 Is Atheism the Null Hypothesis?
 Is Atheism Falsifiable?
 Does Atheism Carry the Burden of Proof?


Atheism has distinct definitions which can be categorized as follows:

• Weak/Soft Atheism: I do not believe "god exists". Weak atheism is the rejection of the positive claim that "god exists".
• Strong/Hard Atheism: I believe "god does not exist". Strong atheism is the acceptance of the negative claim that "god does not exist".
Strong atheists are a subset of weak atheists: those who believe god does not exist form a subset of those who do not believe god exists.

I am a hard atheist, depending on the definitions of knowledge and god: I believe there is no god but do not claim to know that god does not exist. I do not believe there is a god, and in fact go one step further in affirming the belief that there is no god, but I do not claim to know god does not exist. Certainty is not a requirement of knowledge. The necessary conditions for knowledge justified, true, belief, also referred to as the Tripartite Analysis of Knowledge (i.e., classical philosophic definition of knowledge). Note, justified true belief are only the necessary not sufficient conditions for knowledge. What exactly more than justified true belief is sufficient to constitute or define knowledge is an open question in modern epistemology, ever since Gettier cases (i.e., “Gettier Problem”. An American philosopher (Massachusetts) Edmund Gettier published a 2.5 page refutation of the justified true belief theory of knowledge by giving two counter-examples in which justified true belief does not suffice to constitute or define knowledge.

The burden of proof is on the proposition, not on the opposition!

The burden of proof is on the one who makes a claim, regardless of the positive or negative content of the claim. The burden of proof is on the claimant, not the respondent.
Example:

Null Hypothesis: H{0}:= "God does not exist" = "There is no god" = "No god exists"
Test hypothesis: H{T}:= "God exists" = There is a god" = "Some god exists"

There are four possible believe positions here: Let: B[]:= believe [], ~B[]:= do not believe:

B[H{T}]:= I believe "god exists". ------------------ acceptance of a positive claim.
~B[H{T}]:= I do not believe "god exists".------------ rejection of a positive claim.
B[H{0}]:= I believe "god does not exist".----------- acceptance of a negative claim.
~B[H{0}]:= I do not believe "god does not exist.----- rejection of a negative claim.

Note that: ~B[H{T}] is the rejection of the positive test hypothesis H{T} (where "rejecting" = "not accepting"). On the other hand, B[H{0}] is the denial of the positive test hypothesis H{T} (where "denying" = accepting that H{T} is not true (i.e. false)).
One need not satisfy any burden of proof for rejecting a claim, whether positive or negative in content, but for accepting (or asserting) that the claim in question is not true (i.e. false).

~B [H {T}] =/= B [H {0}]:
I do NOT believe "god exists" =/= I believe "god does NOT exist"


Definitions
Let: b{X}:= "I believe {X}" = I accept that X is true.
Then: ~b{X}:= "I do not believe {X}" = I do not accept that X is true (i.e., I reject that X is true)
Let: X:= a proposition, ~X = the negation of X (i.e., not X)
• b{X}:= I believe{X};
• ~b{X}:= I do not believe{X};
• b{~X}:= I believe {~X};
• ~b{~X}: = I do not believe {~X}.

Consider: X = "god exists",
then ~X = "god does not exist".
• H{0} = Null Hypothesis = ~X = "God does not exist."
• H{+} = Positive Hypothesis = X = "God exists."

An hypothesis that can be expressed in terms of an equality relation ("="), zero ("0"), or a negation ("not") is to be chosen as the null hypothesis. In our case, one (H{0}) of the two mutually opposing, exclusive, and exhaustive hypotheses carries the negation operator of formal logic ("not"); therefore, its alternative hypothesis expresses a positive proposition, hence H{+}. Therefore, in this case the negative claim "god does not exist" is to be chosen as the null, and the positive claim "god exists" is to be assigned the (alternative) positive hypothesis.

• b{X}: I believe god exists. ------------------- Theism
• ~b{X}: I do not believe god exists. ------------ Atheism (weak)
• b{~X}: I believe god does not exist.------------ Atheism (strong)
• ~b{~X}: I do not believe god does not exist.----- Rejection of Strong Atheism

• H{0}: God does not exist.
• H{+}: God exists.

