## Monthly Archives: January 2012

### Abortion

This is an e-mail I sent to a friend whose official position on abortion I would hesitate to characterize, but leans towards the pro-life side. The actual conversation we had was via text messaging, so I don’t have a record of her questions but I basically wanted to hit all the points of what it is that I think actually makes killing wrong, why society needs to enforce that, why I don’t consider abortion murder, why I think that abortion is okay, and why it shouldn’t matter (from a legal point of view) if the mother wants an abortion and the father does not.

To put it in some context, she and I had already had a previous discussion where I had explained that I think killing is wrong because it severs the attachments that people have to the world around them, and vice-versa. Her hypothetical example that I mention in my opening is whether that makes it okay to kill a reclusive person who has no such ties. I’ve edited it to remove some personal stuff that I had in a preamble and ending.

If someone is a recluse then killing them is morally wrong. While true that nobody will care about the result (in the hypothetical), we don’t live in a vacuum. What we’re talking about is two different types of morality: legal and metaphysical. If truly nobody cares about the death of our hypothetical person (note that this must include himself), then you’re correct: there is no metaphysical moral issue with killing him. However the legal morality of this is problematic. We create our own society and then we all have to live in it. If we allow people to kill other recluses, then we are also subject to the lawlessness of being killed ourselves. Thus, from a legal moral perspective, it is wrong to kill the recluse.

However, in a very slightly less constrained hypothetical where nobody cares about the man, but the man still has attachments to the world around him (art, music, hobbies, whatever), the metaphysical moral issue reappears. In my opinion, it is once again subject to the premise that the reason killing is wrong is the severing of these attachments. Granted that the man won’t care after the fact, but there’s a difference between caring and being affected. While he won’t realize it, the loss of these attachments is an effect, and I do believe that when making a decision about morality, the opinions that matter are specifically the opinions of those affected (in this case, the man). You could argue that his opinion is voided by his death, but that’s not quite right because we are talking about the moral *decision*. A decision which, by necessity, precedes the action, thus his opinion at the time of the decision is what we base our moral choice on.

Furthermore, while I do believe that it’s severing these attachments that make killing wrong, not all killings are equally wrong. In the same way that snow and dry ice are both cold, one is certainly colder than the other. With this in mind, it is more wrong to kill a person with a lifetime of these attachments than it is to kill an infant whose only attachments may be the love of its parents. This runs counter to the intuition that evolution has programmed us with, but I think the reasoning is enough (for me at least) to trump the gut instinct of biology. Note again, that killing the infant is still wrong, but simply less so than the adult.

This brings us back to abortion. Everything I just said about killing being metaphysically morally wrong because it severs attachments then fails to apply (if one assumes the desire for an abortion on the part of the parents). The fetus has no such attachments. At the point where it is legally allowed to be aborted, it hasn’t developed sufficiently to even have these attachments. As such, even if abortion counts as killing, it is not a form of killing I find morally quarrelsome.

A quick aside: I will not be using the term “murder” because it confuses the issue. Murder is a legal distinction, and has nothing to do with morality. Furthermore, murder is illegal, and is distinct from merely “killing”. This is why killing someone in self defence or by accident is not called murder by our justice system, and since we live in a society where abortion is legal, it too cannot honestly be called “murder”.

Anyways, this whole argument deals with the premise that aborting a fetus is “killing”. I actually deny this premise as I don’t consider the fetus to be anything more than a cluster of cells that will develop into something to which the term “killing” can be applied. Unfertilized eggs and sperm share this description, but we don’t think of menstruation or male masturbation as “killing” any more than we ought to think the term applies to a fetus. There’s nothing magical about the moment of conception, it’s just one step along the way to creating a person. The line between fetus and person isn’t a clear cut one. It reminds me of the question “how many grains of sand does it take to have a pile”. There isn’t a concrete answer. At some point it just stops being a few grains and becomes a pile. While answers will differ in the intermediate stages, we can look at a couple of grains and come to a consensus that they don’t form a pile. We ought to be able to do the same with the early stages of fetal development.

