity). \\
\textbf{Bekenstein Bound} (Weak Form, Proven Cases) & Thermodynamics + QFT in curved spacetime & For a system of energy $E$ confined to radius $R$ , the entropy $S$ satisfies:

$S \leq \frac{2\pi ER}{\hbar c}$

Proven in specific cases using quantum field theory and black hole thermodynamics. \\
\textbf{Positive Energy Theorem} (Schoen–Yau, Witten) & Classical GR & In asymptotically flat spacetimes obeying the dominant energy condition, the ADM energy $E$ satisfies:

$E \geq 0$

with equality only if spacetime is isometric to Minkowski space. \\
\textbf{Topological Censorship Theorem} & Classical GR & In asymptotically flat, globally hyperbolic spacetimes satisfying the averaged null energy condition, every causal curve from past null infinity to future null infinity is deformable to a topologically trivial curve — thus, nontrivial topologies are hidden behind horizons. \\
\textbf{Generalized Second Law (GSL)} (Wall, etc.) & Semi-classical gravity & In any process involving black holes and quantum fields, the generalized entropy (black hole area / $4G\hbar$ + outside matter entropy) never decreases:

$\frac{d}{dt} \left( \frac{A}{4G\hbar} + S_{\text{outside}} \right) \geq 0$

Proven under certain conditions using QFT in curved space. \\
\bottomrule \end{longtable}
The only philosophical systematisation of no-go theorems so far has been put forward by \cite{Dardashti}. Dardashti suggests the formalisation of no-go theorems via the notion of a no-go result: a no-go result is the tuple $<P, M, F$ \lightning $G, B>$ which encodes that some goal $G$ and some physical background assumption $B$ are together logically incompatible with the conjunction of certain physical assumptions $P$ , a certain physical-mathematical framework $F$ and further mathematical structure $M$ ;\footnote{In fact, Dardashti distinguishes between `physical’ framework and `mathematical’ structure but I take this to be a charitable reconstruction as surely any physical framework works with mathematical assumptions. A better distinction is thus drawn between (representing) physical-mathematical frameworks, and mathematical techniques used in deriving statements about it (such as numerical procedures, approximations or asymptotic expansions — things which Dardashti also has in mind with mathematical structure).} the formalisation of no-go results is supposed to be faithful to the logical structure of many no-go results found in practice even if it cannot be claimed to capture all possible forms of no-go results.
Based on this logical formalisation of a no-go result, Dardashti can straightforwardly lay out four major `methodological pathways’ to resolve the inconsistency linked to the no-go result, namely to deny the goal itself (the usual route), some of the physical assumptions, the physical framework, or the mathematical techniques used.\footnote{A more careful formalisation accounts for that $P, G$ and $B$ are all respectively dependent on $M$ and $F$ , i.e. presuppose $F$ and $M$ for their formulation; one might thus rather want to write something like $<M, F; P(M, F)$ \lightning $G(M, F), B(M, F)>$ which (a) still accounts for the assumption of $M, F$ as (essential) part of the no-go result but (b) also renders explicit that $P, G, B$ depend on $M, F$ . Such a dependence is arguably already acknowledged by Dardashti albeit not implemented.} This formalisation makes, in particular, clear that the actual epistemic or pragmatic relevance of what is dubbed no-go theorem in practice does not have to lie in ruling out the supposed `goal’ $G$ but possibly — if $G$ is to be kept — in \textit{considering} the dismissal of other conditions, instead. It is in such a context that Dardashti talks of no-go theorems being `go-theorems’: no-go theorems encourage us to re-assess certain well-established conditions if we take the no-go result itself to be unacceptable. This is a fair observation from practice.\footnote{See also the discussion by \cite[section 2.1]{MitschOthers} who invoke the image of “malleable” (as opposed to sacrosanct) assumptions, and ask us to “[p]icture the assumptions as faces of the dials of a combination lock, and unlocking” (p. 4) it as finding a way to the goal $G$ .} However, the use of `go’ in Dardashti’s notion of `go-theorems’ here is slightly confusing as it is a different sense of go (namely, for \textit{pursuing} a status check of a condition) from that in the standard no-go theorem (`go’ means here \textit{accept} a condition—or, rather, `no-go’, not to accept a condition).
What are the conditions for a no-go theorem to in fact deliver a proper no-go result, though? Inspired by Dardashti’s formalisation, one can propose the following characterisation of `no-go theorem’: \begin{quote} A no-go theorem is an inconsistency statement concerning a set of conditions where the negation of the least established condition that can be dropped to evade inconsistency is called the no-go result. \end{quote} Once another condition is genuinely seen as less established than the condition usually denied, its negation gives technically speaking rise to a new no-go result (even if this is may not be typically explicitly acknowledged in practice). So, the set of conditions that are together inconsistent can be re-read to give rise to a different no-go theorem.\footnote{There is in fact no requirement that only one condition should come out as problematic on such a re-analysis. Sometimes phrasings in terms of dilemma, trilemma or even higher-order lemmas can be apt to account for equal lack of support there are for certain conditions vis-à-vis the relatively more accepted ones.}
More precisely, we might then distinguish between a no-go theorem and a go-theorem (the latter is different from Dardashti’s notion of `go-theorem’) as follows:
\begin{quote} A no-go theorem is an inconsistency statement concerning a set of conditions where the least established condition that can be negated to evade inconsistency is a positive statement; its negation is called the no-go result. \end{quote}
\begin{quote} A go theorem is an inconsistency statement concerning a set of conditions where the least established condition that can be dropped to evade inconsistency is a negative statement; its negation is called the go result. \end{quote}
Again, note that `go-theorem’ here is not meant in Dardashti’s sense sketched above but in direct analogy to that of `no-go-theorem’: in a no-go-theorem, one dismisses a positive statement—in a go-theorem, to resolve an inconsistency between conditions, one should dismiss a negative statement, i.e., accept a positive statement.
With these notions of go-theorem and no-go-theorem in place that are counterparts, we are now in position to immediately explain how it comes there are no-go-theorems rather than go-theorems in the context of general statements: The fact that the clash between conditions is at all relevant requires that all statements involved are well-established. (Only if the clash involves relatively well-established conditions is it at all an actual clash.) At the same time, in a practical context, we can only expect to find a clash between conditions that are not perfectly established but involve some epistemic riskiness. Generally, only positive statements provide specifically enough concrete statements to be somewhat epistemically risky. (Negative statements concern a much bigger possibility space and are thus much less epistemically risky.)\footnote{There is a long tradition of discussing that facts tend to be positive rather than negative. Alternatively one might appeal to some of these positive arguments as well to make a case for how typically in physics positive facts are considered; so that if there are di-, tri-, or multi-lemmas, they will be between positive rather than negative facts. And so they will indeed, once the weakest (positive) fact in that clash of facts has been picked out, be presented as a no-go theorem regarding that fact.}
As a result, then, we might expect a collection of conditions which are each a positive statement; at the very least, we should expect the condition \textit{which is ultimately deemed least likely and then negated} to be a positive statement. This explains why no-go theorems are effectively more common than go-theorems.
Finally, one might take it that positive and negative claims are intertranslatable. This goes, however, against common conceptions on the presence of positive facts.