Three domains of knowledge

It's instructive to first understand what philosophy is. The term, much like "logic", "science", "ethics", etc. is often thrown around to mean things completely unrelated to epistemology. Philosophy isn't about some cliche proverbs on a social media site, or some ridiculous analogies between unrelated things. Philosophy is epistemology -- it is the study of knowledge. Note that it is not the study of human knowledge, or of how knowledge is stored in society -- it is about the fundamental idea of knowledge, i.e. any statement, what mathematics abstractly is, what physics abstractly is, what ethics abstractly is.

If you're confused, it will be clear by the end of this series what kind of questions philosophy deals with.

We first classify knowledge into three fundamental disciplines, as the specifics differ by discipline. Note that philosophy is not concerned with the distinctions between sub-fields of these disciplines -- the distinction between biology, sociology and particle physics is not relevant to philosophy, as the distinctions are based on specific features of the real world. Philosophy must remain valid regardless of any knowledge we know from observation, any moral beliefs, or even any specific logical systems.
• Analytical knowledge (mathematics) -- the study of logical connections between (any abstract) statements, i.e. A implies B.
• Positive knowledge (physics) -- the study of logical connections between empirical statements
• Normative knowledge (ethics) -- the study of logical connections between moral statements
Philosophy is also not concerned with the different methods used by different sub-disciplines in practice, or whether these methods happen to be reasonable, as these are purely the activities of humans. Studying how science is done by humans is a sociological field, and a subject of positive knowledge, not philosophy.

Mathematics

Mathematics is fundamentally about logic, or reason, i.e. given some statement, what are its logical implications? It is important to realise that the logic we're talking about is pure, and independent of any scientific knowledge we might have, and also doesn't refer to the predictions of a scientific theory. For example, if you use a pH meter in an experiment and deduce, using your knowledge about the pH meter that the pH of the solution is equal to what is shown by the pH meter, then your theoretical knowledge isn't reason, it is one of your premises.

Here's what logic is not:
• Theoretical prejudice -- Often, people like to claim that there is a conflict between logic and observation. This is presented as a conflict between "rationalism" and "empiricism". This so-called "rationalism" has nothing to do with real reason -- instead, it is simply conformity to the existing theory.
• Utilitarianism -- Utilitarianism is a specific ethical theory, and it starts with the fundamental premise that "maximise aggregate happiness" is always the right choice to be made. There is no way to rationally argue that this premise is correct, that being a psychotic murderer is a bad thing, or even that acting out of emotion is a bad thing.
• "It's only logical" -- The following is not only not an example of logic, it is also an unsound ethical argument: "A toy is meant to be played with, therefore one should play with a toy"
• Risk-averse behavior -- This is another example of the claim that an action or behavior is somehow logical or illogical. It is not. An action can be logical or illogical with respect to an ethical premise or system (more on this later), not in itself illogical. For the record, all action involves risk, and there is always an optimal amount of risk to take depending on the expected utility of each choice available. (In fact, from a utilitarian perspective, I would argue that people take too few risks. It is worth noting that even the enjoyment attained from uncertainty can be weighed accordingly in a utilitarian calculation.)
• "Logicality" of languages -- it's common among linguistic fanatics to claim their languages as somehow "logical". What they mean is really that their language is intuitive to them, or that it is structured.
I've only listed a few examples, generalise these to other situations whenever you hear or think the word "logic".

We can make the following general observation: the word logical/illogical can never be applied to individual statements, but can be applied to systems of statements, such as arguments.

So in mathematics, one works with some set of statements/premises that do not contradict (i.e. they are consistent) and derive all the logical implications of these statements. The fundamental statements are called axioms, and the statements that derive from them are called theorems. These systems, taken together, are called theories.

The reason that this axiomatic way of doing mathematics is useful is twofold:
1. To a pure mathematician, it makes it easier to identify contradictions in a theory.
2. To an applied mathematician (e.g. a financial analyst trying to model some stock with a certain mathematical theory), this means that he only needs to verify that the real-life phenomenon satisfies a small list of properties (the axioms), and all the theorems of the theory would apply to the physical phenomenon.
The second is essentially why mathematics is so useful in other disciplines -- essentially, we often have unrelated objects in the physical world that follow the same set of laws. In mathematics, one studies the kind of laws that appear often and derive their logical implications -- these logical implications then form more laws that apply to the physical object. The applied mathematician's job is to identify the relevant mathematical object, showing that the physical object satisfies its axioms.

