Saturday, April 12, 2014

Extending the Power of Logical Argument: Coherent Propositions and Truth Functions

While some may question the extensive application of logic (see e.g. my previous post on logical fallacies) to human disputation, it is important to bear in mind logical rigor doesn't come naturally to most humans. We are not all 'Mr. Spocks', after all, nor is it likely our significant others would want that. Hence, when we are confronted by a debate or having to make a logical and coherent case we have to be prepared to summon great energy and often investment of time to ensure: a) we don't mix 'apples' and 'oranges', and b) we consistently use each in its proper reference frame.

A certain level of rigor or discipline then must be applied to ensure coherent, critical thinking, especially taking care that  truth categories are not mixed up and that claims of truth aren't confused with attitudes to truth! Sadly, most who pontificate on matters of religion, or economics,  have probably never taken a logic or philosophy course in their lives, so end up flying by the seat of their pants - and end up crashing and burning.

The purpose of this post is to extend the power of logical argument beyond the basics laid out in recognizing assorted logical fallacies.  I do not promise the going will be 'easy' only that the energetic reader should be much better able to handle him (her-)self in a debate or argument afterward.


1) One cannot assume a "logical deduction" unless there exists a relationship between the premise and the conclusion such that we have a right to expect the conclusion IF we know the premise is true (cf. Russell, Introduction to Mathematical Philosophy, p. 145)

Said relationship then ought to be between analogous truth claims OR attitudes,  but not mixing up both.

2) From (1) it follows that two truth functions which have the same truth value for all values of the argument are indistinguishable. Thus, p and q are the negations of not-p and not-q.

In terms of propositional functions, let q = S(A) and p = S(B). Then if p = f(q) the contradictory hypothesis is p' = f(~q) .

Example: if p = f(q) = "The Bible is the actual word of God verbatim and cannot have either errors or contradictions"

BUT - if one error or contradiction is found in it, we must have: p' = f(~q) or:

"The Bible has a contradiction in it, so cannot be the actual word of God, but of mentally-limited humans."

3) Five basic truth functions exist (op. cit.)and are not all independent:

i) The 'negator': Not-p

Expresses that function of p which is true when p is false, and that which is false when p is true.

Note: The truth of a proposition or its falsehood is referred to as its "truth value". This can be either TRUE or FALSE, i.e. the true value of a true proposition is true, and the falsehood of a false proposition.

Thus: Mars has two Moons, Deimos and Phobos, is a true proposition, and

Saturn does not have 101 Moons, is a true proposition

ii) The disjunction: p OR q:

This is a function whose truth value is 'truth' when p is true and also when q is true. (It can be mechanically-electronically embodied by the logical OR - gate)

But the truth value is falsehood when both p and q are false.

iii) The conjunction: p AND q:

Has 'truth' for its truth value when BOTH p and q are true, otherwise falsehood.

It is denoted by the logical AND gate.

Example: p : "Solar flares erupt with large areas and temperatures in the millions of degrees"

q: "Solar flares result from magnetic instability in the associated plasma"

iv) Incompatibility: ~ (p /\ q)

E.g. p and q are not both true, or the negation of conjunction (iii). It is also the disjunction of the negations of p and q, i.e. not-p or not-q. The truth value appends 'truth' when p is false, and also when q is false. Likewise, the truth value appends falsehood if both are true.

v) Implication: p->q (p implies q) or "if p, then q"

Example: If the New Moon is aligned with the Sun we can get a total eclipse of the Sun

Here: p = alignment of Sun and New Moon

q = total eclipse of the Sun

This can be generalized in various ways, e.g.

"Unless p is false, q is true" OR

"Either p is false or q is true"

4) Further generalizations are possible using (1)-(3), e.g.

a) Negation is the incompatibility of a proposition with itself, or p/p

E.g. "the Apollo astonauts brought back Moon rocks made of green cheese"

b) Disjunction is the incompatibility of not-p and not-q or (p/p)[(q/q)

Example: "Saturn is not a very large or hot star"

c) Implication is the incompatibility of p and not-q, or p[(q/q)

Example: "If the Moon rises tonight, the Sun will not rise tomorrow"

d) Conjunction is the negation of incompatibility, e.g. (p/q)[

Example: "Saturn and Earth are both member planets of the solar system"

5) Gödelian Truth Limits:

Consider a simple statement of logical transitivity:

X = Y

Y = Z

therefore X = Z

What if instead we append an axiomatic statement that reads, in effect: "X=Y is unprovable-in-the-system". If this statement is provable-in-the-system, we get a contradiction, since if it is provable in-the-system, then it can’t be unprovable-in-the-system. This means the original axiom: "X= Y is unprovable-in-the-system" is false. Since in any consistent system nothing false can be proven in-the-system, the proposition is undecidable.