As for father’s rights, I consider that an entirely separate issue. If the father wants to keep the child but the mother does not, it is a tragic situation but the final say should (and currently does) belong to the mother. This goes back to legal morality above. Should we create a society where one person can be forced to carry an unwanted child? Considering all of the risks and efforts involved on the side of the mother (with none being involved for the father) it is unfair for a man to be able to force this upon a woman against her will. Imagine a society where a rapist could force his victim to carry his child. I hope you’d agree that this is not a society we would want to live in. Perhaps you’d suggest building in an exception for rape, but now all of a sudden all a woman has to do to get out of a forced pregnancy is accuse a man of rape, which I think causes more problems than it solves. Even at the end of the day, if abortion services are made unavailable to these women, it won’t stop them from obtaining one; it will simply stop them from obtaining one safely.

I’m prepared to change my opinion on this if it ever becomes possible to bring a baby to term without having to incubate it inside of the mother (artificial wombs or surrogacy transplants, for example) depending on what’s involved with the process. But for the time being, situations such as these are tragic but must fall to the decision of the woman.

### 1/11/2012 – Things I Saw on the Internet Today

Jessica Ahlquist Has Won Her Lawsuit! Fucking hero.

Drama I don’t really know what to say about this apart from that it sucks and makes me want to give Jen McCreight a hug and that anyone who bears witness to this kind of behaviour should verbally smack the offenders in the back of the head.

White House Denies President Obama Travelled To Mars Via Teleport At Age 19 Glad we cleared that one up.

Don’t Copy that Floppy Amazing anti-piracy add from 1992. The best part is how their arguments no longer apply to modern software (owning it, manuals, physical backups).

SMBC on coming out as gay It’s sad because it’s basically true.

### 1/9/2012 – Things I Saw on the Internet Today

In a new segment I’m calling “Things I Saw on the Internet Today” I post links to thinks I saw on the Internet today. Yep, that about sums it up.

Dear Customer Who Stuck Up For His Little Brother – Guy buying a video game for his little brother stands up to his dad who wants him to buy a “manlier” game.

NYU Student Weaves Elaborate Email-Drama – Or as I like to call it: Sociology student forced to go near icky poor people.

Digital Memory in Minecraft – This is SO goddamn cool.

# 3 Axiomatized formal theories

Alright, cards on the table: I found this chapter kind of boring until around the end, so I’m just going to try and skim through it to get to the interesting parts.

## 3.1 Formalization as an ideal

Gödel’s theorems are statements about formal languages. Why do we care about formal languages? It’s pretty straightforward: formal languages allow us to ensure correctness by following specific rules regarding structure and syntax. In a natural language, such as English, you can have sentences with ambiguous meanings. For example:

“John knows a woman with a cat named Amy.”

could have two possible meanings: either John knows a woman who has a cat whose name is Amy, or John knows a woman named Amy who has a cat. This won’t upset us too much in our day-to-day lives. Usually the intended meaning can be inferred from context. But when we’re trying to prove something logically, this requires precision. Thus, we can use a formal language (such as first-order logic or even any given programming language) to create an unambiguous parsing of our sentences.

Even in proofs, mathematics or computer science, however, we don’t always use such rigorous formalization, as such precision can be tedious. What’s important in these cases is more to do with conveying an understanding of a concept. But the underpinnings of the formal language exists, and if one desired should be able to be spelled out with perfect rigour.

## 3.2 Formalized languages

Normally we are interested in interpreted languages, ie: ones with not just a syntax for deriving valid structures, but one where those structures have actual meaning. I could symbolize one version of the John/Amy sentence from above with $Kja\wedge Wa\wedge Ca$ but that sentence is meaningless unless I also inform you that $Kxy$ means “$x$ knows $y$“, $Wx$ means “$x$ is a woman”, $Cx$ means “$x$ has a cat”, and $j,a$ mean “John” and “Amy” respectively.

Thus, we will define a language $L$ as being a pair $\langle\mathcal L,\mathcal I\rangle$ where $\mathcal L$ is a syntactically defined system of expressions and $\mathcal I$ is an intended interpretation of those expressions.

