Short note on valid arguments

On one characterization of deductive validity, a valid argument is any argument where it is "Logically impossible" for the set of premises to have only true members while the conclusion of the argument is false.

One consequence of this definition is that any argument with an inconsistent set of premises, or a contradictory premise will guarantee that an argument is valid. For example, consider the following argument:

1. P -> Q (Premise)
2. Q (Premise)
3. R & ~R (Premise)
4. P (Conclusion)

Ordinarily, this argument would appear to be affirming the consequent and is consequently invalid. However, since premise 3 is a contradiction it follows that it is not the case that every member of the set of premises can be true, so trivially the argument is valid.

Basics of Set Theory

 Set theory basics:

Example set: 

Set of numbers less than 5 but greater than 1 = {2, 3, 4, 5}. Call the set "S".

Notation used:

{2, 3, 4, 5}

Read as: The set that contains the numbers 2, 3, 4, 5.

2 ∈ S and 6 ∉ S

Read as: 2 is an element of set S and 6 is not an element of S. 

Subset: B is a subset of A if all elements of B are elements of A.

Example subset:

Notation used:

B ⊆ A

Read as: B is a subset of A.
Proper subset: B is a proper subset of A if all elements of B are elements of A and A has elements that B does not.

Example of proper subsets:

A = {1, 2, 3, 4, 5, 6}, B = {1, 2, 3}. B is a proper subset of A since every member of B is a member of A and A has members that B does not.

Notation used:

B ⊂ A

Read as: B is a proper subset of A.

Power set:  The power set of any set A is the set of all possible subsets of A.

Example of power set:

S = {1, 2, 3}

Power set of S = {{∅}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}

Notation used:

ℙ(S) 

Set Union: The union of two or more sets is the set that contains all and only members of the sets in question.

Example of set union:

A = {1, 2, 3}, B = {4, 5, 6}

The union of A and B = {1, 2, 3, 4, 5, 6}

Notation used:

A∪B = The union of A and B

Set intersection: The set of all common elements between two or more sets.

Example of set intersection:

A = {1, 2, 3}, B = {3, 4, 5} 

The intersection of A and B = {3}

Notation used:

A∩B = The intersection of A and B.
 

Cherry picking

Cherry picking facts is detrimental in a lot of respects, and here is an example of how it can be detrimental to two different persons doing natural theology.

Person 1 defends the following thesis: The probability of atheism given the evidence is not only higher than the probability of theism given the evidence, but it is more likely than not.

Person 2 defends the exact opposite argument. However lets grant that Person 2 is cherry picking their facts. When Person 1 argues that 'Pr(A/E) > Pr(T/E)' Person 1 is arguing with a set of evidence E that consists of 10 facts, but Person 2 is arguing with a set of evidence E1 that consists of only 5 facts.

If the actual evidence consists of the 10 facts rather than merely the 5, then Person 2 can come up with an unjustified degree of belief in T since the actual evidence is E rather than E1. This is a problem that many people might have when making a cumulative case for or against theism.

For example let's grant that the evidence to be considered in this world was that minds depend on brains, the roles of biological pain and pleasure, the flourishing of humans, the flourishing of non humans, and the existence of rational beings.

The theist might reason that the probability of rational beings existing given theism is extremely high, and is much greater than the probability of rational beings given atheism. But the theist stops there. He does not think to factor in the probability of the total set of evidence given theism, rather the theist makes use of a small subset of the evidence. But by doing so the theist might unjustly find that theism is much more probable than atheism, even if all things considered atheism is more probable given the total set of evidence. If one portion of the evidence greatly favoured theism over atheism but the rest did not, then this could make the probability of theism given the evidence much lower, possibly in such a way that belief in theism is not justified at all.

Three misconceptions about atheism and atheists

Here is a short list of misconceptions about atheism, which entail causes myths about atheists:

1. "Atheism is identical to naturalism". This claim is not true. Atheism is the lack of belief in God, not the lack of belief in all supernatural entities (Whatever those are). Take ghosts for example. A ghost is almost always defined as being supernatural. As a result many super-naturalists (And maybe non super-naturalists) falsely believe that atheists cannot believe in supernatural beings such as ghosts. So the myth about atheism causes a myth about atheists, that being that no consistent atheist can believe in supernatural beings.

