Thursday, August 23, 2018

Moral Ought Facts are Non-Natural

Introduction



In my third Maverick Christian Vlog episode I refer to a scholarly paper which is called A Folk Semantics Argument for Moral Non-Naturalism. In this blog entry I’ll provide some of the technical background so that those of us who aren’t analytic philosophers can better understand it.

Why is it important that morality is non-natural? One reason is that it reveals that there is more to reality beyond the natural, physical world. Another reason is that morality being non-natural makes it so that atheism doesn’t fit in very well with the existence of morality, especially objective morality for reasons I explain in my third vlog episode. In contrast, the existence of an objective and non-natural morality makes perfect sense in a theistic worldview.

Next I’ll explain some philosophy lingo before explaining the math used in the paper.

Philosophical Terminology



Moral semantics is about how to define moral terms. In philosophy, the word “folk” refers to colloquial stuff that laypersons use; e.g. “folk psychology” is (an albeit derogatory) term for beliefs about the human mind that ordinary people accept. In the paper, “folk semantics” with respect to morality refers to what most ordinary people mean when they use terms like “morally wrong.”

A stipulative definition assigns a meaning to a particular word or phrase to be used in a given context (as a philosophy paper). For example, in a philosophy paper one might give a stipulative definition of “fully justified” by saying, “I will say that a belief is fully justified to denote the belief being justified to the point where one can rationally say one knows it to be true.” Stipulative definitions are often used for conveniently assigning a label to some concept and won’t necessarily match the lexical (“dictionary”) definition.

A hypothetical imperative takes the form of something like, “If you want to do X, you should do Y” and describes what to do as a matter of practical necessity to accomplish some goal. For example, “If you want to do well in school, you ought to study” meaning something like, “As a matter necessity, you need to study to do well in school.” The sort of ought used in hypothetical imperatives is called a hypothetical ought.

A category mistake (or category error) is attributing a characteristic to something that it can’t possibly have because it’s not of the right category; e.g. saying that the number six has mass or volume, when the category of abstract objects is such that they can’t have mass or volume.

Set Theory



Some Basics



Sets are collections of stuff where order and duplicates are irrelevant. For example, the followings sets are all identical.

{1, 2, 3, 4}
{1, 2, 2, 3, 4}
{4, 3, 2, 1}

There’s the empty set, sometimes symbolized as {} which is a set that has no members.

To illustrate some set operations, suppose our “universe” consists entirely of natural numbers 1 through 9. Now let A, B, and C be the following:
A = {1, 5, 9}
B = {1, 5, 7, 8}
C = {2, 3}
SymbolExampleExplanation

(element of)
1 ∈ AFor any set S, x ∈ S means that x is an element of S.

(not an element of)
1 ∉ CFor any set S, x ∉ S means that x is not an element of S.

(intersection)
A ∩ B = {1, 5}Given sets S and T, S ∩ T contains all the elements x such that x ∈ S and x ∈ T.

(union)
A ∪ B = {1, 5, 7, 8, 9}Given sets S and T, S ∪ T contains all the elements x such that x ∈ S or x ∈ T.

(subset)
{1, 5} ⊆ BGiven sets S and T, S is a subset of T if and only if each member of S is also a member of T.

(not a subset)
{2, 9} ⊄ BGiven sets S and T, S is not a subset of T if and only if it is not the case that S ⊆ T.


The set “All x such that x > 3” can be symbolized like this:

{ x | x > 3 }

The set “All x ∈ A such that x > 3” can be symbolized as:

{ x ∈ A | x > 3 }

That set described above would be {5, 9}.

Relations



Unlike sets were order and duplicates don’t matter, they do matter in tuples. The following are all different from each other:
(1, 2, 3, 4)
(1, 2, 2, 3, 4)
(4, 3, 2, 1)
Those who have taken algebra might remember the tuple known as the ordered pair:
(2, 3)
(11, -3)
Relations are sets of tuples, with a binary relation being a set of ordered pairs. For example, suppose we have this set:
{Diana, Steve, Barbara}
The relation “taller-than” could consist of this set of ordered pairs, where e.g. Diana is taller than Steve.
{(Diana, Steve), (Steve, Barbara), (Diana, Barbara)}
If we symbolize our taller relation as T then we could say that (Diana, Steve) ∈ T.

