Content - Nate's Notes

# Content Theories imply *consequences*, logical implications—statements that follow from a theory. These consequences can be *logically deduced* from the theory. They are interlocked with and cannot be severed from the theory. They can be of the form "$X$ must happen" (a positive prediction) or "$Y$ must not happen" (a negative prediction). The content of a theory is defined as the set of things that it *forbids*—i.e. the set of things that would contradict at least one of it's consequences. The more that a theory *rules out*, the higher its content. Both positive and negative consequences can factor in to a theories content. Saying "the planet will be here at time $tquot; is a positive consequence. If it is not there, this is a falsifier and it adds to the theories content. Likewise, saying "the planet will not be over there at time $tquot; is a negative consequence. If it *is* there, this is a falsifier and also adds to content. They both increase the range of states the theory excludes—this is the measure of it's content. As a heuristic we can say: > We must seek theories with high content. Note that this will naturally lead us to seek *simpler*, more *universal* theories. This is because they often have higher content. But this still leaves us with two slightly tricky counter examples to deal with. ### Theory: The Angels Push the Planets One may argue that this theory is simple. But note that it is both easy to vary (why angels and not devils?), and it has effectively no content. This is because it has almost no consequences. There is nothing specifically implied by this theory, and therefore it doesn't rule out any conceivable state of the world. Thus we can strongly argue that this is a bad theory because it has almost no content. And in this case another way to say this is that it can nearly explain *anything*. Imagine we are arguing with the proponent of the angels theory. They claim that no no, the angel theory does have consequences and thus content. It's consequences are "all possible planetary motions". This would effectively entail every possible trajectory. Well let us just [take this theory seriously on it's own terms](Take%20Theories%20Seriously%20on%20Their%20Own%20Terms.md). If we do that, we will find that there are *mutually contradictory consequences*. One version will say "Mars will be at location $x$ at time $tquot;, while another says "Mars will be at location $y$ at time $tquot;. This means that the theory is logically entails *contradictions*, which is fatal. Thus if we take the theory seriously we see that it collapses under inconsistency. ### Theory: Epicycles Theory (Ptolemaic Astronomy) This theory on the surface looks much more reasonable. It may appear to say something very specific and look hard to vary. And it may look like it has high content. But is that really so? First off, it actually is very easy to vary. You can always just tack on an additional epicycle in an ad hoc matter to patch up your predictions. And if you allow for that many degrees of freedom, you can "explain" nearly anything you want (consider [Von Neumann's Elephant](Von%20Neumann's%20Elephant.md)). Second, it has low content. This is because with each new conjunction you add (epicycle) you are decreasing the content of the theory. Let's see how this unfolds. Consider two basic theories: $T_1$ and $T_2$, which posit one and two epicycles respectively. $T_1$ has fewer degrees of freedom, makes stronger claims, and has more planetary paths that it rules out. If the observations don't match, the theory fails. $T_2$ can fit a wider range of planetary motions, meaning it rules out fewer possibilities and is therefore harder to falsify. Thus the more complex $T_2$ is logically weaker, because it is compatible with more observational outcomes. Also let us note the difference between a specific instantiation of the epicycle theory, and the meta-theory (i.e. epicycles as a method). A specific instantiation of the epicycle theory (e.g. there are four epicycles with these parameters...) is a reasonably rigid structure, hard to vary, and has reasonable content. But the meta-theory, simply saying "epicycles govern planetary motion", has *zero logical content*, because it forbids nothing! We are back to saying "the angels did it". Compare this with Newton's theory. Newton’s law of gravity posits a rigid, fixed structure. From this, you get precise, testable predictions for future planetary motion. If reality deviates from those predictions — the theory is wrong. That’s high logical content. --- **Logical Consequence** A specific implication (consequence) that follows from a statement. These can be empirical, but fundamentally, they are implications that follow from asserting the theory. So, when you make a statement about the world, you are simultaneously ruling out a potentially infinite number of alternatives. For example, stating _“today is April 28”_ implies it’s not April 27, 29, or any other date. Language carries an implicit richness: even saying a little can logically entail quite a lot. **Logical Content** The set of all logical consequences of a statement **Empirical Content** The set of all empirical statements logically entailed by a statement. The empirical content forms the set of potential falsifiers of a statement. **Explanatory Content** The set of all consequences following from an explanation. These include logical consequences (predictions) as well as explanatory consequences (how does it explain some phenomena outside of it's original scope) which are directly related to [Reach](Reach.md). **Hard to Vary** The details of a theory are not easily modified or replaced without spoiling the explanation itself. Bad explanations are easily variable because there details are barely connected to the phenomena they purport to explain. HTV imposes *constraints*. **Constraints** An HTV theory has constrained, specific predictions. It has a constrained, specific reach. It has a constrained structure—you cannot arbitrarily modify