FREGE, THE LAWS OF LOGIC AND THEISM
BY MIKE ROBINSON
Friedrich Ludwig Gottlob Frege was a renowned logician and mathematician. Significant features of his work can be utilized to advance the notion that the laws of logic are not the product of matter and its supporting capability. Without God as the uppermost mind and the source of logic one weakens, beyond recovery, the justification that one can trust human reason. Frege added that the laws of logic are not “psychologistic,” but logical; they are objective and not subjective. These laws of truth do not come from nor do they only remain in the thoughts of humans. Selected non-theists assert that these laws of logic are not laws—they are not independent and universal. Nevertheless, immutable laws by definition are not subjective and individualistic. Frege was correct; they are unmistakably objective and universal.
In the beginning was the Logos and the Logos was with God and the Logos was God (John 1:1).
Friedrich Ludwig Gottlob Frege (1848–1925) is one of my favorite thinkers. He was a German and is considered a great mathematician, logician, and philosopher. Frege is largely deemed to be the initiator of analytic philosophy and a significant contributor to the philosophy of mathematics. His influence on philosophical issues includes philosophical logic, systems of connotation, linguistic philosophy, philosophy of arithmetic, and mathematical logic. Frege was the originator of axiomatic predicate logic, which he denoted in his book, Conceptual Notation. This was a critical breakthrough in the history of logic. This genius described a system of symbolic logic that moved well beyond Aristotle’s two-millennial-old logic and Frege’s method remains influential to his day. Frege’s subsequent “philosophical masterwork, The Foundations of Arithmetic, drew the attention of both Bertrand Russell and Ludwig Wittgenstein. Frege sought to explain numbers as extensions of concepts while demolishing other theories.”
Not everything can be defined (Frege).
Frege was a “German mathematician, logician, and philosopher who worked at the University of Jena. Frege essentially reconceived the discipline of logic by constructing a formal system which, in effect, constituted the first ‘predicate calculus.’ In this formal system, Frege developed an analysis of quantified statements and formalized the notion of a ‘proof’ in terms that are still accepted today. Frege then demonstrated that one could use his system to resolve theoretical mathematical statements in terms of simpler logical and mathematical notions. One of the axioms that Frege later added to his system, in the attempt to derive significant parts of mathematics from logic, proved to be inconsistent. Nevertheless, his definitions (of the predecessor relation and of the concept of natural number) and methods (for deriving the axioms of number theory) constituted a significant advance. To ground his views about the relationship of logic and mathematics, Frege conceived a comprehensive philosophy of language that many philosophers still find insightful. However, his lifelong project, of showing that mathematics was reducible to logic, was not successful”
Frege’s Logic and the Philosophy of Mathematics
Frege birthed contemporary logic by “developing a superior method of formally representing the logic of thoughts and inferences. He did this by developing: (a) a formal system that served as a basis of modern logic, (b) an analysis of complex sentences and quantifier phrases that showed an underlying unity to certain classes of inferences, (c) an analysis of proof and definition, (d) a theory of extensions which, though seriously flawed, offered an intriguing picture of the foundations of mathematics, (e) an analysis of statements about number (i.e., of answers to the question ‘How many?’), (f) definitions and proofs of some of the basic axioms of number theory from a limited set of logically primitive concepts and axioms, and (g) a conception of logic as a discipline which has some compelling features.”
Additionally “Frege raised a problem with the traditional account and offered a different understanding of identity statements.” Identity statements are “necessarily true by definition.” Which led to the paradoxical question “how can there be contingent identity statements if identity is necessary?” Consequently, Frege both helped disentangle deep paradoxes while simultaneously, sometimes unwittingly, producing others.
Frege was one of the most noteworthy philosophers in the rational maturity of logic. His discoveries and writings regarding the laws of logic and analytic philosophy led the way for the varied schools, forms, and methods of modern logic. One of Frege’s foremost goals was to convert or reduce mathematical truths to logic. He failed in this pursuit, but his work helped develop the modern philosophy of logic.
The laws of logic are not psychological laws of takings-to-be-true, but are laws of truth. They (the laws of logic) are boundary stones set in an eternal foundation.
