God, Deontology, and Slavery

Throughout the Old Testament, Yahweh acts like a deontologist. Here are some passages where God hands down categorical prohibitions against various behaviors (all passages are ESV. bold mine).

Leviticus 20:6
If a person turns to mediums and necromancers, whoring after them, I will set my face against that person and will cut him off from among his people.

Leviticus 20:13
If a man lies with a male as with a woman, both of them have committed an abomination; they shall surely be put to death; their blood is upon them.

Deuteronomy 14:19
And all winged insects are unclean for you; they shall not be eaten.

Deuteronomy 23:24
If you go into your neighbor’s vineyard, you may eat your fill of grapes, as many as you wish, but you shall not put any in your bag.

This is only a small sample of deontological statements in the O.T. But I’d like to look at one more passage:

Leviticus 25:44-46
As for your male and female slaves whom you may have: you may buy male and female slaves from among the nations that are around you. You may also buy from among the strangers who sojourn with you and their clans that are with you, who have been born in your land, and they may be your property. You may bequeath them to your sons after you to inherit as a possession forever. You may make slaves of them, but over your brothers the people of Israel you shall not rule, one over another ruthlessly.

So, what do various apologists have to say about slavery in the bible? Here’s a few sources.

http://www.rzim.eu/does-the-bible-condone-slavery
Jesus lays down a method of interpretation that has to be taken very seriously. He makes it clear that certain Old Testament commandments were to be understood as concessions to the hardness of the human heart rather than as expressions of God’s holy character. [...] The regulation of slavery should therefore be seen as a practical step to deal with the realities of the day resulting from human fall.[...] While the Bible does not reject slavery outright, the conclusion that it actually favours slavery is patently wrong.

http://www.gotquestions.org/Bible-slavery.html
The Bible does not specifically condemn the practice of slavery. [...] What many fail to understand is that slavery in biblical times was very different from the slavery that was practiced in the past few centuries in many parts of the world. The slavery in the Bible was not based exclusively on race. [...] In Bible times, slavery was more a matter of social status. People sold themselves as slaves when they could not pay their debts or provide for their families. [...] Another crucial point is that the purpose of the Bible is to point the way to salvation, not to reform society.

This sort of response is quite common. But distinctly lacking is an explanation as to why there is no categorical condemnation of slavery to be found, as there is for so many other things. Where’s the verse that reads, “You shall not keep slaves as property. This is an abomination.”?

Throughout the Old Testament, God acts as a strict deontologist. But when it comes to the issue of slavery, suddenly his attitude changes dramatically. He’s making concessions, he’s guiding the Israelites, he’s not interested in social reform. Furthermore, the answers given for all this might be somewhat satisfying with regard to domestic “slaves” (I can’t see much wrong with being legally required to work off one’s debt); but when it comes to foreign slaves, it’s clear that there’s strong similarity to the slavery practiced in the American south.

The reason that southern slavery was immoral is not just that it was race-based, although that’s certainly a major point against it. In addition, it was immoral because it was a method of treating other people not as people, but as property. If all the slaves in the American south were white, it would *still* be a morally disgusting practice. So the apologetic that OT foreign slavery wasn’t race-based doesn’t save the practice from immorality. As is clear in Leviticus 25, such slaves were treated as property, and could even be inherited.

I categorically condemn owning human persons as extremely immoral. Why doesn’t Yahweh?

Theism and Intelligent Design

The core of what is known as ‘intelligent design’ is an inference that, when interpreted charitably, can be construed as something along these lines:

Design Inference (DI):
There are certain facts about biology, and especially genetics, such that it is reasonable to infer from these facts that an intelligent agent is at least partly responsible for the diversity of life on earth.

Taken alone, this shouldn’t be too objectionable to scientists, at least at first glance. Scientists infer that things were designed fairly often – arrowheads, stone tools, etc. But of course, this isn’t a complete picture; the scientist’s job is never as simple as this. To see how, consider the design inference in the case of an arrowhead in more detail:

Imagine that you are walking along in the plains of North America, and you stumble across a stone object half-buried in the sand. Upon digging it out, you see that it’s made of stone, triangular, and has crudely sharpened edges. Given this, you conclude that it was almost certainly designed by some kind of intelligent agent. And you’d be reasonable to do so; because while it *could* have been fashioned by some natural process, you’re not aware of any that could actually do so. On the other hand, you can think of several ways in which intelligent agents could rather easily fashion such an object.

