Infinities have different sizes
Cantor argued that the infinite size of the set of positive integers is smaller than the infinite size of the set of real numbers in the interval from 0 to 1. This (a) established the incorrectness of our intuition that all infinities are the same size, and (b) was the first argument that used a diagonal construction.
The argument is a reductio ad absurdum. Suppose the two sets were the same size. Then by the definition of same size, each real number somewhere in the range from 0 to 1 could be paired up with a positive integer and vice versa. Here is how such a pairing might look:
Consider the diagonal number that is produced by using a decimal point and then selecting the diagonal digits that are highlighted and underlined above:
This number might be somewhere down the list, but the following number definitely won’t be. Take each digit of the diagonal number and add one to it, but if the digit is 9 then replace it with 0.
That new number differs from the nth number in the list at least in its nth digit. It should be on the list but isn’t. That’s the contradiction. For any other table that tries to pair natural numbers with real numbers, there is a similar diagonal construction.
The Euthyphro dilemma (Plato’s Socrates)
What gives a moral truth its authority?
Are actions morally good because God commands them, or does God only command what is morally good?
1) the authority of a moral truth is established by God commanding it, orIf 1 is true, then any moral truths established by God are arbitrary. (God could have commanded something different, and the authority of morality is not based on the reasons why God commanded what he did.)
2) God understands the authority and that leads him to make his commands
If 2 is true, then God is not the source of morality. (God must be referring to some kind of moral reasons, principles, or rules in order to know what is morally good. Morality is not arbitrary on this view because it is based on reasons. God is not the source of morality on this view because the reasons, principles, or rules are the source or authority of morality and God is (presumably) using rationality to infer morality from those reasons, principles, or rules.)
This argument is important because, regardless of one’s religious beliefs, it encourages us to think carefully about what makes something morally good or bad. This kind of thinking is required for us to work out how to behave towards others in our diverse world. And it’s cool because I like to imagine God puzzling over trolley problems and experience machine scenarios (“it just is wrong to push the man!”)
I think the best arguments begin with really simple and uncontroversial premises, which are used to develop much more substantive and interesting conclusions. Alan Gewirth (Reason and Morality, 1978) begins with the simple and uncontroversial thought that each of us considers the aims and purposes of our chosen actions as good (otherwise, why do them?). Each of us, then, should insist that others not interfere with these actions. We are all committed to objecting and resenting others when they prevent us from acting on what we consider good.
So, from the first-person point of view every agent is rationally committed to a) demanding that others not interfere with her actions and b) acknowledging that the basis of this demand is a generic feature of human agency. Therefore, c) every agent has the same basis for making this demand as she does, and d) anyone who made this demand for herself while refusing to acknowledge the standing of others to make it, as well, has contradicted herself.
This general commitment makes freedom of action a moral default. It's not absolute, of course, but it establishes a moral presumption and, for Gewirth, a very strong one so that no one's freedom may be restricted without special justification.
G. Randolph Mayes
1. If successful scientific theories are not accurate descriptions of reality, then their success is a miracle.The idea underlying premise 1 is that the best explanation of the success of science is that it provides accurate descriptions of reality. This is called an inference to the best explanation (IBE).
2. The success of scientific theories ain't a miracle.
3. Therefore, successful scientific theories are accurate descriptions of reality.
I find it compelling in naturalistic terms: It seems right, e.g., that the relative success of organisms with minds seems (partially) best explained by the accuracy with which they represent their environment.
But the principle of IBE also strikes me as a confusion. Simply put: we can't infer that a theory is true or accurate because it is a good explanation. Being true is just part of what we mean when we say that an explanation is good.
There are other criteria, too, e.g., prediction, consilience, and simplicity. So we could just remove truth from this list. But then when we say that realism is the best explanation, what we mean is that it possesses these virtues to the greatest degree. Well, maybe so- compared to no explanation at all. But how good is it? What does it predict? What is unifying or simplifying about it? When you really hold it accountable to explanatory criteria, the no-miracles argument starts to feel like magic.
In Ethics: Inventing Right and Wrong, Mackie offers an ‘Error Theory’ for how we have come to be so mistaken about what there is. One part is the Argument from Queerness:
1. If values are objective, then they must be metaphysically similar to other objective things. Objectivity entails that what is objective is real, i.e., is true against the way the world actually is (not against the way we agree it is, or against our practices, desires, perspectives, or semantics). For values to have this kind of objectivity entails “they would be entities or qualities or relations… utterly different from anything else in the universe” (38).
2. If values are objective, then we must be able to know them. But, to know something “utterly different from anything else in the universe,” we’d need “some special faculty of moral perception or intuition, utterly different from our ordinary ways of knowing everything else” (38).
What makes this argument cool, is that Mackie demands that we question how we can know our moral judgments are true. Even obvious prescriptions, like ‘do no harm’, presume an entailment between the fact of pain and the wrongness of it. But this is not self-evident, as intuitionists and moral realists propose. Mackie’s Argument from Queerness jabs at that connection. This is not to say he’s right, but his jab demands we rethink what we take for granted when we declare anything to be wrong or right. It is also cool just ‘cuz of its name.
In Animal Liberation (1975) Peter Singer makes a compelling case for not using or consuming non-human animals when it is not necessary to our survival, because animals experience pain and suffering as we do, they need their lives as much as we do.
We really don’t accept that differences in intelligence between humans make it morally acceptable for some humans to eat or experiment or otherwise subordinate less intelligent humans, because intelligence is not the relevant difference we use to guide our treatment of others. The capacity for suffering is the only relevant difference. In suffering, animals are our equals.
Refusing to acknowledge animal vital interests on the grounds that they are not human or lack other qualities that humans have is a form of discrimination akin to racism and sexism, he calls it speciesism, which has the same recipe-for-exploitation rationale: Ignore or weigh differently the similar, vital interests of members of different groups.
Singer’s argument is cool because he extends the principle of equality that guides our treatment of fellow humans “consider and treat the interests of those who suffer equally” to non-humans. It isn’t fair for racists and sexists to discriminate on the grounds of real or perceived physical or cognitive abilities and then inflict suffering on those whom they subordinate, and it isn’t fair for speciesists. Sure animals eat each other, but unlike animals, we humans are morally responsible for what we do.
Fission and Fusion
Derek Parfit in Reasons and Persons asks us to imagine a fictional case where you are taking a timed exam with two problems left and only enough time remaining to finish one.
But, you can easily trigger a device in your brain—perhaps by raising your eyebrows—that effectively divides your stream of consciousness—and your ability to control your body—in two.
So the left half of you can work on one test problem at the same time as your right half is working on the other test problem.
The way Parfit describes it is great: for example, the left ‘you’ can see in its peripheral vision the right ‘you’ scribbling furiously as the clock ticks down, and can wonder what the right ‘you’ is thinking and feeling. And vice-versa.
This is an illustration of what it might be like to experience a certain kind of ‘division’ or ‘fission’ (in this case, binary fission, of one’s stream of consciousness and bodily control).
Of course, philosophers divide (pun intended) on what such a case might prove or illustrate about personal identity and other matters. (For example, do you literally become two persons? If not, which stream are you?)
But one reason I like the case is that Parfit quickly closes its description by imagining a ‘fusion’ once the test is over. (Perhaps the you—the left ‘you’?—blinks twice?) The individual at the end of the fusion (you?) suddenly remembers everything that happened in both streams.
It’s a great case.