Please note that a hypothesis cannot be a belief position, but it must necessarily be a hypothesis (H{0} = ~X or H{T} = X). The rejection of the null hypothesis is a belief position: ~b{~X} = ~b{H{0}} = not accepting that 'god does not exist', the null hypothesis being 'god does not exist'. Therefore, since atheism is a position on the proposition "god exists", where weak atheism rejects that proposition (fails to accept as true) and strong atheism denies that proposition. Denying X means accepting that the negation of X is true (i.e., accepting X is false), in contradistinction to rejecting X which simply means not accepting X as true.
H{0} can be falsified, but cannot be accepted; it can only fail to be rejected. In statistics, it is incorrect to accept the null hypothesis because of failing to be able to accept the (alternative) positive hypothesis.

H{+} cannot be falsified, but can be accepted if there were sufficient evidence constituting proof for a given standard of proof (degree of certainty), such as a "clear and convincing evidence", "beyond a reasonable doubt", "beyond a shadow of a doubt", "95% confidence interval", etc. If sufficient evidence is gathered, then the null can be rejected. If the evidence is insufficient to reject the null, one does not accept the null, but merely fails to reject the null.

The negative claim is by default the null hypothesis.
Null means zero in German. The null hypothesis be expressed in terms of the number 0:
ex., "The difference between the average IQ of men and that of women is zero"
ave{M} - ave{F} = 0

therefore, ave{M} = ave{F}
The null hypothesis able to be expressed in terms of 0 can be rendered into an equation. Therefore, a hypothesis which posits an equality relation is a null hypothesis. The alternative of the null hypothesis, also called test hypothesis or research hypothesis or simply "alternative hypothesis" must then posit some non-equality relation: inequality (=/=), but also could be strictly greater than (>), strictly less than (<), greater than or equal to (>=), or less than or equal to (<=). The hypothesis that we are testing that is central to our research is the one that does not carry the equal sign (=).

Similarly, a hypothesis that has the negation operator of formal logic "not" is also a null hypothesis. The test hypothesis then is the one of the two contradictories (H{0},H{T}) which does not carry the operator "not".

In statistics, it is said that the null hypothesis is never accepted, only rejected, when the evidence for its alternative test hypothesis (H{T}) reaches a statistically significant level.

Let us entertain the following null and test hypotheses: H{0} := "This drug has no effect" H{T} := "This drug has an effect"
It is easier to devise a controlled experiment to test the hypothesis H{T}, and virtually impossible to devise an experiment demonstrating that a drug does not have an effect; the effect may merely be too weak to be detectable.

That is why the null hypothesis is treated as a default hypothesis in the absence of evidence to the contrary, but the null is never accepted, but merely failed to be rejected. It is a matter of philosophical debate whether the null hypothesis is even provable at all. This is why statisticians always try to reject the null in favor of its alternative test hypothesis because the test hypothesis is provable.

Example:

Null Hypothesis (H {0}): [There is no god] = (God does not exist) = {No god exists}.
Test Hypothesis (H {T}): [There is a god] = (God exists) = {Some god exists}.

By definition: "god": = a supernatural immaterial being existing outside of time and space.

Let us entertain which of these mutually logically negating hypotheses is falsifiable (i.e., able to be proven false, disprovable, refutable).
First note that existence is a temporal condition. Nothing (not a thing) can exist outside of time. If you disagree then imagine god existing for 0 seconds. Moreover, since god is defined to be immaterial (not material) and supernatural (beyond nature), the test hypothesis is that "god exists" is unfalsifiable. The null hypothesis that "god does not exist" is falsifiable and can be falsified if there were sufficient evidence positively indicating proposition H{T} in exclusive concordance: H{T} must be exclusively concordant with the facts of the matter.
H {T}: = "God exists": not falsifiable because the null hypothesis cannot be accepted.
H {0}: = "God does not exist": falsifiable (by proving that god exists) but cannot be accepted since this is the null hypothesis.


I believe "god exists" =/= I do NOT believe "god does NOT exist"
B[H{T}] =/= ~B[H{0}]

One needs to meet a burden of proof for asserting a claim, not for rejecting one!
So, one who disbelieves that god exists has no burden of proof to meet.
One who disbelieves that god does not exist has likewise no burden of proof to meet.
One who asserts there is a god has a burden of proof.
One who asserts there is no god likewise has a burden of proof.

Absence of Evidence =|= Evidence of Absence
• Does this proposition always hold (true), or are there exceptions?
• Is it a logical fallacy to claim: “absence of evidence is evidence of absence”?
• How is this proposition related to the argument from ignorance fallacy? Explain.
• Burden of proof, null hypothesis, negative proof, and “one cannot prove a negative”!