Frege in his work Begriffsschrift observed that the laws of logic are universal and fixed. These laws are non-mutable laws that rule over all thinking, assessments, discernment, applications, assertions, propositions, predication, theories, actions, and conclusions. He labeled these laws the “laws of truth.” This is a way to name them that helps one understand that these laws govern all true things, not just thinking, but they are imposed on all our actions too. Frege wrote that the laws of truth are “the most general laws, which prescribe universally how one ought to think if one is to think at all.” He thought that the laws of logic must be expressed and applied in a painstakingly precise and methodical manner if the logician is to be consistently logical. Frege, standing on Aristotle’s shoulders, pressed the universality and necessity of the laws of logic.
The Christian knows that the laws of truth are universal and necessary. Additionally, they are aspatial, immaterial, immutable, and everywhere in force. Thus they require God, who is aspatial, immaterial, immutable, and everywhere present, universal in knowledge, and necessary, to ground them. Deny God and one denies the only possible foundation for these obligatory laws.
Mario Livio in his book Is God a Mathematician? Esteemed the work of Frege by stating that his “program was extraordinarily impressive.” However, many mathematical and logical geniuses are a bit odd. Some scholars, like Ludwig Wittgenstein, did not always understand Frege’s behavior: “I was shown into Frege’s study. Frege was a small, neat man with a pointed beard who bounced around the room as he talked. He absolutely wiped the floor with me, and I felt very depressed; but at the end he said ‘You must come again,’ so I cheered up. I had several discussions with him after that. Frege would never talk about anything but logic and mathematics, if I started on some other subject, he would say something polite and then plunge back into logic and mathematics. He once showed me an obituary on a colleague, who, it was said, never used a word without knowing what it meant; he expressed astonishment that a man should be praised for this! The last time I saw Frege, as we were waiting at the station for my train, I said to him ‘Don’t you ever find any difficulty in your theory that numbers are objects?’ He replied: ‘Sometimes I seem to see a difficulty but then again I don’t see it.’”
Frege’s Goal for New Logic
Frege’s objective was to “display in a perspicuous way the relationships between concepts and propositions. The goal of the whole of logic is to demonstrate the correctness of deductions without gaps between premises and conclusions, using acknowledged formal and precise rules of inference. Frege’s focus seems initially to have been on determining the correctness (and hence non-synthetic, a priori nature) of mathematical proofs, rather than examining reasoning of all sorts, as had traditionally been the subject of logic.” Additionally Frege’s main interest was to “understand both the nature of mathematical truths and the means whereby they are ultimately to be justified. The appeal to reason: What justifies mathematical statements is reason alone; their justification proceeds without the benefit or need of either perceptual information or the deliverances of any faculty of intuition. The Task: To articulate an experience-and intuition-independent conception of reason.”
The Nature and Prominence of Logic
In his book Begriffsschrift Frege proposed to reveal the “nature and status” of logic. Logic has normative standing. Logic imposes the standards that have the composition that governs all assessments, beliefs, assertions, suppositions, and inference. Frege rightly insisted that the laws of logic are the “laws of truth.” He states that the laws of logic are “the most general laws, which prescribe universally how one ought to think if one is to think at all.” He thought that the laws of logic must be expressed and applied in a painstakingly precise and methodical manner if the logician is to be consistently logical.
Frege further noted that the laws of logic are not “psychologistic,” but logical; they are objective and not merely subjective. These laws do not come from nor do they only remain in the minds of men. Many modern non-theists assert that these laws are not laws; they are not objective and universal. Yet immutable laws by definition are not subjective and must be universal. If one tries to reject the objective universality of the laws of logic, let the person attempt to write one sentence or think one thought or walk one step without utilizing these laws. Frege was correct; they are clearly universal and objective.
The Eminence of Frege
Richard Mendelsohn writes: “Frege had created … a formal language in which he axiomatized higher-order quantificational logic; derived many theorems of propositional logic, first-order logic, and second-order logic; and defined the ancestral relation. For his work represents a milestone not only in the history of logic and thereby, in the history of philosophy, but also in the history of modern thought, for it was one of the first sparks in a hundred-year explosion of research into the foundations of mathematics, and into the application of mathematical representation of structures other than numbers and shapes.” This son of a theologian was a logical mastermind and a philosophical trailblazer. Mendelsohn adds: “No only did Frege create modern quantificational logic, but he also provided the theoretical framework for many subsequent philosophical developments in logic as well as in speculative philosophy.”