Relations between different sets are also possible. Suppose we have these two sets:
L = {Reed, Scott, Clark}
F = {Sue, Jean, Lois}
And the “is-husband-of” relation is a relation from set L to set F; e.g. Reed is the husband of Sue:
H = {(Reed, Sue), (Scott, Jean), (Cark, Lois)}
An inverse of a binary relation R goes like this:
R-1 = {(y, x) | (x, y) ∈ R}
For example, the inverse of the “is-husband-of” relation would be the “is-wife-of” and be this:
H-1 = {(Sue, Reed), (Jean, Scott), (Lois, Clark)}
A relation from set A to set B is a function if each member of A is paired off with exactly one member of B. The “input” part of a function is the domain (set A) and the “output” part is called the range (set B). For instance, the “is-husband-of” relation is a function because each member L is paired off with exactly one member of F, with L being the domain and F being the range, whereas an “is-husband-of” relation would not be a function if there were polygamous marriages.

Suppose relations S and T are the following:
S = {(1, 2), (10, 11)}
T = {(2, 3), (11, 12)}
A composition of two relations S and T can be symbolized as S ∘ T, and when the relations are binary, the set of ordered pairs in such a composition goes like this:
{(x, z) | (x, y) ∈ S and (y, z) ∈ T}
In our example, S ∘ T would be the following:
{(1, 3), (10, 12)}
Suppose relation V is the following:
V = {(1, 2), (1, 3), (1, 9), (2, 3), (2, 4)}
Because the relation is binary, V(x, ⋅) is { y | (x, y) ∈ V }

Examples:
V(1, ⋅) = {2, 3, 9}
V(2, ⋅) = {3, 4}


Formal Logic



Deductive Arguments



A deductive argument tries to show that it’s logically impossible (i.e. self-contradictory, like a married bachelor) for the argument to have true premises and a false conclusion, and thus that the conclusion follows from the premises by the rules of logic. If it’s logically impossible for an argument to have true premises and a false conclusion the argument is deductively valid or valid. An example of a deductively valid argument:
  1. If it is raining, then my car is wet.
  2. It is raining.
  3. Therefore, my car is wet.
The above example uses a famous rule of logic called modus ponens which has this structure:
  1. If P, then Q
  2. P
  3. Therefore, Q.
Another famous rule of logic is called modus tollens where “not-Q” means “Q is false.”
  1. If P, then Q
  2. Not-Q
  3. Therefore, not-P.
An argument is deductively invalid or invalid if it is not deductively valid. An example of an invalid argument:
  1. If it is raining, then my car is wet.
  2. My car is wet.
  3. Therefore, it is raining.
In logic lingo, a deductively valid argument with all its premises being true is called a sound argument. And since a valid argument having true premises guarantees the truth of its conclusion, a sound deductive argument has a true conclusion.

Basic Symbols and Rules of Inference



Here’s a summary of how the connectives in propositional work where p and q represent propositions (claims that are either true or false):

Type of
connective
EnglishSymbolic
Logic
When it’s true/false
Conjunctionp and qp ∧ qTrue if both are true; otherwise false
Disjunctionp or qp ∨ qFalse if both are false; otherwise true
ConditionalIf p, then qp → qFalse if p is true and q is false; otherwise true
NegationNot-p¬pTrue if p is false; false if p is true


As suggested in the above table, the symbols →, ¬, ∨, and ∧ are called connectives. It’s a somewhat misleading name since ¬ doesn’t connect propositions even though the other three connectives do. Still, it’s a popular label a lot of logic textbooks use. While the terminology varies among writers, I’ll call a single letter a simple statement and one more or more simple statements with one or more connectives is called a compound statement. For example, “¬P” and “A ∧ B” are compound statements.

The type of conditional (pq) being used here is called a material conditional. A material conditional is equivalent to “It is not the case that the antecedent (p) is true and the consequent (q) is false,” such that the only way for a material conditional to be false is for it to have a true antecedent with a false consequent. A material conditional might seem like a pretty weak claim (in the sense that it doesn’t claim very much), since the antecedent and consequent don’t even have to be related to each other for a material conditional to be true. Thus, “If there is a married bachelor, then Minnesota is awesome” constitutes a true material conditional since it is not the case that we have a true antecedent (there is a married bachelor) with a false consequent (Minnesota is awesome). But it turns out that a material conditional is enough for modus ponens and modus tollens to be valid rules of inference, since in a true material conditional if the antecedent is true, then the consequent is true as well.