An important note regarding axioms -- it is often said that an axiom is a "self-evident truth", i.e. an obvious truth not regarded as requiring proof.

This is a complete misrepresentation of what an axiom is. An axiom is indeed stated without proof, but its choice is completely arbitrary $(*)$, not based on its "obviousness". In order to illustrate our point, let's take the example of Euclidean geometry.

Euclidean geometry is based on five axioms, of the nature of "a line is defined by two points", etc. While we use terms like "line" and "point", it must be noted that Euclidean geometry itself is a completely abstract system, and a line or a point do not actually have any basis in reality. This cannot be emphasised enough: the whole "drawings on a board" thing we associate with Euclidean geometry is simply an application of visual geometry -- drawings on a board happen to be described by this abstract axiomatic system called Euclidean geometry, if we correspond the "line" in the abstract system to a "line" as we see on our board, etc. Fundamentally, Euclidean geometry is just a bunch of abstract objects and the laws they follow, how these follow from our five basic axioms. A course on Euclidean geometry that contains diagrams and geometric intuition is akin to a course on differential equations that explains DEs based on their applications in circuits and harmonic oscillators -- pedagogically useful, but that doesn't mean DEs are intellectually equivalent to some physical systems.

However, Euclidean geometry is not the only axiomatic system in the world. Even an axiomatic system similar to Euclidean geometry but with one axiom different would still be perfectly reasonable, assuming there are no contradictions between the axioms. It would not, however, model a flat plane, but a different structure -- perhaps some sort of a curved surface. The fact that a certain physical object satisfies a set of axioms must be demonstrated, not accepted based on some "obviousness".

The point is that any consistent axiomatic system is acceptable, so you don't need to "choose" axioms. Axioms need not be obvious at all -- for a simple example, Euclid's axioms might be replaced with the Pythagoras theorem, and Euclidean geometry, including the 5 statements of Euclid, will arise as theorems.

$(*)$ Note that here, by arbitrary we mean there is no such thing as an axiom being right or wrong -- there are reasonable ways in which the actual axiomatic systems we study are chosen, such as practical applicability. This was the point of our earlier statement about applied mathematics.

Finally, we come to the question that links mathematics to our other mentioned disciplines: physics and ethics are both sub-fields of mathematics where the axioms are chosen based on certain special conditions.

Physics - logical positivism

Here we define physics: physics is the study of our universe. Mathematics gives us the description of every mathematically possible universe, and we can employ a specific axiomatic system whose axioms describe a certain real-life system, to describe the real-life system. An example of such a real-life system is the physical universe -- in principle, we may study the specific axiomatic system whose theorems (i.e. predictions) agree with our observations about the universe.

On a sidenote, we may also extract approximate "effective theories" -- these are often called models -- of special physical systems based on a completely different axiomatic system -- for example, a theory of particle physics, a theory of the solar system, a theory of biology, etc.

This is the important point: the origin of all positive knowledge is in observation. Note that this observation need neither be a rudimentary observation without any equipment or a sophisticated experiment, it need neither be a deliberate experimental observation or a standard fact we know from reality, like "humans exist" (whose application is called the "anthropic principle").

Before we go any further, we must clarify what kind of statements are actually meaningful, especially in physics.

The word "meaningless" is often thrown around aimlessly. Here, we define the word more clearly, by listing examples of statements that are not meaningless:
• Practically pointless questions aren't meaningless. For example, "what is the sum of the averages of the phone numbers, treated as if they were in a factorial number system, of right-handed people with a detatched earlobe and a prime number rounded income in their respective currencies, in Bangalore and New York?"
• Physically wrong statements aren't meaningless, neither are morally wrong prescriptions. For example, "there are no women in Australia" is physically untrue, which can be verified by finding a woman in Australia. However, the statement is not meaningless, as its meaning is quite clear, and something that can be verified/falsified with an empirical observation.
• Mathematically wrong statements aren't meaningless. For example, "John and Jojo are older than each other" is mathematically impossible under the axioms that define age. But it isn't meaningless -- just false, because it contains a contradiction.
A meaningless statement is one that is semantically and gramatically OK, but does not actually convey any meaning. For example, "What is the speed of pi?" is a meaningless statement, because in the axioms of a system that defines pi, no attribute called "speed" is associated with a number, or with pi. Similarly, "Is Caesar a prime number?" is meaningless.

Let's try to think about meaninglessness in the context of positive knowledge. Since the origin of all our positive knowledge is in observation, any physical statement is fundamentally a statement of what exactly we observe with our senses. For example, the statement "there is a table here" can in principle be reduced to a statement of the sort of "my eyes observe a certain pattern of light originating from this region...", where "light" is also expanded in its definition to refer to exactly what we observe, etc.