Starting with $\mathcal L$: it will be based on a finite alphabet of symbols (I’m pretty sure you can get away with relaxing the requirement to an effectively enumerable alphabet, but the book says finite so we’ll go with finite). Some of these symbols will make up $\mathcal L$‘s logical vocabulary, for example: connectives, quantifiers, parentheses, identity… Other symbols will constitute $\mathcal L$‘s non-logical vocabulary: predicates and relations, functions, constants, variables… We will also need a system of rules for determining which sequences of symbols are well-formed formulae of $\mathcal L$ (referred to throughough the text as its wffs).

For example, in first-order logic, our logical vocabulary is $\{(,),\wedge,\vee,\neg,\rightarrow,\leftrightarrow,=,\forall,\exists\}$. Our non-logical vocabulary is a bit more complicated in that it needs to potentially address infinitely many variables. There are two ways to do this. The way I like to think of it is having $\{f^i_j,P^i_j|i,j\in\mathbb N\}$ which gives you an infinite set containing all of your variables $f^0_0,f^0_1,f^0_2,\ldots$, all of your $k$-place predicates $P^k_0,P^k_1,P^k_2,\ldots$ and all of your $k>0$-place functions $f^k_0,f^k_1,f^k_2,\ldots$. But, of course, this results in an infinite alphabet of symbols. The book chooses to accomplish this by having your variables be something like $x,x^\prime,x^{\prime\prime},\ldots$ which operates on a finite alphabet ($x$ and $^\prime$). In either case, we will typically just denote variables as $x,y,z$, predicates as $P,Q,R$, functions as $f,g,h$ and so on. As stated in the previous section, we don’t always need to use perfect rigour but it’s important to understand how it would be accomplished. The union of the logical and non-logical vocabularies form the language’s alphabet. To see how first-order logic determines its wffs, see wikipedia. It is important for our purposes that determining whether a sentence is a wff is effectively decidable.

We then use $\mathcal I$ to set the interpretation of our language. It cant do this by manually setting the truth conditions for each wff (there are too many of them). What it does, rather, is to determine the domain of quantification, the which terms are applied to particular predicates, and the rules for determining the truth of a sentence. For example, $\mathcal I$ might set the domain of quantification to the set of people, set the value constants $m,n$ to Socrates and Plato respectively, and the meaning of the predicate $F$ to mean “is wise”. Then $\mathcal I$ continues by indicating which predicates are true of terms predicates, for example that $F$ is true of both $m$ and $n$. Finally, $\mathcal I$ sets up rules for determining whether wffs are true. For example a wff of the form $A\wedge B$ is true iff $A$ is true and $B$ is true. This can be tedious but straightforward. Again, however, it is important that this process be an effectively decidable one.

## 3.3 Axiomatized formal theories

The following things are required to construct an axiomatized formal theory $T$:

(a) First, some wffs of our $T$‘s language are selected as its axioms. There must be an effective decision procedure to determine whether a given wff is an axiom of $T$. This doesn’t mean that $T$ must have a finite number of axioms: we can have an axiom schema which tells us that sentences of such-and-such a form are axioms. For example, in Zermelo-Fraenkel set theory any wff of the form $\forall z\exists y\forall x(x\in y\leftrightarrow(x\in z\wedge\phi))$ (where $\phi$ can be substituted for (essentially) any property) is an axiom, giving the theory infinitely many axioms.

(b) We also need some form of deductive apparatus or proof system in order to prove things in $T$. I’ve talked about proof systems before and demonstrated two: truth tables and Fitch-style calculus. The exact system used for $T$ is irrelevant so long as it is effectively decidable whether a derivation from premises $\varphi_1,\varphi_2,\ldots,\varphi_n$ to a conclusion $\psi$ is valid. Note that this is different from having an effective procedure to actually create this derivation. All is has to do is determine whether a given proof which has been provided is valid for teh system.

(c) Thus, given an axiomatized formal theory $T$, since we can effectively decide which wffs are $T$-axioms, and whether a derivation in $T$‘s proof system is valid, it follows that it is also decidable whether a given array of wffs forms a $T$-proof (ie: which proofs are sound in $T$).

(d) For the remainder of this book, when we discuss theories, what we really mean is axiomatized formal theories. Many times in logic “theory” simply means any collection of sentences, thus we must be careful to make this distinction here.