2. "Atheism is identical to materialism". This is similar to (1.), and it also false. While all materialists are atheists (I mean materialists about everything in existence, not simply particulars such as human persons), it is not true that all atheists are materialists. An atheist can consistently believe in the existence of non material things just in case their existence is not dependent on God(s). For example an atheist can believe that abstract objects exist, and some would argue that if abstract objects exist, then only atheism can be true (i.e. Abstract objects are generally defined as necessarily existent and outside causal relations, as a result of their necessity and a-causality some people argue that they are defined out of being created by God, or affected by God, thus God could not be the creator of all things, thus God does not exist).

3. "Atheism entails scientism". This is false. While atheism might be highly correlated with scientism, it isn't true that atheism by definition entails scientism, An atheist can consistently believe that it is possible to not only discover truth sans the sciences, but that non scientific investigation is actually better at discovering truth than it's counterpart. The reason is simple: Atheism only entails that God does not exist, not that fairies are not planting knowledge within us that science cannot investigate.

Considering the above three is important for at least a few reasons. One reason is that the "Theism vs atheism" debate often uses arguments refuting naturalism, materialism, or scientism in an attempt to refute atheism. But atheism is a different view. Atheism is entailed by both naturalism and materialism, but it entails neither of the former. So any argument refuting atheism would refute both naturalism and materialism, but the opposite would not. This is not to say that discussing problems with the three former views is not relevant to the theism vs atheism debate (For example lets grant that I am an atheist because I am a materialist. If a theist found good reasons to reject my materialism, I would have less reasons to reject theism, or I would need entirely different reasons depending on the circumstances).

A second reasons for considering these misconceptions is that some people might assume propositions such as the following:

1. "Materialism (or naturalism) excludes objective morality. Atheism either entails or is identical to materialism. So atheists reject objective morality, or should to be consistent". An atheist can believe the first statement listed, but still consistently believe that objective morality exists (Or at least there are no obviously sound/strong cogent arguments that I know of which suggest otherwise). For example philosopher Richard Swinburne seems to believe that moral truths are necessary and independent, and as a result they are not contingent on God. If what Swinburne thinks is true, then atheism does not exclude morality since morality is not dependent on God. This issue is definitely an important one. Hardly any atheist would wish to be told they believe that child rape is morally okay, or that they are some sort of amoral monster, or that given their worldview there is no reason to be anything except an amoral monster.

Now here is a few truths many atheists want others to know:

1. Some atheists are stupid, and don't believe theism for stupid reasons.

2. Not all atheists hate religion. Many believe that religion should stay, and some simply don't believe in religious tales without any hate.

3. Atheists do not necessarily believe they think they know everything. One of the reasons many people they should be atheists is because we do not know everything. i.e. They do not believe because they would rather scientific inquiries to provide answers prior to using God to fill a gap in knowledge.

4. Darwin is not the God of atheism. Darwin's work might be one of the reasons that some people are atheists, but atheists don't worship the man (Or at least I've seen none who would).

Bayes Rule again

Let's grant that we order banana trees from two countries; Guatemala and Honduras.

60 percent of the trees ordered are from Guatemala, so the other 40 percent is from Honduras. 6 percent of the trees from Guatemala have tarantulas on them, and only 3 for Honduras.

Question: What is the probability of the trees being from Guatemala given that there are tarantulas on them?

So we're looking for "Pr(G/T) = ?". The equation will look like the following:

Pr(G/T) = Pr(G/b) * Pr(T/G) / Pr(G/b) * Pr(T/G) + Pr(~G/b) * Pr(T/~G)

"Pr(G/b)" is the prior probability of G given our background information. I decided to add in the "b" because it looks like a lot of people tend to do that. In this scenario there are two options; Guatemala and Honduras. Since there are only two, then "~G" can only be Honduras as there are no alternative possibilities. So Pr(~G/b) = the prior probability of Honduras given our background info.

Now let's plug in some numbers.

(1) Pr(G/b) = .6 since 60 percent of the trees come from Guatemala.
(2) Pr(T/G) = .06 since only 6 percent of Guatemalan trees have tarantulas.
(3) Pr(G/b) * Pr(T/G) = .036
(4) Pr(~G/b) = .4 since 40 percent of the trees come from Guatemala.
(5) Pr(T/~G) = .03 since only 3 percent of Honduran trees have tarantulas.
(6) Pr(~G/b) * Pr(T/~G) = .012
(7) .036 + .012 = .048
(8) .036 / .048 = .75

The final answer is .75 which means the probability of the trees being from Guatemala given tarantulas is 75 percent.

