Revisiting The Realm Of Mathematical Logic

 

                                                           Bertrand Russel in 1957

One of Bertrand Russell's major contributions was to put philosophy on a mathematical footing, and with it the use of symbolic logic - which actually began with George Boole's "laws of thought", e.g.

Symbolic Logic Intro: George Boole's Laws Of Thought

 Russel himself rose to much prominence as a mathematical philosopher through his propositional functions. Much of this content was articulated in Russell's book: An  Introduction to Mathematical Philosophy,  and interested readers can find much more background material there.

Preliminaries:

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, op. cit., 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 inerrant on all truth - physical and spiritual"

BUT - if one  contradiction is found - say by geological science, i.e. "The world is 4 billion years old not four thousand", we must have: p' = f(~q)

"The Bible has at least one false (physical) claim, so cannot be inerrant"

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 astronauts 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.

In a more generic sense, the applications of Gödel’s theorem go well beyond mathematical formulae or arithmetic axioms to encompass any statements which can be framed in those abstract terms. This is so important to grasp that it behooves me to give examples, starting with a simple statement of logical transitivity:

X = Y

Y = Z

therefore X = Z


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

           Thus the statement, X = Y is unprovable-in-the-system is true.

            With these preliminaries, let’s examine the logical structure applied to the symbolic sequence of statements:

 X = Y

But, Y Ç  Z = Æ

Then, X /\ Z (contradiction)


Example:  Epimenides’  All Cretans are liars paradox, which itself perpetuates a causal loop with no closure.

Take the statement X= Y: All Cretans are Liars  

Take the alternative statement X = Z: All Cretans are Truth tellers.

 Clearly, Liars (Y) cannot also be Truth tellers (Z), hence the intersection of the set of all Liars with the set of all Truth tellers must be an empty set (Æ), i.e. there are no members in both.

More technically, if the speaker is a Cretan, then the statement is clearly ipso facto unresolvable. If a Cretan, he exists within the so-called abstract, formal system. Yet he’s definitely making a statement (meta-) about the system. Is he lying?  Or, is he telling the truth? This, alas, cannot be resolved, so an undecidable proposition exists as the Gödelian Incompleteness Theorem (II) indicates.  Note that while simple and interesting, the Cretans= Liars example only serves to illustrate what a meta-statement is.  It doesn’t help us grasp the basis for a supernatural or religious concept.

 Is there a way out of the meta-loop? Yes!  Provided one uses realism-based science to assess statements. For example, in the Einstein equation, E = mc², scientific epistemology allows us to regard E, m and c as constructs, connected via operational definition to the P- (perceptual) facts of energy, mass and the speed of light, c. Thus, we expect a correlation like:

                                 C « P

This re-affirms logical closure, physical significance and no meta-linkage. For instance, the operational definition of mass is accomplished by comparing inertias for two different bodies. We use the ratio of their detected accelerations to masses: m2/ m1 = a1/a2 and Newton’s 2nd law, say in a collision experiment. See e.g.

 Introduction to Basic Physics (Mechanics) Pt. 2 

In effect, even if a scientific or research hypothesis includes some open or meta-statements (evidently leaving enough room open for undecidable propositions) there are nevertheless empirical checks and tests that can close the system parameters. Nothing similar exists for supernatural claims embodied in religious concepts.

       Consider the statement:

This consecrated bread wafer is the body of Christ

          Here we have neither P-fact nor C –construct. There is no confirmatory device for example, to demonstrate that the consecrated bread is a human body. The statement is thus open-ended. It also permits the real cause of religious experiences to be attributed to hallucinations or maybe a micro-seizure in part of the brain’s temporal lobes.  Michael Persinger demonstrated the latter using an electrical helmet to stimulate subjects’ temporal lobes[1].

        Note also that we can’t even identify unique and distinguishing attributes that point to the validation of the preceding religious claim. Without venturing into the realm of P-facts, the set of C-constructs (bread, body of Christ) is ripe for self-reference as well as the intrusion of incompleteness.

        What if one ignores this and assigns attributes willy-nilly? Say by insisting: Well you cannot detect the body because you are only able to ascertain the physical accidents using scientific analyses[2]. In this case the claimant commits reification. He imposes his preconceived percepts on what is in reality an open-ended field. For such an open field, discussion is fruitless, since it ends up being a mental Rorschach for the benefit of the proponent

The Gödel Incompleteness theorem (I) can be applied to all or most incomplete statements. A good way to test such statements is to try to write them in some symbolic language form then compare them. Consider the two statements below, pertaining to material evidence in the JFK assassination:

p: All of the bullet fragments recovered were from a 6.54 mm Mannlicher-Carcano rifle

q: All of the bullet fragments recovered were from a 7.65 mm Swedish Mauser rifle

We write the symbolic form taking both into account:

~ (p /\ q) e.g.

Two statements p and q are contrary if they cannot both hold. Note, however, that two contrary statements may both be false! Note also that all contradictory statements are contrary, but many contrary statements are not contradictory, i.e.:

Lee Harvey Oswald was observed in a 6^th floor window of the Texas Book Depository

And

A swarthy man was observed in a 6^th floor window of the Texas Book Depository’

            

They are not contradictory because a man was observed in a sixth floor window. Thus, swarthiness is not fundamental to the falsity of the statement because a shadow may have fallen at the time of the observation engendering a darker hue.

Does this application allow for contradictions because of the latent incompleteness? Would such a partially true statement be unprovable, say in a court of law, for an eyewitness?  In the case of the solar flare example, it’s not very likely unless the flare was homologous, e.g. occurring almost simultaneously at two nearby locations. Then the L1 statements aren’t refined enough to separate the heliographic locations. The residual statements would then remain unprovable only if no higher resolution observations were forthcoming.

What does all of this say about any truth claim?

It says that in general truth claims must be treated with great skepticism. At any rate, one must always assume the initial claim for truth is partial, or at the L1 level. The claimant must be then pushed as far as possible to disclose the maximum content of the truth as he understands it, especially if the truth claim is made dogmatically

Now, given all these basics, let's see if we can apply them to the assertions or alleged propositions made by one Wordpress blogger  with whom I had an exchange some years ago:


"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).

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.

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[1] Persinger, M.:  The Neuropsychological Bases of God Beliefs.

[2] According to the Catholic doctrine of Transubstantiation, the “accidents” refers to the outer, physical signs. For example, that the wafer when chemically tested yields starch presence, not protein.