Logic


On pages 148-155 of Naming and Necessity, Saul Kripke concludes his book by setting forth an argument against type-type identity theory (the view that that any given mental state is identical to a brain state). Kripke argues that a mental state (pain) cannot be identical to a brain state (C-fiber stimulation) because if the aforementioned states are identical, then they must be so necessarily; and since we can conceive of one state existing without the other, they cannot be identical. Below, I shall set forth a general argument (with explanatory comments) which demonstrates the basis of Kripke’s reasoning and then cite Krikpe’s example from the text (also with explanatory comments).

1. If X is a rigid designator and Y is a rigid designator, and if the statement “X is identical to Y” is true, then the statement “X is identical to Y” is necessarily true.

• An example of the above is “Mark Twain is identical to Samuel Clemens”. Since both of these names are rigid designators, then “Mark Twain is identical to Samuel Clemens” is true in every possible world in which these rigid designators pick out an individual. On the other hand, the statement “Mark Twain is identical to the author of The Adventures of Tom Sawyer” is composed of the rigid designator “Mark Twain” and the non-rigid designator “the author of The Adventures of Tom Sawyer”. While these two designators may be identical in the actual world, it is possible that someone else could have written The Adventures of Tom Sawyer. Thus, the latter example of identity is not necessary.

2. X is a rigid designator.

3. Y is a rigid designator.

4. However, “X is identical to Y” is not necessarily true.

5. Therefore, “X is identical to Y” is not true.

Below is a reconstruction of Kripke’s specific example involving “pain” and “C-fiber stimulation”:

1. If “pain” is a rigid designator and “C-fiber stimulation” is a rigid designator, then the statement “pain is identical to C-fiber stimulation” is necessarily true.

• Kripke anticipates that some people will object that the identity statement above cannot be necessary since it is known a posteriori. However, Kripke points out that the statement “pain is identical to C-fiber stimulation” is analogous to other necessary identity statements that are known a posteriori such as “water is identical to H2O” or “heat is identical to molecular motion”. Besides this, Kripke demonstrates early on (pp. 35-37) that some necessary truths are known a posteriori (like certain mathematical proofs that require a great deal of calculation).

2. “Pain” is a rigid designator.

• Once again Kripke anticipates objections to the above premise. He asserts that “pain” is a rigid designator because it picks out what it refers to by an essential property (the sensation of painfulness). It is impossible to conceive of pain existing without the property of “feeling painful”.

3. “C-fiber stimulation” is a rigid designator.

4. However, “pain is identical to C-fiber stimulation” is not necessarily true.

• It is possible to conceive of a world in which one can exist without the other.

5. Therefore, “pain is identical to C-fiber stimulation” is not true.

I must confess that the above argument appears to be a powerful refutation of identity theory. It seems to me that the only way out of the conclusion is to either deny Krikpe’s semantics or argue convincingly that either “pain” or “C-fiber stimulation” or both are not rigid designators.

(Click to see Part 1)

Once again, Aristotle was one of the first recorded ancient thinkers to discover the law of non-contradiction. It is important to note that Aristotle did not create this law, no more that Isaac Newton created the law of gravity; he merely discovered it as an unchanging principle of the universe.  In Metaphysics Aristotle states, “For the same thing to be present and not be present at the same time in the same subject, and according to the same, is impossible.” [i]  This then is the law of non-contradiction, one of the first principles of knowledge.[1]

      The law of non-contradiction can be expressed simply as such: A cannot be both B and non-B at the same time and in the same sense.  In this equation the letters A and B are variables.  We may insert a number of different words in this equation to illustrate the law of non-contradiction.  For example, an object (A) cannot be both square (B) and round (non-B) at the same time and in the same sense.  Now let’s examine that proposition with the aid of figure 3 below. 

      In this diagram A represents “object”, and B represents “square”.  Non-B represents anything that is not “square” (i.e. a circle).  Non-B is what is called the “complimentary class” because it completes the proposition.  To deny the law of non-contradiction would be the same thing as to make a square-circle, which is logically impossible (we cannot even conceive of such a shape).  

