Propositional Logic and the Nervous System

The nervous system is inherently logical, exhibiting properties that correlate to logical functions. Many somatic reflex arcs seem to operate according to logical principles. We find negations, conjunctions, sequences of cause-and-effect events that mirror conditional statements and Modus Ponens arguments, feedback loops that can be modeled by the rules of implication as well as structures possessing the properties of exclusive disjunctions which are thought to help us to ignore superfluous signals.

A Modus Ponens argument is one of the form “If P then Q. P. Therefore, Q.” This argument is replicated in the withdrawal reflex which protects a vertebrate from harm. It depends on a reflex arc, at least three neurons in series whose structure can be said to recapitulate the conditional statement “if P then Q.” For instance a specialized nerve ending (a nociceptor) in the finger may connect to a neuron whose cell body is located in the spinal cord’s grey matter. From this point, a motor fiber may innervate the biceps brachii, the main flexor involved in a total retraction of the arm (1). The physio-logical structure is akin to the first premise of the Modus Ponens argument, the conditional statement.

The second premise P, then, is akin to a painful stimulus. When the nociceptor is electrically depolarized by a stimulus which damages tissue, an impulse is sent to the central nervous system causing a secretion of chemical messengers in the spinal cord’s grey matter. The conclusion Q is the retraction of the arm. As a result of the spine’s reception of the signal P, one or more interneurons is depolarized, ultimately resulting in increased excitation of the motor neurons innervating the flexor. The striated muscle cells are depolarized and the sliding protein filaments begin to shorten. The forearm is pulled about the elbow whose angle decreases, moving the finger away from the painful stimulus. The whole argument can be summarized as follows: If the nociceptor is excited then the flexor will withdraw the limb. The nociceptor was excited. Therefore the flexor withdrew the limb.” A Modus Tollens argument arises if it is the case that there is no painful stimulus ( ¬P). Then the conclusion is ¬Q; there is no withdrawal of the arm. It is interesting to note that the nociceptor’s ending is structurally undifferentiated and basically identical to those of interneurons populating the spinal cord and the brain (2).

The crossed extensor reflex can also be illustrated with propositional logic. It consists of a withdrawal reflex as well as the contraction of the opposite limb’s extensor (3). The unmodified withdrawal reflex was a Modus Ponens argument but now our conditional is adapted to be “if P then Q and R” where R is the extension of the opposite limb which does not receive a painful stimulus. The painful stimulus P must now be applied to the toe. We will also have to change our conclusion to “therefore Q and R.”

P → (Q ˄ R)

P

∴ (Q ˄ R)

Now our argument can be summarized in the following way: “If the toe is hurt, the flexors which bend the knee will contract, retracting the leg, and the extensors in the opposite leg (quadriceps) will extend. The toe was hurt. Therefore, the afflicted leg was flexed and the other leg was extended.” The crossed extensor reflex prepares the other leg to bear the weight of the body. The remarkable thing is all this happens without the brain. I believe many people would be more logical if they lost their heads.

The stretch reflex, that mechanism whereby skeletal muscles autonomically limit the degree to which they are stretched, can be modeled with the rules of implication. When a muscle’s length is increased, certain nerves called muscle spindle fibers begin to fire. The frequency at which they fire is correlated with the length of the muscle. These signals go to the spine where they effect excitation of the motor neurons which contract that same muscle, shortening it and restoring its length (4). Now let P be the signal of the spindle fibers and Q be the excitation (depolarization) of the motor neurons which contract the muscle being stretched, helping to restore its original length. Let ¬Q be the inhibition (hyperpolarization) of the motor neurons.

( P → Q ) ↔ ¬( P ˄ ¬Q)

This is to say “Only if it is not the case that the spindle fibers are excited and the motor neurons are inhibited is it true that if the spindle fibers fire, the motor neurons are excited.” In other words, “It is the case that if the spindle fibers fire then the motor neurons are excited if and only if it is not the case that the spindle fibers are excited and the motor neurons are inhibited.” If the spindle fiber signal inhibited the motor neurons which contract the muscle in which the spindle is housed, the muscle may be stretched to a dangerous degree and, not to mention, the biconditional would be invalidated.

Whereas the spindle fibers prevent a muscle from being stretched too much, the Golgi tendon organs prevent it from developing too much force – from contracting too much. These structures are located in the tendons which connect skeletal muscles to bones and they send nerve fibers into the spine. Increased tendon tension mechanically deforms their neurons, generating a signal which relates force to frequency. When the Golgi tendon organs send signals to the spine, the motor neurons innervating the muscles between the tendons in which they are housed receive a hyperpolarizing, inhibitory signal (5). This in turn effects a decrease in tension. With this opposite effect, we can let P be the signal from the Golgi tendon organ while ¬Q continues to represent the inhibitory effect of the signal on the motor neuron: the consequent. Then we must rearrange our conditional to state “if P then not Q,” which is to say that the motor neuron is inhibited if a signal from the Golgi tendon organ is received. The following is a slightly modified rule of implication statement.

( P → ¬Q ) ↔ ( ¬P ˅ ¬Q)

This is to say that “only if it is the case that the Golgi tendon organs do not signal or the motor neuron is inhibited is it true that it is the case that if the Golgi tendon organs signal then the motor neuron innervating the muscle is inhibited.” It can also be said that “it is the case that if the Golgi tendon organs signal, the motor neuron is inhibited if and only if it is the case that the tendon organs don’t signal or the motor neuron is inhibited.” The validity of the whole statement, which we may equate to normal function of the Golgi tendon organ reflex arc, depends on the two halves of the biconditional being either both true or both false. For the inclusive disjunction to be true, either the tendon organs must reduce their activity (¬P), the motor neuron must receive inhibitory input (¬Q), or both. A true disjunction therefore suggests a reduction in the muscle’s developed tension. With a true disjunction, validity demands a true conditional as well and we appropriately see a negated Q denoting inhibition of the motor neuron, the proper consequence of neuronal input from the Golgi tendon organs.