Now, given all these basics, let's see if we can apply them to the assertions or alleged propositions made by a recent blogger:

Example (1):

"All TRUTH is exclusive ! The truth that "two plus three equals five" is very exclusive too . It does not allow for any other conclusion ."

The basic and most fundamental error is that he mixes up an absolute general statement, with a particular one. "All Truth is exclusive" is in fact unprovable by the same set of axioms or axiomatic statements that would prove "2 + 3 = 5". The reason is that the first is a meta-statement about the WHOLE Truth system! To be more concise, the offender is rendering an ALL-inclusive truth statement about what he insisted is an absolute exclusivity! He claims "all truth is exclusive" but his statement itself is all-inclusive and implied to be the exception to his own proposition! Thus it violates the Gödelian Truth Limits (5).

It helps here to present a more concrete illustration and example, to show why the most general statement is false. We take here the original claim of Epimenides in his “all Cretans are liars” paradox:

"All Cretans are Liars"

If the speaker is a Cretan, then the statement is ipso facto unresolvable. If Cretan, he exists within the so-called abstract, formal system. Yet, he’s making a statement (meta-) about the system. Hence, is he lying? Or is he telling the truth? This cannot be resolved. An undecidable proposition, as Gödel’s Incompleteness Theorem (II) applies.

In like manner, the statement "All Truth is exclusive" emerges as an undecidable proposition at the very least. (And it may also be false in the framework earlier considered).

Scott Soames in his monograph Understanding Truth further clarifies the issue of more and less general schema to arrive at truth, and what is “materially adequate” p. 69:

The characterization of individual instances of (different) schema has consequences for more general definitions of truth. If such instances (e.g. L1 statements) are thought of as partial definitions, then the task of defining truth for an entire language may be seen as finding a way of generalizing the partial definitions so as to cover every sentence of the language.”

In other words, in order to make the claim: "All Truth is exclusive" one would have to: a) have access to ALL COMPLETE truth statements that can exist in the universe, and b) show that no two of them exhibit disjunction, or incompatibility of other problems. In addition, one would have to articulate his proposition as compatible with the claim: in other words, one cannot formulate an all-inclusive truth statement to apply to one for which "All Truth is exclusive". This is not feasible, because no human mind is capable of formulating an absolute exclusive truth statement via an all-inclusive truth claim.

Consider the following statements referring to solar flares, and note the L1 hierarchy that presents:

1) A class X solar flare occured on the Sun last Tuesday.

2) A class X-7 solar flare occurred on the Sun at 22h 33m GMT last Tuesday.

3) A class X-7, optical class 2B solar flare occurred on the Sun at 22h 33m GMT last Tuesday.

4) A class X-7, optical class 2B solar flare occurred on the Sun at 22h 33m GMT last Tuesday and lasted a total duration of 1440 seconds.

5) A class X-7, optical class 2B solar flare occurred on the Sun at 22h 33m GMT last Tuesday, peaked 543 seconds after inception, and lasted a total duration of 1440 seconds.

Now, are ALL of the above statements (referencing the same event) EXCLUSIVELY true? Or better, are they all EQUALLY true? If not, why not? Can one therefore have true statements which do not express the entire truth but rather only a partial truth? Obviously, the answers are in succession: NO - they can't all be exclusively true (on account of factual intersection), NO - they are not all equally true (since they embody different levels of knowledge about the event), but YES - one can have TRUE statements which only convey a partial truth.

Now, if one can have partial truth statements then one can still access truth or true valued assertions at some level.

Let's go on:

Example (2):

"The same is true for value statements , such as "Racism is wrong" and "People should not be cruel." These views do not tolerate any alternatives . "

The problem here is that he has not proven the values are actual truth values, as opposed to the subjective values of a moralism in his own mind. In particular, he's violated Precept (1): "One cannot assume a "logical deduction" unless there exists a relationship between the premise and the conclusion such that we have a right to expect the conclusion IF we know the premise is true". But he has formulated no premise, because he's not defined either "racism" or "cruel" - he simply assumes everyone knows exactly what he means! Hence, he leaves us no testable proposition that can be put into any logical relationship, far less link a premise to a conclusion.