This is pretty simple stuff, but this elementary use of Bayes is still useful to know. 

I found this example (Or something like it) in Ian Hacking's book Intro to Probability and Inductive Logic.

Bayes Rule

Here's a basic and common example of the use of Bayes Theorem:

1 percent of women tested for breast cancer will have the disease. Only 80 percent of the women with cancer will have a positive mammography. An additional 9.6 percent of women will have a false positive mammography. What is the probability of having cancer given a positive mammography?

Let C = Has cancer, ~C = no cancer, M = Positive mammography
The probability of cancer given a positive mammography would be labelled as "P(C/M)". So we're looking to find the answer to "P(C/M = ?)".


The Bayes Rule to solve this in plain English is: The probability of cancer given a positive mammography is equal to the prior probability of cancer multiplied by the probability of a positive mammography given cancer, then divided by the same number with the addition of the prior probability of no cancer multiplied by the probability of a positive mammography given no cancer.

So the symbolized rule is the following: P(C/M) = P(C) * P(M/C) / P(C) * P(M/C) + P(~C) * P(M/~C)

When we aren't using the "/" symbol we are referring to the prior probability of something. The prior probability of an event of hypothesis is basically the probability of it with our current knowledge without updated evidence. In this scenario the updated evidence would be the test results. Since only 1 % of women tested will have cancer, then the prior probability before evidence is 1 %. So "P(C)" = the prior probability of having cancer.

Here are the numbers:

P(C) = .01 since 1 percent of women tested will have cancer.

P(M/C) = .8 since 80 percent of the women with breast cancer will have a positive mammography.
P(~C) = .99 since 99 percent of women tested will not have cancer.
P(M/~C) = .096 since 9.6 percent of women without cancer will have a positive mammography.

With these above numbers we can solve the equation:

P(C) * P(M/C) = .01 * .8 = .008
P(C) * P(M/C) + P(~C) * P(M/~C) = .01 * .8 + .99 * .096 = .008 + .09504 = .10304
Now you divide the top number by the bottom, so it is .008 / .10304 = .07764
The end result is that the probability of having cancer given a positive mammography is only 7.8 percent. This is higher than the prior probability of 1 percent, but still nothing to be too worried about.


I think everything up there is all correct, but here is a crucial link anyways: http://yudkowsky.net/rational/bayes/

You can Prove a Negative:

Both some atheists and theists alike claim you can not prove a negative. By this, they mean you cannot prove something does not exist.

This is untrue, and is based on poor understanding of the basic laws of logic.

- For example, one can safetly say a squared circle does not exist. This is due to the law of non contradiction. It is the case that something cannot be both what it is, and what it is not simultaneously. Example:

-> 'A' cannot be both 'A' and 'B' simultaneously, as to be 'A' it must not have the nature of being 'B' which has the nature of being itself and not 'A'. - So, 'A' can only be 'A', and 'B' can only be 'B'. They cannot exist as one simultaneously.

-> For a circle to exist, in its nature is to be completely round with no edges. A square has the nature of having four corners and in no way round. For a circle to be both a circle and a square simultaneously, it would need to have the nature of being both completely round and have 4 sides at the same time. But this is abusrd. The nature of a circle and square are distinct from one another, and to be both simultaneously would show a contradiction. To be more clear, both the nature of a circle and square cancel the other out, so they cannot exist as one thing.

-> Due to the above, you can prove that something does not exist. If something has contradictory properties as a squared circle would, then its the case that this cannot exist.

-> This is important to the theist/atheist debate. The relevancy is due to the existence of certain God's having properties that are contradictory with reality or itself.

Example: Some Atheists say a God cannot both have freewill and be omniscient. This is because to be omniscient, a God would know every possible action prior to its realization. If this is true, then a God would necessarily know every action it will commit before the actions realization. If that is true, the God does not and cannot have freewill. God cannot both be free to act but know how God will act.

Example2: Some atheists claim 'An omnibonevolent God cannot exist if there is gratuitous evil'. The argument claims that if a God is all loving, then gratuitous evil would not exist. This is because an all loving God would want gratuitous evil to not exist, and being omnipotent would allow it to prevent such evil. So the Atheist would say the existence of gratuitous evil is inconsistent with a God that is both Omnibonevolent and Omnipotent.

Due to the above examples, it is clear that people try to show that God has attributes that are inconsistent with reality, and therefore 'Proving a Negative' is both possible and very relevent to the question of 'Does God exist?'