But you’re not satisfied yet. You’re a curious person; you want to know what kind of intelligent agent would make such a device, and furthermore, for what purpose. So to get things started, you rush back home and list as many potential designers as you can think of:

1. pre-U.S. native americans
2. extraterrestrials
3. modern-day humans
4. ancient egyptians
5. a perfect, supernatural, arrowhead deity

To begin narrowing things down, you first examine the object in greater detail. You look at it under a magnifying glass, and see that it’s been fashioned rather crudely, likely with primitive tools. You also notice that it’s made out of granite. With these facts in mind, you consider what sort of arrowhead design you’d expect to see given each of the potential explanations above:

1. triangular, made of granite, built with primitive tools, found in North America
2. made of some advanced metal, built with advanced technology
3. made of carbon steel, built with modern tools, razor sharp, three-bladed
4. made of granite, built with primitive tools, found in Africa
5. made of the best possible substance for making arrowheads, designed perfectly, completely unlike anything humans have made

Next, you consider why each of these potential designers would be motivated to construct an arrowhead:

1. to hunt game, or to shoot at targets
2. no reason apparent. surely, aliens would have more effective methods and devices
3. to hunt game, or to shoot at targets
4. to hunt game, or to shoot at targets
5. as a gift to its followers, so they could hunt game, or shoot at targets

From this analysis, we can conclude that probably (in fact, almost certainly), the best explanation for the arrowhead is that it was designed by pre-U.S. native americans. They had a good reason to construct arrowheads, *and* this arrowhead is the kind of arrowhead we’d expect, *and* it was found in the location we’d expect, if pre-U.S. native americans were the designers. In contrast, none of the other potential designers fit as well with the specifics.

Now, let’s move on to looking at design in nature. For the sake of argument, I’m going to grant the design inference at the beginning of this post, and look at which sort of designer is the best explanation of said design. Here are some options:

1. extraterrestrials
2. simulation
3. God
4. time travelers (causal loop)
5. non-God supernatural being

As in the case of the arrowhead, we shouldn’t just stop at concluding “some sort of designer”. Explanations, if they are to be good explanations, need to be more robust than that. So, like the arrowhead, let’s examine the details of the designed ‘object’ – human bodies.

So what relevant features can we point out about human bodies? Well first off, they’re physical. They have a lifespan of roughly 80 years. They’re extremely vulnerable to being damaged by temperature or by physical trauma. They’re not very efficient.

Given 1, 2, and 4, we’d probably expect these things (or similar things) to be the case. Extraterrestrials would have to work with physical matter, and aren’t perfect. Given that we’re a simulation constructed by some other sort of physical being, we might expect such beings to roughly model what they’re familiar with in the simulation. And given a causal loop where a human time traveler goes back and designs life in order to “complete the loop” of his own existence, it’s *exactly* what we’d expect.

5 is a toss-up. A non-God supernatural being may or may not still have internal or external limitations that necessitate working with physical matter. But with 3, God, there are no such limitations. While it’s true that given God, we’d expect to see minds that are capable of being moral agents, physical bodies that are inefficient and easily damaged seem not only superfluous, but contrary to what we’d expect such a being to design.

So, absent some reason to think that God would want to create inefficient physical bodies that are easily damaged, intelligent design advocates should not be silent on the identity of the designer. Rather, they should reject God as said designer. But this is a large bullet to bite for the theistic I.D. advocate, as it entails that God did not create the universe with humans in mind. Thus, the design inference, if it is a rational inference to make, is evidence against theism.

Brendon Eich: What Now?

Ok, I feel like I should write a more serious piece on this. So, what happens now for Brendon Eich? Well, a few things.

First, he’s going to find work somewhere else. Given his previous positions, it’ll probably be a higher-up position; something where he has some form of decision-making power in a company. Thing is, he’s probably not going to want to work for a liberal-minded company. If he does, the same thing will happen again – the internet will put pressure on that company to fire him. That’s *if* such a company will even want to hire him in the first place. They’ll likely not want that sort of attention, it’s bad for business.

So what will he do? Probably work for a more politically conservative company. Something like Chick-Fil-A (just as an example). Here, the controversy will actually benefit them. He can be trotted out as a martyr for the religious right. And furthermore, there won’t be any pro-equality community guidelines keeping his decisions in check. Ironically, he’ll be *more* capable of discrimination than he would have been at Mozilla.