It must be possible to reduce every physical statement to a statement of an observer's observation (we'll call this positive language), or it is meaningless. Therefore, the following questions/statements are meaningless:
• "Do quarks really exist, or are protons just elementary just behave exactly like they would if they were made up of quarks?"
• Are we brains in vats?
• Superdeterminism (a conspiracy theory stating that the laws of physics conspire to tamper with all our observations so we never discover their truth -- this is meaningless, because the laws of physics are fundamentally about what we observe)
• Self-awareness/qualia
• What was there before the universe
• Did the big bang really happen, or did a bearded guy in the sky just arrange the universe last Tuesday as if it had? (last Thursdayism)
These are metaphysical statements, and all metaphysics is meaningless. When we talk about the big bang or any historical event having happened, what we are essentially saying, in the language of positivism, that our observations today "look like" (based on some theoretical narrative about how things evolve with time) that historical event happened. Our only actual knowledge is of the present, of this very instant -- the past is merely a metaphysical construction.

For another example, consider our memory of past events -- all we know, right now, is that we have such a memory -- that some neurons in our brain are connected in such-and-such a way. When we express this by talking about our past, e.g. when we say "I slipped on a a banana peel yesterday", what we're doing is essentially a linguistic trick, or training our minds to create a past.

Note that this understanding of history doesn't mean that it is impossible to determine historical facts. Historical narratives are simply convenient metaphysical gauges to describe precise statements about what one would observe if one made certain archaeological digs, etc. In principle, it would, for example, be possible to precisely map the consequences observable in the modern day of different historical theories.

In other words -- metaphysics is meaningless, because the only real physical meaning (knowledge) is what we directly observe.

We therefore have a definition of meaninglessness: any statement that is neither analytical (of the nature "A implies B"), positive (of the nature "I will observe...") or normative (of the nature "I should act ...") is meaningless.

Note how we never use the word "exist", like "the only meaning that exists is what we directly observe" -- this is because the meaning of the word "exist" varies by discipline. In mathematics, it means logical consistency; in physics, it means empirical reality; in ethics, one might say it means moral acceptability. Metaphysics does mathematically "exist", in the sense that it does not create logical inconsistencies if metaphysical objects are considered abstract mathematical, logical structures.

In fact, there is nothing wrong with holding such metaphysical beliefs as a gauge to view the world through for the sake of personal comfort -- it is not the case that viewing oneself as the only observer and moral actor and the rest of the universe as existing merely in one's observations is the only correct gauge to view the world. My own metaphysical frame contains things like, "mathematics is the system of all possible universes, our universe is just one in this platonic realm, there is one consciousness which keeps instantly swapping through people's heads, we are not brains in vats, the many-worlds interpretation of quantum mechanics". Logical positivism just means recognising that all metaphysical gauges are equivalent.

You might've noticed we've often talked about the meaninglessness of questions. It seems philosophically interesting to understand what a question actually is. A question is just another way of writing a statement. Rather than stating a statement X (and responding with "True" or "False"), we may write "Is X true?" and respond with "yes" or "no".

Why would this be useful? Suppose we want to make several statements, perhaps $\aleph_0$ or even $\aleph_1$ -- here's an example of the latter:

"John weighs 500N."
"John weighs 499.92N."
"John weighs 0.001N."
...

Where only one of the statements may be true. It is more efficient to write "What does John weigh?", and the answer would select which of the statements is true.

We now turn our attention to the question of verification. Logical positivism makes it clear that the statement S implies the answer to "How is S to be verified?" by a few simple logical arguments.

In order to test a physical theory, one must know its predictions. Only if a theory makes testable predictions is it meaningful -- this, we know from logical positivism.

There are two ways a theory or model might make predictions -- either in the form of $P\to Q$ or in the form $P=Q$, where $P$ is the theory and $Q$ is the prediction. The latter case is a characteristic of simplistic models whose predictions are not generic but all its predictions are about events within a finite amount of time. $P\to Q$ is more useful, and most of our standard scientific theories (which we take seriously) fall into this category.

(See related: Hume on the fallacy of induction)
(Note also, that if your prediction only holds true until a finite time, then it is useless, in fact meaningless afterwards, i.e. it is meaningless as soon as you know it is true $(**)$.)