Next time we’ll finish up chapter 3 where it actually gets interesting.

## 2.4 Effective enumerability

Similar to effective computability and effective decidability, effective enumerability describes which sets can be enumerated using a strictly mechanical procedure (such as our Turing machines described in secion 2.2). Now it is easy to imagine a program that enumerates a finite set: it just spits out the elements of the set and then halts. Consider the program:

$\texttt{for (int i=0;i<5;i++)\{}\\\texttt{\indent print(i);}\\\texttt{\}}$

This program will enumerate the set $\{0,1,2,3,4,5\}$. But what does it mean to effectively enumerate an infinite set? Consider the program:

$\texttt{for (int i=0;;i++)\{}\\\texttt{\indent print(i);}\\\texttt{\}}$

We can get an intuitive sense that this program will enumerate the set of all natural numbers, $\mathbb N$, but can we formalize the notion of a non-terminating program being used to enumerate an infinite set? This isn’t discussed in the book, but I like to think of it as follows:

A program $\Pi$ is said to list an element $e$ iff there, after some finite number of steps of execution $\Pi$ prints out $e$. If $\Pi$ lists every element in a set $B$ then $\Pi$ is said to enumerate $B$. Thus, $B$ is effectively enumerable iff there exists a program $\Pi$ which enumerates it.

So in our program above, we can now see formally that every element of $n\in\mathbb N$ will be listed after exactly $n+1$ steps (if we ignore steps involved in managing the loop logic.

An interesting fact is that every finite set is enumerable. To observe this, consider a finite set $\{n_0,n_1,\ldots,n_k\}$. This set is then enumerable by the program:

$\texttt{print(}n_0\texttt{);}\\\texttt{print(}n_1\texttt{);}\\\ldots\\\texttt{print(}n_k\texttt{);}$

Thus, given any finite set, we can in essence simply store every element in the set into memory and spit them out on demand.

## 2.5 Effectively enumerating pairs of numbers

What follows is a useful theorem that we will use time and time again.

Theorem 2.2: The set of ordered pairs of numbers $\langle i,j\rangle$ is effectively enumerable.

Proof: Consider a table of ordered pairs of numbers:

 $\langle 0,0\rangle$ $\rightarrow$ $\langle 0,1\rangle$ $\langle 0,2\rangle$ $\rightarrow$ $\langle 0,3\rangle$ $\ldots$ $\swarrow$ $\nearrow$ $\swarrow$ $\langle 1,0\rangle$ $\langle 1,1\rangle$ $\langle 1,2\rangle$ $\langle 1,3\rangle$ $\ldots$ $\downarrow$ $\nearrow$ $\swarrow$ $\langle 2,0\rangle$ $\langle 2,1\rangle$ $\langle 2,2\rangle$ $\langle 2,3\rangle$ $\ldots$ $\swarrow$ $\langle 3,0\rangle$ $\langle 3,1\rangle$ $\langle 3,2\rangle$ $\langle 3,3\rangle$ $\ldots$ $\downarrow$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\ddots$

We can use a computer to zig-zag through this table (in the order indicated by the arrows) which will eventually arrive at every possible combination. Since the table will be stored somehow in memory, we must generate it on the fly since we can’t operate on an infinite table, but we can make the table of arbitrarily large size and increase it as needed due to the infinite memory capabilities of our idealized computer system. Thus, we have created a program which enumerates the set $\mathbb N^2$.

QED

The book mentions that you can explicitly create a computable function $f:\mathbb N\rightarrow\mathbb N^2$ that will do this without the use of the table, however it doesn’t go into details and even if it did, this exercise can be quite messy. The way I actually know how to do it doesn’t involve the table at all and actually goes the other way, $\pi:\mathbb N^2\rightarrow\mathbb N$. It looks like:

$\pi(x,y)=2^x(2y+1)-1$

You can also define inverse functions $\pi_0,\pi_1$ such that $\pi(\pi_0(n),\pi_1(n))=n$, but this is complex and involves $\mu$-calculus which I’m not ready to get into yet.

That wraps up chapter 2. Next time: axiomatized formal theories and why we should care.