it. **Structure** you cannot sever a theories predictions (consequences) from the explanatory content. They are tightly linked/interlocked. **Reach** [Reach](Reach.md), [Reach and Constraints](Reach%20and%20Constraints.md) **Complexity** The concept of complexity can be linked to *compositionality*. Greater compositionality $\rightarrow$ greater complexity. **Simplicity** The concept of simplicity can be linked to *universality*. Universal statements logically imply less universal statements. If knowledge is our objective, simple statements are better than more complex ones because they tell us more—they imply more consequences. # Summary We are seeking theories with *high content*. This naturally leads us to look for *simple* theories, for simple theories have more consequences that we can check. ##### Counter Example 1: Epicycle Theory Of Planetary Motion Consider the Ptolemaic (epicycle) theory of planetary motion. It had high content for it had many consequences that could be checked. We may also say that it was hard to vary, it was specific. Why then do we view it as a worse theory than Kepler's Elliptical theory? Putting aside the fact that Kepler's theory made better predictions, there is another reason. The reason is that Kepler's theory is *simpler* and thus has more content—it has more consequences that are implied, and thus more ways it could be checked and falsified. The Ptolemaic theory was more complex and ad hoc, meaning that while it implied many consequences, it implied *less* than Kepler's theory. This theory is also *easy to vary*—just add more epicycles to fit new data. It is ad hoc, makes it less explanatory than a simpler, more universal statement with greater content. Each adjustment reduces its falsifiability and makes it more descriptive than explanatory. ##### Counter Example 2: Angels Pushing Planets Consider another theory of planetary motion—that the planets move in their respective orbits because the Angels push them. This theory has almost no logical (and effectively zero empirical) content. It says very little about *how* or *why* the angels push the planets. Do they push them in square orbits? Triangular? Circular? What energy source powers them? Do they operate in shifts? All of these are compatible with the statement "the angels push them", for this statement rules out almost nothing—it has very few consequences. To be clear, it does have *some* consequences. It tells us that the planets aren't moved by demons, or aliens. And it tells us that there is a *reason* for the planets to be moved—due to angels. But again it is so vague it has no empirical content and hardly any logical content. And it is easy to vary! We can swap out the angels for demons, or have the angels pulling instead of pushing the planets. There is no structural coupling between the explanation and the phenomena of interest (the planets moving). We want theories that have specific consequences about the phenomena of interest, and if we change the theory the consequences should change as well! ##### Content vs Explanation It’s possible for a theory to have high content (many consequences) and still be a bad explanation—either because: * It’s too specific (overfitted or ad hoc), or * It’s not structurally tied to its implications (e.g. vague agents like angels). You can think of this in terms of a subclass relationship: some high-content theories are better than others because they are simpler, less arbitrary, and tightly coupled to their predictions. ##### What Does It Mean to Remove Explanatory Content? Removing explanatory content can take two forms: 1. Popperian view: Reducing content means reducing the number of things a theory implies. This weakens its testability. 2. Deutschian view: Making a theory easier to vary—by loosening the link between explanation and consequence—also reduces explanatory content, even if the raw content remains the same. Removing explanatory content means either reducing testable consequences or weakening their connection to the core claim. In both cases, you’re left with a theory that’s less useful, less predictive, and less falsifiable. ##### Ptolemy vs Kepler Popper’s View: * Epicycles are less falsifiable because the model can always be tweaked by adding more epicycles to fit the data. * Kepler’s laws are simpler, make bolder predictions, and are more testable—hence, they have higher content. * The theory with more content is better because it risks being wrong in more situations, making it more scientifically valuable. Deutsch’s View: * Epicycles are easy to vary: if observations don’t match, you just add a small epicycle, or adjust the radii. * The theory is ad hoc: no deep explanation for why planets move that way, just a recipe that mimics observed paths. * Kepler’s laws, by contrast, are constrained by an underlying explanation (eventually Newtonian gravity), making them harder to vary. Popper is evaluating in terms of testable consequences; Deutsch is judging the quality of the explanatory structure. ##### The angels push the planets Popper’s View: * Very low content: it doesn’t entail any specific observations; it’s compatible with everything, hence not falsifiable. * Because it doesn’t logically constrain outcomes, it says almost nothing about the world. Deutsch’s View: * It’s extremely easy to vary: swap angels with demons, or make the force intermittent or whimsically directed. * The explanation is epistemically empty—it doesn’t explain why the planets move the way they do, and allows endless arbitrary variation. Again, you’ve got it right: Popper calls it scientifically vacuous; Deutsch calls it explanatorily empty. # Explanatory Content ##### Removing Explanatory Content * Could we say that the explanatory content is the reach? * Reach? Hard to vary? * The explanatory power and the logical consequences (predictions) are deeply linked; you cannot severe the predictions from the theories’ other content... because that would spoil the theories’ explanatory power * In DD's framework, explanatory content implies _consequences_ that were outside the control of the original creator. For instance, when Einstein came up with general relativity, it had consequences (_reach_) that he did not foresee and consider. It was a consequence that was out of his control. While this was not a logical consequence, it certainly was a certain kind of consequence. ### **Explanatory Content and Constraint** So what is explanatory content? It might be best defined as a statement that has **consequences tightly interlocked with it**—consequences that can’t easily be derived from other, unrelated statements. That is, they arise _because_ of this specific explanation and not just any tweak of it. Take a counterexample: > _“The planets move in elliptical orbits because angels push them.”