Speaking of which, here are those rules of inference I’ve already mentioned in symbolic form:

modus ponens
 
In English In Symbolic Logic
If p then q
p

Therefore, q
p → q
p

∴ q
modus tollens
 
In English In Symbolic Logic
If p then q
Not-q

Therefore, not-p
p → q
¬q

∴ ¬p


In the convention I’m using, the lower case letters p, q, r,...z are placeholders for both simple and compound statements. Thus, below is a valid instance of modus tollens.
  1. (A ∧ B) → C
  2. ¬C

  1. ¬(A ∧ B) 1, 2, modus tollens
It’s worth noting that the order of the premises doesn’t matter when using rules of inference. So below is also a valid use of modus tollens.
  1. ¬C
  2. (A ∧ B) → C

  1. ¬(A ∧ B) 1, 2, modus tollens
Some rules of inference can be used in more than one way. Examples include disjunctive syllogism and simplification.

Disjunctive Syllogism
 
In English In Symbolic Logic
p or q
Not-p

Therefore, q
p ∨ q
¬p

∴ q
p or q
Not-q

Therefore, p
p ∨ q
¬q

∴ p
simplification
 
In English In Symbolic Logic
p and q

Therefore, p
p ∧ q

∴ p
p and q

Therefore, q
p ∧ q

∴ q


Before moving forward, I’ll introduce a quick example of how to use some rules of inference. Suppose we wanted to get C from premises 1 and 2 below:
  1. A ∨ (B ∧ C)
  2. ¬A

  1. B ∧ C 1, 2, disjunctive syllogism
  2. C 3, simplification
Not too hard, right? After learning the above rules of inference, you might even have mentally “seen” that C followed from premises 1 and 2 above. Hopefully you are familiar enough with the symbols by now for me to remove the training wheels of english translation. Some more rules of inference:

conjunction
 
p
q

∴ p ∧ q
hypothetical syllogism
 
p → q
q → r

∴ p → r


Equivalences



In propositional logic, two statements are logically equivalent whenever the connectives make it so that they’re always the same truth-value (i.e. both true or both false). Some rules of propositional logic are themselves equivalences, such as these:

equivalencename of equivalence
 
p ⇔ ¬¬pdouble negation
 
p → q ⇔ ¬q → ¬ptransposition (also called contraposition)
 
¬(p ∧ q) ⇔ ¬p ∨ ¬qDe Morgan’s laws
¬(p ∨ q) ⇔ ¬p ∧ ¬q


Equivalence rules can be used to replace stuff “inline” whenever their equivalence appears. As an example of how to use some equivalences, suppose we want to prove ¬C ∨ ¬D from premises 1 and 2 below:
  1. A
  2. (C ∧ D) → ¬A

  1. ¬¬A → ¬(C ∧ D) 2, transposition
  2. A → ¬(C ∧ D) 3, double negation
  3. ¬(C ∧ D) 1, 4 modus ponens
  4. ¬C ∨ ¬D 5, De Morgan’s laws

Conditional Proofs



The conditional is symbolized as p → q where p is called the antecedent and q is called the consequent. The conditional proof aims to prove that a conditional is true, with the antecedent of the conditional being the conditional proof assumption which is often used to help show that if the antecedent is true then the consequent is true also. The structure of a conditional proof takes the following form below:

conditional proof
 
a) p conditional proof assumption
b)
 ...
 q
c) p → q a-b, conditional proof


For example, suppose we want to prove A → (B ∧ C) from premises 1 and 2 below:
  1. A → B
  2. A → C

  1. A conditional proof assumption
    1. B 1, 3, modus ponens
    2. C 2, 3, modus ponens
    3. B ∧ C 4, 5, conjunction
  1. A → (B ∧ C) 3-6, conditional proof
Notice that the validity of a conditional proof does not rely on the conditional proof assumption actually being true; rather it relies on the fact that if it is true then it properly leads to the consequent. Nothing in the proof above, for example, relies an A actually being true.

Predicate Logic



To give an example of predicate logic, consider the following symbolization key:

B(x) = x is a Bachelor.
U(x) = x is Unmarried.


The letters B and M in these examples are predicates which say something about the element they are predicating. Sometimes parentheses aren’t used; e.g. Bx being used to mean “x is a bachelor.” The symbol; ∀ means “For All” or “For Any” such that the following basically means “All bachelors are unmarried:”

universal quantification
 
In English In Symbolic Logic
For any x: [if x is B, then x is U] ∀x[B(x) → U(x)]


The domain of discourse is the set of things we’re talking about when we make statements like ∀x[B(x) → U(x)], such that the “∀x” means “For any x in the domain of discourse (i.e. set of things we’re talking about here).” We can let an individual lowercase letter signify a specific element in our domain of discourse; e.g. c can signify a guy named “Charles” and we can let B(c) to signify c is B (i.e. Charles is a bachelor).