The statement “Absence of Evidence is not Evidence of Absence” is correct. The mere absence that something is present is not evidence that something is absent. Lacking evidence for (the presence of) something does not constitute evidence of the absence of that thing.
Negative Proof and the Burden of Proof (Onus Probandi):
The burden of proof is on the proposition, not the opposition! The one who makes a claim carries the burden of proof, regardless of the positive or negative content of the claim.
One way in which one would attempt to shift the burden of proof is by committing the fallacy "the argument from ignorance" or "the argument from personal incredulity".

Negative proof by reductio ad absurdum (reduction to absurdity), such as a proof by contradiction or proof of impossibility, are typical methods to fulfill the burden of proof for a negative claim.

A proof by contradiction is a valid rule of inference called modus tollens (also proof by contraposition):
Negative Proof and the Argument from Ignorance:
The argument from ignorance: "some proposition X is true because it has not (yet) been proven false," or "some proposition is false because it has not (yet) been proven true."

To assert that "absence of evidence is evidence of absence" is a logical fallacy called the argument from ignorance.
"Something is concluded to be absent because it has not been proven to be present," or
"Something is concluded to be present because it has not been proven to be absent."
Example:

Proposition G: "God exists" is accepted as true because G has not been proven false.
That is, the lack of sufficient evidence capable of constituting proof of god's existence is not sufficient evidence constituting proof of the non-existence of god.
Proposition G: "God exists" is concluded to be true because its negation ~G: ("God does not exist”) has not been proven true.



Negative Proof through Negative Claims:
A negative claim is the opposite (negation) of an affirmative (positive claim). A negative claim asserts the non-existence or exclusion of something. For a positive claim, only a single example is required to demonstrate the positive claim.
Negative Proof through Negative Evidence:

Absence of evidence: ex., no careful research has been done.
Evidence of absence: ex., an observation that suggests there were no dragons in my garage.
The difference between absence of evidence and evidence of absence lies in whether investigation (i.e. scientific experiment) would have detected the phenomenon if it were there.

Modus Tollens (relies on the contrapositive of the original implication to be equivalent to it).
Premise (1): P -> Q.
Premise (2): ~Q
_____{then}_______
Conclusion: ~P.

(which reads...)

Premise (1) If P, then Q.
Premise (2) Not Q.
_______{then}_______
Conclusion: Not P.

Please note that ‘arguing from ignorance’ for ‘absence of evidence’ is not necessarily fallacious!
Example:

"This drug has no long-term risks, until proven otherwise."
Were such an argument to rely imprudently on the lack of research to promote its conclusion would be considered an informal fallacy (argument from ignorance).
However, in carefully designed scientific experiments, even null results can count as evidence of absence, though not conclusive proof (in and of themselves). A hypothesis may be falsified if a vital predicted observation is not found empirically.

Therefore, in cases where there should be evidence if the hypothesis were true, absence of evidence can count as evidence (not proof) of absence, depending on the detection power of the experiment (including instruments), the confidence of the inference, limiting confirmation bias, etc. Therefore, the argument from ignorance for "absence of evidence" is not necessarily an informal fallacy.

The negative claim is by default the null hypothesis. Null means zero in German.
The null hypothesis be expressed in terms of the number 0:

ex., "The difference between the average IQ of men and that of women is zero"
ave{M} - ave{F} = 0
therefore, ave{M} = ave{F}

The null hypothesis able to be expressed in terms of 0 can be rendered into an equation. Therefore, a hypothesis which posits an equality relation is a null hypothesis. The alternative of the null hypothesis, also called test hypothesis or research hypothesis or simply "alternative hypothesis" must then posit some non-equality relation: inequality (=/=), but also could be strictly greater than (>), strictly less than (<), greater than or equal to (>=), or less than or equal to (<=). The hypothesis that we are testing that is central to our research is the one that does not carry the equal sign (=).
Similarly, a hypothesis that has the negation operator of formal logic "not" is also a null hypothesis. The test hypothesis then is the one of the two contradictories (H{0},H{T}) which does not carry the operator "not".