Since predictions can be tested directly, we try to invert the prediction to derive the truth of the theory from the truth of the prediction. If $P=Q$, then $Q\to P$ and $\neg Q\to\neg P$. If $P\to Q$, then we cannot prove $P$ from $Q$ (as $Q$ may arise from other causes), but we can use the contrapositive, $\neg Q\to\neg P$. This is called falsification.

$(**)$ Falsification as the only valid test
Suppose we have the statement "John is 1 metre tall". It is tempting to believe that mere verification is sufficient to test this statement, as in -- if one measures John's height and he turns out to be 1m tall, then we know for a fact that the statement is true. Let's analyse the statement more closely, however. In simplified positive language, it reads "if one measures John's height, then one would see the measurement 1m." This is a prediction. If one "verifies" the statement at some point in time, it is still not shown that it will be true a few seconds later.

"Okay," you say, "What if the statement is: If the time is 31 July 2016 at 12:10:00:... and one measures John's height, one will see the measurement 1m? Will verification not be enough then?" The problem with the statement is that taken literally, it becomes meaningless as soon as the time has passed. The statement can be rewritten in positive language, however, to include another clause: "at any point after this time, one will have the memory of seeing the measurement 1m (and it may be demonstrated, by observing the memories of others, that the observer's memories have not been tampered with), or may do some experiments, such as looking at the arrangement of the particles in the universe right now to work backwards towards the arrangement of the particles back then and determine the marking on the ruler, to verify that this is true." In this phrasing it becomes clear that our statement is once again a prediction, and can only be falsified.

Note that predictions are often probabilistic, and even have to be, as a result of quantum mechanics. In this case, any test is not definite, but probabilistic -- for this purpose, we have tools like confidence levels, Bayesian interference, etc., which we will cover later. In the previous example, for example, it wouldn't be possible to deterministically determine the earlier arrangement of particles in the universe, but one can, in principle, produce a schema to assign probabilities to the different possible earlier particle arrangements that might've been, hence assigning a probability for the statement to be true.

There are critiques of positivism, including from Popper, claiming that positivism claims that $P\to Q$ means $Q\to P$. This is a strawman argument, as this is not actually claimed by positivism.

Another critique of positivism takes the form of "positivism itself is neither an analytical, positive or normative statement". The problem with this criticism appears immediately -- without positivism, not only metaphysics, but also statements like "Is pi a nice person?" become meaningful. The criticism is wrong, anyway -- positivism is a law of logic, and an analytical statement, as philosophy must be. One may say that the specific claims of positivism, like "Is pi a nice person?" is a meaningless statement, are a restatement of the axioms that define "pi", which makes no mention of a property called "nice person".

An important point is made here regarding the relation between philosophy and mathematics (analytical knowledge). Philosophy discusses knowledge in general, as does mathematics. Indeed, philosophy is essentially "popular mathematics", like what popular science is to science. XKCD puts it best: philosophy's just math sans rigor, sense and practicality. (When a comic strip does a better job explaining your domain than your textbooks, you know the field is in trouble)

To be fair, xkcd gets the physics right too -- it's math constrained by precepts of reality -- and physics isn't in trouble.

Ethics

We make the following analogy between physics and ethics, both of which are subfields of mathematics. This table gives you all the philosophy you'll ever need.

Discipline Mathematics Physics Ethics
Scope Everything Predict observations Prescribe actions
Premises Axioms Postulates Principles
Reasoning Logic Logic Logic
Conclusions Theorems Predictions Prescriptions
Criterion to choose axiom None Observation Personal acceptance
Logical systems Mathematical theories Physical theories Ethical theories
Agent None Physical observer Ethical actor/Moral agent
Convenient simplifications
that eliminate the agent (like
a ZIP file, it's smaller but it
needs to be
unpacked)
None (no agent to kill) Physical noumenalism
(e.g. "the table is there", rather than
"I detect light from...")
Ethical noumenalism
(e.g. "Drugs should be decriminalised", rather than
"I should consider political possibility of drug
decriminalisation positively by such-and-such
amount in my voting, and write things in support
of decriminalising drugs")
Objects Mathematical objects
(e.g. vectors)
Physical objects
(e.g. displacement, energy, black holes)
Ethical objects
(e.g. rights, duties)
Practical connection None Experiments Moral-dilemmas
Third-person arguments None Thought experiments Thought moral-dilemmas