A rule of predicate logic called Universal Instantiation allows us to instantiate a universal quantification (a ∀x[...] statement) for a specific individual, like so:
  1. ∀x[B(x) → U(x)]
  2. B(c)

  1. B(c) → U(c) 1, universal instantiation
  2. U(c) 2, 3, modus ponens
There’s a somewhat complicated rule called universal generalization (also called universal introduction) to get a universal quantification statement. Roughly, the idea is that if a statement contains some variable that is a placeholder for anything in the domain of discourse, we can generalize this to get “For any x, such-and-such holds true.” The universal generalization rule is fairly complicated (you can only use it under certain specified conditions) but the gist of universal generalization should be enough to follow along this proof.
  1. ∀x[A(x) → B(x)]
  2. ∀x[B(x) → C(x)]

  1. A(t) → B(t) 1, universal instantiation
  2. B(t) → C(t) 2, universal instantiation
  3. A(t) → C(t) 3, 4, hypothetical syllogism
  4. ∀x[A(x) → C(x)] 5, universal generalization
And that should be all the technical stuff you need to know to read A Folk Semantics Argument for Moral Non-Naturalism. You still might not find it an easy read if you’re not used to analytic philosophy, but at least you have the background knowledge even if applying it is a bit tricky.

6 comments:

  1. I read through your proof and thought I would offer some commentary.

    I am not an expert in analytical logic set theory, and have no particular desire or motivation so to be. So, my remarks are not on those grounds.

    What I find to be problematic in this description as I understand it is that there is a disjunction or "disunity" between a moral law and the physical repercussion. In other words, the objection is somewhat teleological. This teleology, however, extends from the vertical space as well as the horizontal space. That is to say, from a dimension of timelessness a linear dimension of time can be deduced such that the cause and effect of the timeless dimension become apparent. In such a system, there is a direct mirroring between the physical occurrence and the divine occurrence.

    Set theory is fine for examining things that occur on coordinate systems, but it is not so good for examining things that come from systems that are sans axis. Indeed, the axis unfolds from such a dimension.

    This view, of course, obliterates the argument you advance because the argument is presupposing a separation presumably as an argument against atheism. Of course, the argument against atheism are the fruits atheism yields--not separating out dimensions that cause some kind of spiritual dementia.

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    Replies
    1. I am not an expert in analytical logic set theory, and have no particular desire or motivation so to be.

      That's too bad. Set theory is awesome! 🙂 Certainly anyone interested in analytic philosophy should at least learn the basics (and there isn't much in the above blog article that isn't basic vis-à-vis set theory).

      This view, of course, obliterates the argument you advance because the argument is presupposing a separation presumably as an argument against atheism.

      For the argument to be “obliterated” there needs to be a false premise, because if my deductive argument has all true premises then the conclusion follows whether you like it or not (such is the magic of logic). So if you think the argument is unsound, which premise is false?

      The argument as such isn't really an argument against atheism so much as an argument against ethical naturalism (the view that ethical facts are natural facts). That said, I do think ethical non-naturalism does pose serious problems for atheism, though such problems are outside the scope of the paper.

      What I find to be problematic in this description as I understand it is that there is a disjunction or "disunity" between a moral law and the physical repercussion.

      The paper doesn't say either way whether such a disunity exists.

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    2. "Why is it important that morality is non-natural? One reason is that it reveals that there is more to reality beyond the natural, physical world. Another reason is that morality being non-natural makes it so that atheism doesn’t fit in very well with the existence of morality, especially objective morality for reasons I explain in my third vlog episode. In contrast, the existence of an objective and non-natural morality makes perfect sense in a theistic worldview."

      This statement implies that the division between morality and the natural world exists as a presupposition or at least a rationale for making the argument you advance.


      "For the argument to be “obliterated” there needs to be a false premise, because if my deductive argument has all true premises then the conclusion follows whether you like it or not (such is the magic of logic). So if you think the argument is unsound, which premise is false?"

      The gift of the spirit is discernment. YHSVH, the Messiah, does not bid us to "use the magic of logic" to prove the things of heaven. Rather, he bids us use the gifts of the spirit in the form of discernment granted by His sacrifice.

      So, if your goal is to convert atheists who use rational thought, great, but you are already stepping outside the bounds of the Messiah by trying to rely on "logic". It is already stated that logic, in and of itself is insufficient, or else it would have already have been done and there would have been no need of a Messiah. This premise obliterates the reason for your argument at all.

      Additionally, the fact you try to postulate morality as separate from the physical world would deny the logical consequence of having a physical savior in flesh and blood form mediate for moral choices which have physical repercussions--namely in the form of sin. This is what I meant by my original objection. You will notice neither of these objections necessitate a rigorous understanding of set theory or logic--just as the simple disciples of YHSVH did not possess such a tool.