      The law of non-contradiction can also be used to verify the Correspondence Theory of truth as so: A statement (A) cannot be both true (B) and false (non-B) at the same time and in the same sense.  There is as much probability as a statement being both true and false in the same way, as there is an object being round and square.  It is a logical impossibility. 

God and Logic
      It may come as a shock to many people to know that God, himself, cannot break the law of non-contradiction.  That is, God cannot create a square-circle, or make a statement that is both true and false, or create a rock so heavy that He cannot lift it. 

      Many people respond to this fact by saying, “How dare you say that God cannot do something!  God can do anything He chooses. He is all-powerful!”  But the Bible, itself, plainly states that there are some things that God absolutely cannot do.  Hebrews chapter six lists two things that God cannot do.  Verse 13 states that God cannot swear by any name higher than His own, and verse 18 states that God cannot tell a lie. 

      Both of these verses give a clue to the nature of the things that God cannot do.  God cannot and will not do anything that contradicts His nature.  For example, truth is an essential property of God’s nature so it is impossible for God to lie.  In the same sense, logic is an essential property of God’s nature; He cannot do anything that is logically impossible.  God is not subordinate to logic, but rather, logic is a part of Who He is[2], and so He cannot do anything that would contradict that part of Him.  “He cannot deny Himself.” (II Timothy 2:13).

 Dialectical Logic
      Another so-called form of logic is known as Dialectical Logic.  Whereas non-contradictory logic states that reality must be either/or, dialectical logic states that reality is both/and.  In other words, non-contradiction (or excluded middle) says that a statement is either true or false, but dialectical says that a statement can be both true and false.

      Some critics of the either/or form of logic point out that it is a Western (American, English, etc.) form of logic, and that when you are dealing with Eastern (Oriental) concepts you have to use the both/and form of logic.  This argument is flawed in two main areas

      In the first place, you cannot assign the laws of the universe by different cultures.  Mortimer J. Adler says that, “The fundamentals of logic should be as transcultural as the mathematics with which the principles of logic are associated.  The principles of logic are neither Western nor Eastern, but universal.”[ii]  In essence, Adler is pointing out that the laws of mathematics do not change in varying cultures.  “Two plus two equals four” is the same in America, Russia, Korea, Australia, and everywhere else for that matter.  The same thing applies to the law of non-contradiction.

      Secondly, you cannot argue against the law of non-contradiction without using it, and this, in turn, is self-defeating.  Perhaps a story would illustrate this point.  Ravi Zacharias tells of an instance when was debating a professor who embraced the Dialectical logic of the Hindu religion.

     As the professor waxed eloquent and expounded on the law of non-contradiction, he eventually drew his conclusion:  “This [either/or logic] is a Western way of looking at reality.  The real problem is that you are seeing contradictions as a Westerner when you should be approaching it as an Easterner.  The both/and is the Eastern way of viewing reality.”

     After he belabored these two ideas on either/or and both/and for some time, I finally asked if I could interrupt his unpunctuated train of thought and raise one question.

     I said, “Sir, are you telling me that when I am studying Hinduism I either use the both/and system of logic or nothing else?”

     There was pin-drop silence for what seemed an eternity.  I repeated my question:  “Are you telling me that when I am studying Hinduism I either use the both/and logic or nothing else?  Have I got that right?”

     He threw his head back and said, “The either/or does seem to emerge, doesn’t it?”

     “Indeed, it does emerge,” I said.  “And as a matter of fact, even in India we look both ways before we cross the street – it is either the bus or me, not both of us.”[iii]

       From this humorous story we see the undeniability of the law of non-contradiction.  Any attempt to deny it is self-defeating.  The more you try to disprove it, the more you prove its necessity in rational argument.

      The consequences of denying the law of non-contradiction are dreadful (see fig. 4[iv]).  There would be no basis for reality if it did not exist, and our universe would be uninhabitable.        

(Fig. 4)

Eight Results of Denying the Law f Non-Contradiction

 By James Sullivan (see endnote) 

•1. To deny the necessity and validity of the Principle of Contradiction would be to deprive words of their fixed meaning and render speech useless.