(Phrases in red are non-standard) We'll soon get into talking about what exactly we mean by each, but let's first clarify a few things regarding ethics:
• Every action raises an ethical question -- Even a question like "should I push this table?" or "should I eat more celery?" or "should I attend this university?" is a question of decision-making, and is considered an ethical question.
• Ethical axioms cannot be derived through reason -- Unlike in physics, where postulates are tested through sense perception, there is no rational way to argue for or against a fundamental principle like "maximise aggregate happiness". This often undermines the relevance of science to moral decision-making, although science still does play a role in the reasoning from the axiom to the conclusion.
• The axiomatic system should still mathematically exist -- In other words, the axioms should be precise and mathematically consistent. For example, if one can demonstrate a situation where there is only a choice between stealing and lying, it would not be consistent to adopt "do not steal" and "do not lie" as simultaneous axioms. However, one may start with a more fundamental axiom and derive the two statements as theorems under specific circumstances which do not overlap in the region of contradiction.
• The fact that ethical axioms may validly be chosen based on any arbitrary system of personal acceptance does not mean one may not use non-arguments to convert people into one's moral ideology -- Indeed, such a non-argument may be viewed as a process that has nothing to do with reasoning or convincing, but some kind of therapy administered to the subject's brain. Whether this is moral or not depends on your ethical principles.
• A lot of pseudo-philosophical questions starting with "should" are actually ethical ones -- E.g. "Should we use the scientific method...?", "Should we hold metaphysical beliefs...?", etc.
• "No time to think about ethics" is a meaningless statement -- Ethics is about all decision-making, and if you believe, for example, that in extreme situations it can be justified to do something you would otherwise consider immoral, then it means you consider that thing to be moral in such a situation, whether you like it or not. It means that your morality is actually derived from some more fundamental principle -- i.e. if you're willing to compromise on "do not lie" in specific situations, then it means that your fundamental ethical axioms do not include "do not lie".
• The scope of ethics is ONLY to prescribe actions -- It is not to judge the morality of a person, for instance, which is meaningless (in the strict sense of the word) gossip, not ethics. Neither does ethics simply mean the overall utility of your actions throughout your life should be zero, unless that's what your ethical principles prescribe (i.e. that's not what ethics fundamentally is about, but those might be your ethical principles).
• It might be immoral to think about ethics -- For example, if an ethical calculation took so long to solve that doing so would be so inefficient it would reduce utility more than any of the moral choices would, a utilitarian would prescribe that the moral agent simply made a guess, or intuitively decided on an action. An ethical theory provides, in principle, the most moral course of actions through a moral agent's life -- inefficiencies in resolving these prescriptions are justified, and one should only seek to minimise them. This is analogous to technological limitation in testing a physical theory. It is also analogous to a lot of economic decisions where it might be too expensive to find out precise individual information about a transaction, so one may make generalisations based on certain characteristics of the transaction -- e.g. looking at someone's credit record, etc.

In order to understand ethics, we consider the age-old trolley problem and perform a Socratic dialogue on a solution.

Introduction: Our two characters are Simplicio and Socrates. Simplicio, who thinks the fat man should be killed, believes that his moral beliefs are completely rational, in that he can argue for them without making any unphysical or unmathematical assumptions.

Socrates: Why would you push the fat man onto the track?
Simplicio: The expected value of his life is worth less than the expected value of the combined lives of the 5 people who'd otherwise die.
Socrates: Where is the logical implication from that to "I should push the fat man onto the track"?
Simplicio: Well, if I didn't, I would effectively be killing the five people instead.
Socrates: So? There is still a difference -- you feel more responsible for the death of someone you push onto the track.
Simplicio: But acting based on that principle would be selfish.
Socrates: Is there a law of logic which says "do not be selfish"?
Simplicio: Well, if you had a lever that would deliver unlimited pleasure to you while dooming the rest of humanity to eternal despair, would you pull it?
Socrates: You're not making direct logical arguments now, but I'll play devil's advocate: suppose I would.
Simplicio: If you did, why would you be convincing me to agree with your moral ideas? Wouldn't you want me to adopt a moral principle that makes me work for your benefit?
Socrates: Perhaps I have a moral principle that says "maximise self-interest, except when arguing with Simplicio, then maximise the humour of the readers instead". But no matter, this is irrelevant. You are not making logical argument.

The argument never reaches its conclusion, because Simplicio has a fundamental ethical principle -- "maximise aggregate happiness" -- he refuses to disclose.

We'll now briefly discuss the "convenient simplifications that kill the agent" and "objects" mentioned in our table.