      This leaves your argument bankrupt on the grounds of Biblical precepts. YHSVH never said to go about "arguing with atheists". If anything, He said if your simple teachings are not accepted, you should kick the dust off of your heels.

      If you are trying to be academic, then I suppose your argument is fine, but then what is theology doing in there? You will have to pick between books with the approach you are here using. Either you will be a disciple of YHSVH and so follow His words, or you will follow the World of Man and try to glorify yourself as some sort of "Atheist savior" by using methods the Bible suggests won't work. Either way, it makes the argument advanced here purposeless.

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    3. YHSVH, the Messiah, does not bid us to "use the magic of logic" to prove the things of heaven....you are already stepping outside the bounds of the Messiah by trying to rely on "logic"....YHSVH never said to go about "arguing with atheists".

      Arguing along the lines of "Jesus never told us to do x so we shouldn't do x" is terrible reasoning. Jesus also never told us to use flowers at a wedding, but that hardly implies that God forbids it.

      Additionally, the fact you try to postulate morality as separate from the physical world would deny the logical consequence of having a physical savior in flesh and blood form mediate for moral choices which have physical repercussions--namely in the form of sin.

      Moral properties like moral wrongness not being a completely physical property does not imply that Jesus wasn't necessary for salvation.

      It is already stated that logic, in and of itself is insufficient, or else it would have already have been done and there would have been no need of a Messiah. This premise obliterates the reason for your argument at all.

      Insufficient for what? Unfortunately, much of what you say is vague and is deficient in lucid coherency (I suggest learning to write more clearly). If you intended to say that logic by itself is insufficient for salvation, then I agree. But I never said nor implied otherwise, and it certainly doesn't obliterate the reason I constructed the argument for ethical non-naturalism. The reason I made the argument is because I think ethical non-naturalism is true and that it's better that we have an accurate understanding of what morality is. As a bonus, this argument also has some value for apologetics.

      It would be a mistake to think that apologetics is not Biblical. First there's the value in destroying bad arguments against the faith. 2 Corinthians 10:5 says, "We demolish arguments and every pretension that sets itself up against the knowledge of God, and we take captive every thought to make it obedient to Christ." Second, there are multiple instances in the New Testament where followers of Jesus employ reason and evidence for the faith, such as in Acts 17:22–34. We see Paul using apologetics also in Acts 9:22. In Acts 14:3 evidential confirmation for the message of Paul and Barnabas was done via miraculous signs and wonders. We see that apologetics convinced some people in Acts 17:2-4. Acts 19:8 says that Paul argued persuasively about the Kingdom of God.

      You could argue that apologetics is good only for those who have some belief in God, but why on earth should anyone believe this? It's not as if such arguments can't be convincing; C.S. Lewis didn't believe in God at one point but reason convinced him, and the Christian intellectualism of C.S. Lewis had a profoundly good effect for the faith.

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    4. "Arguing along the lines of "Jesus never told us to do x so we shouldn't do x" is terrible reasoning. Jesus also never told us to use flowers at a wedding, but that hardly implies that God forbids it."

      If you wish to rest on Paul, then the statement everything is permitted but not everything is beneficial is the axiom upon which I would base my statement.

      Furthermore, you completely skip the part where I talk about the discernment of the spirit in order to argue with me. You are of a divisive spirit, and your works will only serve to augment that divisiveness. Whatever other reason you think you are doing the work you are doing, this is enough to understand it has nothing to do with the Messiah as He has already come to make the division. Put simply, you are "puffed up" and are trying to use your intellect in ways the spirit will neither need nor desire. Because of this, your criticism of my writing style I will take as an attack of pride rather than as constructive criticism.

      Like Paul, you will need to be knocked from your high horse. You are already blinded.


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    5. If you wish to rest on Paul, then the statement everything is permitted but not everything is beneficial is the axiom upon which I would base my statement.

      That claim is also insufficient to justify not using logic to argue for Christianity. And it ignores the evidence I gave suggesting that apologetics is beneficial; e.g. the conversion of C.S. Lewis and the conversions mentioned in Acts 17:2-4.

      You are of a divisive spirit, and your works will only serve to augment that divisiveness.

      Divisive how? Like going on someone else's blog to insinuate he's a bad Christian on the basis of insufficient evidence?

      Put simply, you are "puffed up" and are trying to use your intellect in ways the spirit will neither need nor desire.

      I make no claims of grandeur, and if you think the intellect isn't to be used for apologetics, what of the Biblical evidence I offered? You completely ignored that.

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