He may also get invited to speaking engagements for conservative events & news stations. That’s more money in his pocket, and more fame to his name. And this whole event will just fuel the fire of the religious right’s cries of “we’re the ones being discriminated against!” So, the boycott of Mozilla will likely turn out to be counterproductive. Good job, internet.

Mission Accomplished?

Mission accomplished, internet – Brendon Eich has resigned as C.E.O. of Mozilla. This is a great day for LGBT rights – bigotry will not be tolerated!

Mission accomplished, right? Right?

Well…no. Of course, Bredon Eich isn’t going to remain unemployed forever. Eventually, he’ll find new work, likely in the tech industry. Our outrage over his bigotry can’t end here. Keep an eye on him, and pressure his next employer just like we pressured Mozilla. And the next one, and the next one. Because bigots don’t have a right to be employed.

You don’t even have to wait for him to take a new position, internet. You see, Javascript was designed by Brendon Eich, and lots of sites use Javascript. Millions of them! It’s time for us to pressure all of our favorite websites to stop using Javascript. Because who wants to use a scripting language designed by a bigot? Anyone that does clearly isn’t supportive of LGBT rights; using Javascript sends the wrong message to the LGBT community.

Surely, everyone who was opposed to Eich being named C.E.O. of Mozilla will continue the fight against bigotry; because this is the best way to promote equality. To victory, and don’t forget to tweet about it as much as possible!

(note for the thick-headed: this is clearly a parody meant to raise a point about the internet’s reaction to the Brendon Eich ‘controversy’. If you think I’m serious, I’ll laugh at you.)

Two Roads Diverged In a Leibnizian Wood

Blogger and sombrero enthusiast Stephen J. Graham has written an excellent post about the nature of argument. This post is not a rebuttal; rather, it is a companion piece.

Stephen first comments on responses to the Kalam cosmological argument, saying that even if an atheist cannot think of an objection to either of the premises, he may still reject the argument for other reasons – one example being that he objects to the conclusion itself on the ground that it’s incoherent.

Let’s look at the Kalam for a moment:

1. Everything that begins to exist has a cause.
2. The universe began to exist.
3. Therefore the universe has a cause.

Now, imagine that you’re an atheist who has encountered this argument for the first time, and that you, prior to the argument, you agreed with the premises independently but disagreed with the conclusion. But now, you just cannot think of any rebuttal to the premises (and, obviously, the conclusion follows from the premises). Must you accept it? Not necessarily.

What arguments such as this are meant to do is merely to show a logical relationship between propositions. They’re *not* really useful for necessarily inclining people to accept their conclusions. Consider a list of some of the atheists’ beliefs prior to the argument:

{1, 2, ~3}

The argument, when examined correctly, should motivate one not to change one’s mind about believing ~3; but rather change one’s mind about at least one of these. That might not be ~3; it could be 1 or 2. Imagine it as a sort of “weak” reductio – “well, I didn’t realize that 1 and 2 implied 3; since I have good reasons to think ~3, I should change my mind about either 1 or 2″. The argument is essentially a starting point for examining how these different beliefs relate to each other.

Stephen points to two different approaches in responding to arguments. The first, the “high road”, is to just object to one of the premises. The second, the “low road”, is to “shift” the structure of the argument. One way to do this is known as a “Moorean Shift“. Stephen gives an example by making a counterargument to Justin Schieber’s ‘GodWorld’ argument for atheism:

(5) If GodWorld is the unique best possible world, then the Christian God would maintain GodWorld.
(6) The Christian God has not maintained GodWorld
(7) Therefore GodWorld is not the unique best possible world.

Let’s call this argument ‘WorldGod’. If WorldGod is sound, then GodWorld fails. And vice-versa, if GodWorld is sound, then WorldGod fails. How are we to decide which of these is the better route? First, let’s reformulate GodWorld into a simpler, three-premise form:


(1) If the Christian God exists, then the Christian God would maintain GodWorld.
(2) The Christian God has not maintained GodWorld.
(3) Therefore, the Christian God does not exist.

Stated like this, we can clearly see how GodWorld and WorldGod interact with each other. Notice especially that the second premise in each argument is identical. Now, deciding between these two arguments becomes much easier: which premises are you more sure of; (5)+(6), or (1)+(2)?