_ That theory can be “varied” in arbitrary ways—say, the angels pull them instead—and you’d still get the same prediction. The consequences are **not uniquely tied** to the structure of the statement. It’s **easy to vary**. What we want are theories where: - The **predictions are specific** to the theory’s structure - You **can’t tweak the theory** without breaking its predictive power This ties back to explanatory content: we want statements that are **hard to vary**, whose consequences are **inseparable** from the explanation itself. --- ### **Clarifying the Epicycle Example** Now, back to the epicycle theory. You might argue that it’s hard to vary because of the detailed structure of cycles-upon-cycles. But it’s still ad hoc. You can always add more epicycles to make the model fit, which means the theory remains flexible at the cost of being **explanatorily shallow**. That’s the problem you need to confront. It starts to become more descriptive than explanatory. High content isn’t enough. What matters is whether the theory is constrained in a way that makes it both **predictive** and **explanatory**. ### Removing Explanatory Content Theories have *content*. That content is the set of all *consequences* implied by the theory. When explanatory content is removed, this is *chopping off consequences*—specifically those that make the theory checkable. ### Description vs Explanation Recall the distinction between [Description](Description.md) and [Explanation](Explanations.md) ([Description Is Not Explanation](Description%20Is%20Not%20Explanation.md)). Now consider the example of the following explanation: "there is a copper atom on the tip of winston churchills nose in parliment square because, based on the deterministic laws of motion and initial conditions at the big bang, the atom was guaranteed to end up there". This is supposedly an explanation in terms of laws and conditions. But it doesn't really feel like an explanation. It feels more like a description. Or, consider a cellular automata that is started based on a single, simple rule. As show in "the game of life" by John Conway, this can lead to complex behavior, with emergent entities such as "gliders" coming about. In a real sense, any type of behavior that emerges in this game can be "explained" by the following statement: "the behavior emerged because we applied the cellular automata rule and let the system run". But that could be used to explain _anything_ in the system. And if something can explain _anything_ then it really explains nothing. One reason these both feel like bad explanations is that they contain *no empirical content*. While they do constrain what will happen, there is nothing that could happen which would rule out these explanations. They are immune to criticism. A theory is tested by attempting to find refutations or counter examples. The statement "this behavior emerged because the rule was applied" is practically irrefutable within the context of the simulation. What experiment could you perform within the cellular automata world (or by observing it) that would show this statement to be false? There is none, because the statement is essentially a description of the simulation process itself. It's like a tautology about the simulation's mechanism. We are bordering on saying "the simulation created an output constrained by it's rules, because the simulation will create outputs constrained by it's rules". Good explanations are linked to empirical content. Explanations that explain everything tend to have low empirical content because they don't rule out any observable outcomes within their defined scope. The statement doesn't rule out the absence of a glider, given the rule was applied; it just states that whatever emerged did so because the rule was applied. This is directly related to [If a Theory Can Explain Anything, It Explains Nothing](If%20a%20Theory%20Can%20Explain%20Anything,%20It%20Explains%20Nothing.md). A theory that can explain anything has effectively no empirical content. Since it forbids nothing, its class of potential falsifiers (the observable statements incompatible with it) is empty. Put another way: it has minimal informative logical content about the world of experience. It's logical content in the sense of saying something specific about reality is negligible because it doesn't "take the risk of being falsified by any fact". In both of these above examples, they are bad explanations because they have no empirical content, but they are about *empirical phenomena*—we can observe them. Note that for non empirical (philosophical) theories, we can criticize them rationally via viewing them as a proposed solution to a set of problems. We can then ask: * Does it solve the problem it was intended to solve? * Does it solve the problem better than competing theories? * Has it merely shifted the problem without truly solving it? * Is the solution simple? * Is the theory fruitful in suggesting new ideas or problems? * Does it contradict other philosophical theories needed to solve other problems? (i.e., logical consistency with other intellectual frameworks) --- Date: 20250508 Links to: Tags: References: * [Is P(Doom) Meaningful? Epistemology Debate with Vaden Masrani and Ben Chugg - YouTube](https://youtu.be/zKz-t_l5yHg?t=5524)