Questions then occur: E.g. : Is drawing or portraying President Obama as a witch doctor with a bone through his nose, racism? If not, why not ? The first step in setting out a premise that portends a "truth value" is therefore to carefully define one's terms, don't assume everyone knows what you mean!

So, IF he defines both "racism" and "cruelty" and then integrates them into a premise, tied to a conclusion that relates them- we may then consider whether "These views do not tolerate any alternatives". (In which case, depicting President Obama as a witch doctor with a bone through his nose may not meet the criterion of an "alternative" to him, hence can be "tolerated" - but as he's framed his case, we can't say so because it's too vague.)

Example (3)

"All truth CLAIMS are exclusive . For example , if humanism is true , then all non-humanisms are false . If atheism is true , then all who believe in God are wrong . Every truth claim excludes its opposite ."

As I already showed (Example (1)) the first statement is an undecidable proposition by virtue of en embedded contradiction. Thus, we behold an all-inclusive truth statement intended to apply to the proposition that "all truth is exclusive"! Thus it is a meta-statement about truth claims, not a statement within an axiomatic system concerning a class of truth claims. Thus, it is a non sequitur to flip from this most general, undecidable proposition to saying "thus if humanism is true all non-humanisms are false". In addition, it's an oversimplification, since one can have a conjunction (3(iii)) such that a truth value exists when BOTH p and q are true. (And again, we note he's neither defined "humanism" nor "non-humanism" so we've no idea what he means.)

But let's use the conjunction of propositions to show discrete and particular possibilities that defy his simple conclusions. It is known that in its most generic sense "humanism" means :

That philosophy or coherent attitude supportive of human values, i.e. those values that elevate, sustain and integrate a society beyond its merely physical or mechanical operations

Thus, our society provides physical infrastructure - such as roads, sewers, water mains, etc. and also mechanical contrivances (buses, or cars, trains) to help the people survive but humanism allows support systems beyond these basics, for example, decent and affordable health care. The reason is that a society without quality affordable health care (accessible to all) cannot function to its optimum. If too many are ill, then they can't be productive and if not productive, the society deteriorates because too few must work to care for the many who are ill or otherwise disabled.

The underlying conjunction here (p and q) is:

p: Humans are worthwhile beings to support in other ways besides purely mechanicalq: All humans have the same essential needs for humane support, so all should thereby be provided with said humane support

Since (p -q) defines the conjoint truth value, then "non-humanisms" would have to mean non-humans and would thereby be ab initio excluded from the proposition putatively concerning HUMANS. Hence, the claim of "non-humanisms" implicit in a human domain is a falsehood since it wouldn't apply to humans anyway! (Though one can argue that an "extended humanism" would also confer health care on animals in need, or at least certain classes of them, i.e. personal pets)

The statement about atheism is also invalid, because it reduces atheism to one meaning, ignoring all others. (For example, implicit atheism, and agnostic atheism). It would be like me writing: "If evangelical Christianity is true , then all other Christians who believe in God are wrong". This is arrant nonsense. Moreso since it hinges on the claimant providing the necessary and sufficient conditions for his (exclusive) deity to exist - thereby showing why its conditions supersede those of other Christians. To append a truth value to the statement then, a specific definition of atheism is needed, not an open-ended term that invites any reading.

Finally, the appended proposition:

"Every truth claim excludes its opposite ."

Has already been shown invalid, since it omits consideration of the fact that conjunction is the negation of incompatibility, e.g. (p/q)[. If one can have any conjunction, uniting propositions of truth p and q, then clearly a contingent truth claim p does not exclude its co-contingent q. As a more or less trivial example:

p: "Every photon of light is a particle"

q: "All light is waves"

In fact, as noted before in a previous blog:

Both statements can be true, provided the optical limits of observation are made clear. It is this very example, underscored by quantum mechanics, which displaces the either-or exclusionary absolutism of the claim. In other words, the actual experiments of QM show that observables in the form of propositions can be BOTH and not merely one OR the other! This is the very basis for the development of quantum logic, see e.g.

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