But how could we decide which of those sets of propositions we’re more sure of? Well – other arguments, for one. To a Christian, (3) will seem very implausible, since the Christian will think there are other arguments to motivate ~(3); and thus will reject (1)+(2). And vice versa for the atheist – he’ll reject (7) for other reasons, and thus (5)+(6).

But really, in the end, it’s going to boil down to just the antecedents of the first premises: “The Christian God exists” vs. “GodWorld is the unique best possible world”. Which of these do you think is more likely? Personally, I’ll place my bets on the latter. All things considered, the former seems to be pretty unlikely; however there are many things to consider, which is why we should never place too much stock in a single argument.

So what can we conclude from all this? Simply put, that argumentation isn’t nearly as simple as just offering a sound argument. In fact, I could go back and forth with someone examining confidences in (1) through (7) for days. In the end, it’s really just a matter of your various confidences in all the relevant propositions. So, next time you hear someone say that a conclusion “logically and inescapably follows” from some set of premises (this is especially said about the ontological argument), keep in mind that this is not the case at all. Arguments are a tool and nothing more. They don’t necessitate; they merely motivate.

A Fine-Tuning Argument For Atheism

Here is a diagram to explain two different types of fine tuning; narrow and robust (if the image is too small to read, click on it for a larger version):

Tuning1

If narrow fine tuning is the kind of fine tuning there actually is, theists claim, then the probability of life coming about naturally is low; and this motivates the fine tuning argument for theism. This seems to carry at least some argumentative force. But this tweet from Justin Schieber (#FF) got me to thinking:

tuning2

To say that such-and-such combination of cosmological constants doesn’t permit life is not to say anything about what’s logically necessary, but about what’s physically necessary (or so it seems – more on this in a moment). In other words, if the theist is right and God exists, fine tuning would be robust – and yet, the theist will argue that fine tuning is narrow as evidence for God.

But the theist cannot have it both ways. If we’d expect to see robust fine tuning were God to exist, then narrow fine tuning is actually evidence against God. From this, we can construct the following argument:

1. If God is omnipotent, then he can create life under any cosmological conditions.
2. If God can create life under any cosmological conditions, then fine tuning is robust.
3. Fine tuning is not robust.
4. Therefore, God cannot create life under any cosmological conditions. (2,3 modus tollens)
5. Therefore, God is not omnipotent. (1,4 modus tollens)

Premise 1 is just a consequence of omnipotence – it’s logically possible that God create life under any cosmological conditions – that some conditions are impermissible to life is a physical claim, not a logical one. Premise 2 is definitional, and premise 3 is the claim that motivates fine tuning arguments.

The theist clearly wants to keep 3 – if he does not, then he loses the fine tuning argument. He likewise wants to keep 1, as denying it would be to deny God’s omnipotence. So the theist must attack premise 2. How could this be done?

One response is to argue that the argument equivocates between different senses of robustness. When we talk of narrow fine tuning, the theist can say, what we mean is that a small set of values is physically necessary for life; but it’s still true that these values aren’t logically necessary. Thus the argument should look like this:

1'. If God is omnipotent, then he can create life under any cosmological conditions.
2'. If God can create life under any cosmological conditions, then fine tuning is logically robust.
3'. Fine tuning is not logically robust.
4'. Therefore, God cannot create life under any cosmological conditions. (2,3 modus tollens)
5'. Therefore, God is not omnipotent. (1,4 modus tollens)

The theist can then easily say that premise 3′ is false – fine tuning is logically robust, and it just so happens that God decided that fine tuning should be physically narrow. Furthermore, the theist might say, God wanted fine tuning to be physically narrow in order for it to act as evidence that He exists; a sort of divine signature.

Initially, I found this objection convincing. But after some reflection, this is no longer the case. Consider what things would look like if we admitted the logically/physically necessary distinction:

tuning3

The problem here is that making a distinction between logical and physical robustness brings in modal considerations. This allows a premise like this:

6. There is a possible world W1 in which all values of cosmological constants are logically life-permitting, and yet life can never arise in W1 because it is physically impossible.

This highlights a difficulty for the theist. Given that, regardless of what the constants are when this world’s universe forms, life is logically possible; what does it even mean to say that life is physically impossible in W1? When discussing our world, the point is often made that if the constants were different, then life would be physically impossible.