In statistics, it is said that the null hypothesis is never accepted, only rejected, when the evidence for its alternative test hypothesis (H{T}) reaches a statistically significant level.
Let us entertain the following null and test hypotheses:
• H{0} := "This drug has no effect"
• H{T} := "This drug has an effect"
It is easier to devise a controlled experiment to test the hypothesis H{T}, and virtually impossible to devise an experiment demonstrating that a drug does not have an effect; the effect may merely be too weak to be detectable.
That is why the null hypothesis is treated as a default hypothesis in the absence of evidence to the contrary, but the null is never accepted, but merely failed to be rejected. It is a matter of philosophical debate whether the null hypothesis is even provable at all. This is why statisticians always try to reject the null in favor of its alternative test hypothesis because the test hypothesis is provable.
• Null Hypothesis (H{0}): [There is no god] = (God does not exist) = {No god exists}.
• Test Hypothesis (H{T}): [There is a god] = (God exists) = {Some god exists}.
By definition: "god": = a supernatural immaterial being existing outside of time and space.
Let us entertain which of these mutually logically negating hypotheses is falsifiable (i.e., able to be proven false, disprovable, refutable).
First note that existence is a temporal condition. Nothing (not a thing) can exist outside of time. If you disagree then imagine god existing for 0 seconds. Moreover, since god is defined to be immaterial (not material) and supernatural (beyond nature), the test hypothesis is that "god exists" is unfalsifiable. The null hypothesis that "god does not exist" is falsifiable and can be falsified if there were sufficient evidence warranting belief in H{T}, where evidence is points of data positively indicative of and exclusively concordant with the hypothesis that god exists.
H{T}: = "God exists": not falsifiable because the null hypothesis cannot be accepted.
H{0}: = "God does not exist": falsifiable (by proving that god exists) but cannot be accepted since this is the null hypothesis.









Is Atheism the Null Hypothesis?
Atheism has distinct definitions which can be categorized as follows:
• Weak/Soft Atheism: I do not believe "god exists". Weak atheism is the rejection of the positive claim that "god exists".
• Strong/Hard Atheism: I believe "god does not exist". Strong atheism is the acceptance of the negative claim that "god does not exist".

Strong atheists are a subset of weak atheists:
those who believe god does not exist form a subset of those who do not believe god exists.

Definitions
Let: b{X}:= "I believe {X}" = I accept that X is true.
Then: ~b{X}:= "I do not believe {X}" = I do not accept that X is true (i.e., I reject that X is true)
Let: X:= a proposition, ~X = the negation of X (i.e., not X)
• b{X}:= I believe{X};
• ~b{X}:= I do not believe{X};
• b{~X}:= I believe {~X};
• ~b{~X}: = I do not believe {~X}.

Consider: X = "god exists",
then ~X = "god does not exist".
• H{0} = Null Hypothesis = ~X = "God does not exist."
• H{+} = Positive Hypothesis = X = "God exists."

An hypothesis that can be expressed in terms of an equality relation ("="), zero ("0"), or a negation ("not") is to be chosen as the null hypothesis. In our case, one (H{0}) of the two mutually opposing, exclusive, and exhaustive hypotheses carries the negation operator of formal logic ("not"); therefore, its alternative hypothesis expresses a positive proposition, hence H{+}. Therefore, in this case the negative claim "god does not exist" is to be chosen as the null, and the positive claim "god exists" is to be assigned the (alternative) positive hypothesis.
• b{X}: I believe god exists. ------------------- Theism
• ~b{X}: I do not believe god exists. ------------ Atheism (weak)
• b{~X}: I believe god does not exist.------------ Atheism (strong)
• ~b{~X}: I do not believe god does not exist.----- Rejection of Strong Atheism
• H{0}: God does not exist.
• H{+}: God exists.


Please note that a hypothesis cannot be a belief position, but it must necessarily be a hypothesis (H{0} = ~X or H{T} = X), so a proposition. The rejection of the null hypothesis is a belief position: ~b{~X} = ~b{H{0}} = not accepting that 'god does not exist', the null hypothesis being 'god does not exist'. Atheism is a position on the proposition "god exists", where weak atheism rejects that proposition (fails to accept as true) and strong atheism denies that proposition. Denying X means accepting that the negation of X is true (i.e., accepting X is false), in contradistinction to rejecting X which simply means not accepting X as true.
H{0} can be falsified, but cannot be accepted; it can only fail to be rejected. In statistics, it is incorrect to accept the null hypothesis because of failing to be able to accept the (alternative) positive hypothesis.

H{+} cannot be falsified, but can be accepted if there were sufficient evidence constituting proof for a given standard of proof (degree of certainty), such as a "clear and convincing evidence", "beyond a reasonable doubt", "beyond a shadow of a doubt", "95% confidence interval", etc. If sufficient evidence is gathered, then the null can be rejected. If the evidence is insufficient to reject the null, one does not accept the null, bu