However, for a false disjunction, it must be the case that both elements ¬P and ¬Q are false, to wit, that P and Q are both true. This would mean that the Golgi tendon organ is sending a signal (there is force developed in the muscle) and that the motor neuron is being excited. The Golgi tendon organ is very sensitive (6) so if the muscle developed even a slight force, a signal which would inhibit the motor neuron would normally be sent. Moreover, the muscle only develops force if it is excited by a motor neuron. As expected, we see that, given a false disjunction, the entire biconditional statement can only be true if the conditional is also false. To determine when the conditional (P → ¬Q) is false, we construct the following truth table:

P¬Q( P → ¬Q )
TrueTrueTrue
FalseTrueTrue
TrueFalseFalse
FalseFalseTrue
If the Golgi tendon organ signals, the motor neuron is inhibited.

This means our conditional is only false if P is true and ¬Q is false (and Q is true). It is not the case that a signal from the Golgi tendon organ causes excitation of the motor neuron but, restricting our investigation to the conditional, all other cases are physiologically viable. If it were the case that the disjunction (P ˅ Q) is true, it could not be the case that signals from the Golgi tendon organ inhibit the motor neuron – assuming the tissues are otherwise functioning properly – because this reflex arc is a negative feedback loop. The Golgi tendon organ’s signal causes inhibition of the motor neuron.

Nervous function which mirrors propositional logic is not limited to the motor neurons which innervate skeletal muscle. It is also found in those neurons responsible for tactile sensation. When a stimulus affecting an area of skin is received, lateral inhibition allows us to sense where exactly we have been touched. Just under the surface of the skin are the receptive fields of nerve fibers concerned with detecting touch. These fields consist of heavily bifurcated nerve endings which, when mechanically deformed, depolarize and relay information about mechanical contact to the central nervous system. Like trees in a forest, their branches overlap greatly but we can still discern the location of contact with a high precision by virtue of lateral inhibition, it is thought. The neurons whose receptive fields receive the highest pressure, presumably those in the center of the area affected by the stimulus, send the highest frequency signals. These signals are not merely destined for the central nervous system. They arrive at the local, surrounding sensory neurons and secrete inhibitory chemicals which have a hyperpolarizing effect. These inhibited neurons also receive the mechanical stimulus but, since they are on the periphery of the affected area, fire at a slower rate which is further retarded by inhibitory signals from the neurons in the center of the affected area. Responding to the mechanical pressure, the less stimulated peripheral neurons send inhibitory signals to the more stimulated central neuron as well but their low frequency signal is not sufficient to inhibit the central neuron (7).

Lateral inhibition is very similar to exclusive disjunction. In a true exclusive disjunction of the form (A ˅ B) ˄ ¬(A ˄ B), the two elements A and B can never both be true because of the negated conjunction. For all other cases, the exclusive disjunction is true. If we let A and B represent the signals from two neurons whose receptive fields are just below the surface of a mechanically stimulated area of the skin, parallels between lateral inhibition and exclusive disjunction can be drawn. Because the more stimulated neuron will inhibit the surrounding neurons, acting as a high-pass filter, the conjunction (A ˄ B) is negated. Only one signal will predominate: either A or B (and not A and B).

Ascending the clades we see spinal cords, an aggregation of neurons towards the mouth and light-sensitive organs, encephalization and, in the animals we consider to be the smartest, evermore developed cerebrums. The human brain has between ten billion and one hundred billion neurons. One neuron may receive inputs from between one thousand and one hundred thousand other neurons (8). In addition to those described above, inherently logical reflex arcs responsible for the function of many organs incorporate fibers going to and from the brain (9). The cerebral cortex is associated with sapience. language, intelligence and reason and it may be that some of its structures, on whatever scale, recapitulate the functions of propositional logic as well. It may be possible that thought is facilitated in part by such structural arrangements. By that token, it also may be that such structures, through chronic ignorance, self deceit and excitation of descending fibers, can be conscripted to counter these inherently logical reflex arcs at every level.

(1) Widmaier, Raff & Strang – Vander, Sherman, & Luciano’s Human Physiology: The Mechanisms of Body Function – Chapter 10: Control of Body Movement – Local Control of Motor Neurons pg. 319-320

(2) Widmaier, Raff & Strang – Vander, Sherman, & Luciano’s Human Physiology: The Mechanisms of Body Function – Chapter 7: Sensory Physiology – Specific Sensory Systems pg. 219

(3) Widmaier, Raff & Strang – Vander, Sherman, & Luciano’s Human Physiology: The Mechanisms of Body Function – Chapter 10: Control of Body Movement – Local Control of Motor Neurons pg. 320

(4) McMahon, Thomas A. – Muscles, Reflexes and LocomotionChapter 6: Reflexes and Motor Control pg. 147-148

(5) Ibid. pg. 146-152

(6) Ibid. pg. 149

(7) Widmaier, Raff & Strang – Vander, Sherman, & Luciano’s Human Physiology: The Mechanisms of Body Function – Chapter 7: Sensory Physiology – General Principles pg. 211-213

(8) Malmivuo, Jaakko and Plonsey, Robert – BioelectromagnetismChapter 5: Synapses, Receptor Cells, and Brain5.4.2 Brain Anatomy pg. 114

(9) Malmivuo, Jaakko and Plonsey, Robert – BioelectromagnetismChapter 5: Synapses, Receptor Cells, and Brain5.2.3 Reflex Arc. 109

This was written as the final paper for an informal logic class.

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