The problem for the theist is that this sort of reasoning implicitly invokes the idea of possible worlds – the only reason that life would be physically impossible if the constants were different is because of how the actual world works. This other possible world doesn’t have to play by the rules of the actual world, precisely because the constants are different. When we examine which values of the constants could permit life, we’re peering into modal space, and to do so is to ponder what’s logically possible. Consider also this premise:

7. There is a possible world W2 in which the constants are the same as the actual world's constants, but W2 is physically not life-permitting.

This doesn’t seem like a possible world at all; whether a world is life-permitting or not just is a function of the constants. So it seems that physical life-permittingness reduces to logical life-permittingness. Thus, the original argument stands. What options does the theist have now?

Well, the theist isn’t going to want to accept the argument’s conclusion. The only remaining option is to deny premise 3, and say that fine tuning is robust. But if this is the case, then the theist’s fine tuning argument for God fails. Thus, the theist’s confidence in God’s existence should drop accordingly.

Evil and Ontological Arguments

There are two rough views on the ontology of what philosophers call ‘possible worlds’ – modal realism, which maintains that these worlds are actually existing things, just as real as the actual world; and modal fictionalism, which maintains that these worlds are not real in the same way as the actual world is.

Theists who wish to make Plantinga-style ontological arguments must rely on a premise which looks something like this:

(1) There is a possible world W1 in which an entity has the property of being maximally excellent in all possible worlds.

The property of being maximally excellent in all possible worlds is often shortened to the property of ‘maximal greatness’. On its face, this premise seems plausible. But consider the implications if modal realism is false. If this is the case, then what this premise means might look something like this:

(2) There is a set of compossible propositions W2 which maximally describe a way things could have been, in which the proposition "entity X has the property of maximal greatness" would have been true were W2 the case.

If this is the premise motivating the ontological argument, then the argument just fails outright – because so what? The S5 system of modal logic just can’t get us to the conclusion that some entity actually exists on modal fictionalism, because the premise in question would just be saying something about an abstract object (the set of propositions) that exists in the actual world. To claim that it can would be to beg the question in the worst way.

‘But wait!’ the theist might say, “modal realism is true, not modal fictionalism, so the argument goes through after all.” This may or may not be the case, although the ontological argument is certainly in a better position if modal realism is true – since it would be positing something much more substantial than the existence of a proposition that could have been true.

However, let’s assume that this saves the ontological argument, and see where it leads. If modal realism is true, we can also posit this premise:

(3) There is a possible world W3 in which there are substantially many more moral and natural evils than in our world, and these evils obtain to a substantially higher degree.

This world would truly be a ‘hell on earth’. In this world, rape and murder happen en masse every second of every day. Trillions of humans are in constant, blinding pain for their entire lives. The universe is filled to the brim with gratuitous suffering that serves no greater good whatsoever.

Given modal fictionalism, luckily, this world doesn’t actually exist – it’s just another set of propositions. But given modal realism, this world *does* actually exist, and it’s just as real as ours. There are actual persons suffering, on the grandest of scales, for no reason. This is the problem of evil write large. In this world, there is no soul-making, no greater good – all people do is suffer. Even worse, there are a near-infinitude of worlds which very closely resemble this one, differing only in minor, inconsequential details.

What response could a theist have to this? The traditional theodicies wouldn’t seem to work. Perhaps skeptical theism – but for all we know, God has good reasons for these worlds? That’s even less plausible here than it is when considering just the actual world.

One response may be to concede that such worlds are indeed a problem, but that since the ontological argument is deductive in nature, then it overrides the confidence-lowering nature of this argument from modal evil. But I think this is mistaken. While it’s true that the ontological argument is deductive, this cannot override all evidential concerns, no matter how strong, and provide certainty. Even with deductive arguments, we must still examine how confident we are in the argument itself. In the case of an evidential vs. a deductive argument, we could still be more confident in the evidential one – especially if the deductive argument has such horrifying implications.

The best solution for the theist, I think, is to drop ontological arguments altogether. This avoids the massive problem entailed by (3), and puts the argument from evil back to where it is currently, while only losing the force from one style of argument – cosmological, teleological, and moral arguments (and many others) would still be available, and these do not have the implications that ontological arguments do.

Freewriting

Ok, let’s try a bit of philosophical freewriting. I’m just going to do a stream of consciousness here, and it will likely come out rambly as fuck and make no sense. But perhaps there’s value in just writing down one’s thoughts as they come.