•2. Reality of essences must be abandoned; there would be becoming without anything that becomes; flying without a bird; accidents without subjects in which to inhere.

•3. There would be no distinction between things.  All would be one.  Ship, wall, man would all be the same thing.

•4. It would mean the destruction of truth, for truth and falsity would be the same thing.

•5. It would destroy all thought, even opinion, for its affirmation would be its negation.

•6. Desire and preference would be useless, for there would be no difference between good and evil; there would be no reason to go home, for to go home would not be different from staying where one is.

•7.  Everything would be equally true and false at the same time, so that no opinion would be more wrong than any other even in degree.

•8. It would make impossible all becoming; change, or motion.  For all this implies a transition from one state of being to another; but if the Principle of Contradiction is false, all states of being are the same.

 Characteristics of Truth
      We will conclude this post with a few of the main characteristics of truth.  These characteristics are essential properties of what truth is and how it works.

 Truth is Exclusive
      All truth claims are absolutely exclusive.  When a statement is true, then by definition, it excludes everything else that opposes it.  For example, if the statement, “Socrates is a man” is true, then that statement excludes all other conceptions of what Socrates is.  Even the statement, “No truth is exclusive” is an exclusive statement, because it excludes any conception of truth except the one stated.

 Truth is Immutable
      Truth does not change; it remains the same no matter what.  There are a few objections to this, but the one most commonly stated is that truth changes with time.  For example, the statement, “Abraham Lincoln is president” was true in 1863 but is not true in 2003; this is seen as a valid contradiction of two equally true claims.  This objection is easily refuted because it is based on confusion of the law of non-contradiction

      The law of non-contradiction teaches that two opposing statements cannot both be true in the same time and the same sense.  Time is an essential context to a truth claim.  To quote Norman Geisler and Ronald Brooks, “The spatial and temporal context of statements is an inherent part of the context which determines the meaning of that assertion.”[v]  The truth claim is understood in context of the time it was made.  So the statement “Lincoln is president” (as said in 1863) is an absolute, unchangeable truth.

Truth is Objective
      Truth is objective not subjective. That is, truth exists outside of us.  Our opinions concerning a statement or idea do not make them true or false.  The opposite of this belief is known as relativism.  Any conception of truth outside of its being objective will ultimately lead to a logical contradiction, and is therefore impossible.

 Conclusion
      In this post we have looked at the definition of truth, the logic on which it is based, and some of its essential properties.  It is imperative that we understand the meaning and nature of truth if we hope to defend the absolute claims of Christ and the Christian faith.  Who knows, but that some of those who are reading this will not be faced with a Pilate of their own someday, who will ask them with earnest and hopeful desire, “What is truth?” 


[1] The first principles of knowledge are the self-evident starting points on which all knowledge is based (i.e. the law of non-contradiction).  These principles do not contain any content of knowledge, but are necessary for knowledge to exist.  To those who would try to explain away these first principles, C. S. Lewis objects, “It is no use trying to ‘see through’ first principles.  If you see through everything, then everything is transparent.  But a wholly transparent world is an invisible world.  To ‘see through’ all things is the same as not to see.” (The Abolition of Man, (Harper Collins, 2001), p. 81)

[2] It is interesting that the Greek term used to describe Jesus in John 1 is the word logos, from which we derive our word Logic.

 


[i] Aristotle, Metaphysics 4.3 1005b, (Prometheus Books, 1991), translation by John H. McMahon

[ii] Adler, Mortimer, J., Truth in Religion, (Macmillan, 1990), p. 36

[iii] Zacharias, Ravi, Can Man Live Without Go?, (Word Publishing, 1994), p. 129

[iv] Sullivan, James, B., “An Examination of First Principles in Thought and Being in the Light of Aristotle and Aquinas”, Ph. D. Dissertation, Catholic University of America, Washington, D.C., (Catholic University of America Press, 1939), pp. 121-122

[v] Geisler, Norman, R.; Brooks, Ronald, M., When Skeptics Ask, (Victor, 1990), p.256