Often, it is annoying to have moral discussions while expressing every moral prescription in fundamental, agent-specific terms, especially when that isn't the point of the discussion. For an example, take the statement "drugs should be decriminalised". In most modern democracies, there is no one individual with the power to decriminalise drugs -- rather, power is quite distributed, and it wouldn't even be accurate to say that the statement is equivalent to saying that people in parliament should vote for laws decriminalising drugs. For example, one may say "drug decriminalisation would be beneficial on its own, but doing so would make my political party very unpopular, thus reducing the likelihood of being able to pass/block other, more significant legislation, therefore it has an overall negative effect".

What one usually really means when they say they support drug decriminalisation or that drugs should be decriminalised, is that anyone in power -- and that includes a voter -- should assign positive utility to the direct consequences of decriminalisation of drugs while making political decisions (such as voting). The reason this specific set of consequences (the direct ones) are often useful to distinguish is that the method one uses to evaluate these consequences is very different (and therefore the arguments actually made are very different) from the method one uses to evaluate the other consequences, such as political pragmatism -- an argument on these consequences can thus simply be a debate on the positive (in the sense of positive vs. normative) consequences of drug decriminalisation from some ethical premise.

The other consequences are then irrelevant to the debate of "Drugs should be decriminalised". This way, one has simplified our language and made communication a lot easier, even though the word "should" is fundamentally attached to the actions of an ethical actor, i.e. "I should... one should..."

Note, however, that it is still important to ensure that such imprecise language actually means something, i.e. can be converted into a precise form (we'll call this "normative language"), in terms of the actions of an ethical actor and the observations of a physical observer. Make this a practice in your own argumentation (you should!).

An important point to be mentioned is that your moral beliefs are fundamentally about your actions. As an example for why this is important -- consider the Non-Aggression Principle, which many libertarians cite as the ethical basis for their political beliefs. However, if your fundamental ethical principle is the Non-Aggression Principle, that wouldn't actually imply that you should be a libertarian, as it only means that you shouldn't violate anybody's rights, not that you should act politically to stop other people from violating people's rights. The ethical principle you should cite might be "minimise aggression", but "non-aggression" is clearly not it.

One often hears words like "rights" and "duties" thrown around. These do not seem to correspond to anything in our ethical vocabulary so far.

These terms have meaning within the structure of an ethical theory. For example, in a rights-based ethical theory, the word "right" is interpreted in the following way: "People have a right to free speech" = "One should not coercively suppress a person's speech."

The word "duty" seems to refer to moral obligation in general, but is usually used within the structure of an ethical theory which makes predictions discretely (more on what this means later).

These, which we call ethical objects are constructed to simplify the language of ethical arguments, but it is necessary to always keep in mind the real, fundamental meaning of the statement referring to an ethical object. It is always essential, for a statement to be meaningful, that a statement containing an ethical object can actually be written in normative language.

Finally, we discuss a few specifics:
• Continuous and discrete ethical theories -- Some ethical theories provide moral prescriptions in every situation -- an example is utilitarianism. Others, like rights- and duties- based ethical theories only prescribe or unprescribe a specific action or discrete set of actions. E.g. any action is equally moral, as long as you don't violate other people's rights. Or -- any action is equally moral, as long as you do your duty.
• The traditional bifurcation is "consequentialism vs. deontology". However, this is problematic. Ethics is fundamentally about making choices between actions, and saying that a certain action can be universally immoral requires the choice of a universal non-immoral incation to compare it against. To demonstrate that such an inaction does not exist, consider the following thought-moral-dilemma: you're driving a car with your foot on the accelerator when a child steps onto the road. If and only if you release the accelerator, the child won't die. Is it inaction to not move your foot or is it inaction to not continue to accelerate the car?
• Furthermore, deontology and consequentialism, as usually defined, cannot actually be distinguished, as explained in the next point.
• Utility functions -- A convenient universal formalism to express ethical theories in -- similar to the least-action formalism in physics -- is the "utility function". Essentially, this is stated in a principle similar to utilitarianism, with "maximise utility", except with any arbitrary definition of utility. Since all ethical theories can be stated in this formalism, a simple test to show that something is or isn't a meaningful ethical theory.
• Resolving Buridan's ass -- Buridan's ass is a thought moral dilemma where a donkey is presented two symmetric (i.e. equal utility) ethical choices, and it can't choose -- this is often stated as a criticism of utilitarianism. However, there is no real paradox, as there are plenty of ethical theories, including the amoral "do whatever" and all discrete ethical theories, which do not always provide ethical prescriptions between two ethical choices. Indeed, the donkey should make its choice between the two randomly.