For example, I’m a bit disappointed in the lack of any philosophical inclination in the general atheist community. I mean, I’ve read a *lot* of really great works; for example, ‘Theism and Explanation’ by Greg Dawes. It’s just such a good book. And it’s not just valuable for discussions about whether theism is true. It has a lot of good info about how we examine proposed explanations in a more general sense.

I’m also baffled about atheism’s attitude toward Christian apologists; most notably William Lane Craig and (more recently) Alvin Plantinga. Don’t get me wrong, I don’t find their arguments convincing either. But I see a lot of what I can only describe as *vitriol* about them. They’re stupid, they’re liars, etc. When I read or watch them, I don’t see any of this – I just see philosophers who happen to have gotten things wrong.

Shouldn’t we all try being a little more intellectual about this?

…Hmm. I think I’m stuck. I just wrote the above in one go, and I’m not going to edit it. Kind of hit a wall there; I don’t know where to go next. Oh well. This was interesting anyway. Perhaps I’ll do it again.

Additional thought: I suppose I’m baffled about a lot of stuff that goes on in the atheism community. But I’ll save that for another time.

Catharsis

There’s some people I really like on Twitter. I’d even consider many of them friends, just as much as I would someone I knew in “real life”. But even so, I feel like there’s no real community for me on Twitter. When I joined, I assumed that I’d fit in at least fairly well with the atheist community. But I just can’t. Too many of the most popular people in that community gained their popularity by being antagonistic, rude, or even downright nasty. “Hahaha religion is stupid!” gets way more retweets and followers than a reasoned argument. Sometimes, hundreds and hundreds. Yet my tweets encouraging people to be more charitable only ever get a handful.

Is that really where peoples’ priorities lie? Most of these people would probably call themselves ‘humanists’. I call bullshit on that. If you delightfully retweet insults about the latest religious apologist that’s happened to pop up on the atheist collective radar, but you skip over promoting the call to care about the plight of the homeless, you’re not acting humanistically at all.

If you’re reading this and you’re tempted to speculate about who inspired it, don’t bother. It isn’t any one person. It’s a whole mass of people. The atheist/skeptic/humanist community, as a whole, is failing badly. We need to stop the petty arguments with the religious, stop the insults, stop the jokes about how stupid it all is.

We need to start making the world a better place. Organize. Instead of getting into that next argument over gay marriage, go write an email to your representatives about why they should support it. Instead of talking about how much of a charlatan William Lane Craig is, go read what atheist philosophers are writing about the Kalam argument in journals. Instead of getting into a shouting match with some hapless creationist about evolution, go tutor your nephew so he gets an A in biology.

Do something that matters. It’s a better use of your time. And stop giving the time of day to people with negative output.

Playing With Logic

Many people use the rule of inference known as ‘modus ponens’ in debate, but few have really thought about why it works. So to start, here’s what modus ponens looks like:

1. P → Q
2. P
3. Q (1,2 modus ponens)

Intuitively, we can see why it works, but proving it is another matter. To prove it, first we have to play around with premise 1. As it turns out, premise 1 is logically equivalent to:

4. ¬(P & ¬Q)

To see why, consider English ‘translations’ of both 1 and 4. 1 can be written as “If Spot is a dog, then spot is a mammal”. 4 can be written as “not(spot is a dog and spot is not a mammal)”.

So, let’s see what happens if we use 4 and 2 as premises, and then assume ¬Q for the sake of argument:


2. P (Premise)
4. ¬(P & ¬Q) (Premise)
5. ¬Q (Assumption for reductio)
6. P & ¬Q (2, 5 Conjunction Introduction)
7. (P & ¬Q) & ¬(P & ¬Q) (4, 6 Conjunction Introduction)
8. ¬¬Q (Proof by contradiction)
9. Q (8, Double Negation Elimination)

Yes, that looks a bit intimidating, so I’ll explain. First, 2 and 4 are the premises of modus ponens. 5 is to see what happens if we use those premises and ¬Q. 6 and 7 show us what happens. In 6, we’re merely combining 2 and 5 (that’s what ‘conjunction introduction’ is). And the same in 7 – we’re combining 4 and 6.

But notice what happens in 7 – we get a contradiction! Well, if 5 leads to a contradiction, it must be false (in classical logic, anyway). So we get 8. Finally, 9 is just eliminating the double negative.

Modus ponens is highly useful, not just because it can be used to prove things, but because it allows us to simplify things. Imagine if we had to write out “(¬(P & ¬Q)) & P” in every step of a long proof. No one wants to do that.