Frege An Atheist?
Nearly all logic before Frege was a logic of terms. This “traditional” logic, initiated by Aristotle, had the advantage of … being simple.
Although Frege was the son of a fine theologian he is sometimes claimed by atheists. He may have been an atheist, but a better label would probably be an agnostic or a subtle non-professing atheist in the manner of David Hume. Both men rejected religious commitment and did not display any religious sensibility. Yet contrary to Hume, Frege apparently affirmed an ontological argument for existence which I suppose can be extended to God’s existence (many scholars deny this and affirm the antithesis). Frege pondered the ontological question: “To what then do we refer when we speak of the existence of something?” Oderberg further notes: “No one before Frege talked of an ‘is’ of identity.”
Frege’s Ontological Argument for Existence
Frege’s Ontological Argument for Existence (OAE henceforth) can be summarized:
The primary premise is that the idea of not existing is mathematically zero. Since a person can ask what a number is, this implies that the person’s existence is higher than a zero. Hence the person is the antithesis of not existing, therefore the person exists.
The Ontological Argument for the Existence of God (OAEG henceforth) was first offered for the Christian God by Anselm. God is “that than which nothing greater can be conceived.” Considering God can be conceived in the mind and He is thought to exist, His existence is greater than not existing. So that “which nothing greater can be conceived” necessarily exists. This argument has been confuted forasmuch as many consider it to be mere play on words (Plantinga has proposed a contemporary and more robust version).
To summarize Frege’s OAE: He asserts that one exists because one can conceive of one’s existence, whereupon this existence is greater than zero and not as great as infinity. On the basis of mathematics, existence is that which is greater than zero, and not as great as that which is greater than the infinite (greater than one can conceive). Since you exist and are greater than zero, then you must exist. Frege did not expand this to the existence of God, but that implication seems to follow. Paradoxically the not-religiously-denominated Frege later employs the same wording with reference to the existence of God. He claims that those who assert or reject the existence of God imply that their notion of God cannot be caused by the empirical sense of an object extended in space (there is no immediate perception of God).
Frege and Identity Statements
Frege argued that within an important operation of the Law of Identity “every object has to be identical to itself.” J.P. Moreland and William L. Craig disclose that Frege viewed identity statements as “statements about language, and they assert that a certain relation holds between the two referring expressions used in the statement, namely, they are coreferring expressions, i.e., they each name the same thing [e.g., car/automobile].” Identity requires and presupposes the Law of Identity even if identity statements can be ambiguous. Frege may not have submitted to theism but he knew that mathematics required a fixed and powerful ground. He wrote: “I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.” Frege projected in Grundlagen der Arithmetik that there is a “logical inference from n to n + 1.”
The Laws of Logic and Thought
The fundamental laws, the laws of thought, [are] those operations of the mind by which reasoning is performed (George Boole: The Laws of Thought).
Livio noted that “according to Frege, even statements such as 1 + 1 = 2 were not empirical truths, based on observation, but rather they could be derived from a set of logical axioms.” Another concern is the meaning of existence and its relation to the ontological status of various items (mental objects, forms, ideas, material objects, etc.). In one passage Frege seems to not just connect existence with self-identity but to equate the two, tossing Aquinas’ distinctions (and countless others) out the ontic window. Additionally, “Frege and Russell were … later to claim that algebra stems from logic.”
Frege contended that the laws of truth are not psychologistic, but are necessities of logic; they are objectively true and in force. These laws are not bound to the fleeting subjective opinions or thoughts of men. They are necessarily utilized by all men, but a particular man or set of men (and their particular brains) lack the ontic capacity to ground these laws. Thus I draw from this that only an immutable and universal power source can ground the laws of truth and this can only be God. The always-in-flux cosmos lacks an unchanging nature to ground the laws of logic. Nonetheless, many modern non-theists assert that these laws are not laws; they are not fixed and universal. Yet Frege was correct; they are surely fixed and universal.
Frege published his first revolutionary work in logic in 1879.
Thus the laws of logic are not material laws that may change forasmuch as truth must utilize these principles. Posit them as mere brain accessories or cerebral tools and this will place them in the subjective psychologistic realm. This cannot be true because these laws are objective and necessary. Thus the principles of logic are not mere human conventions or limited to subjective governance. One must be antecedently committed to their independence from the human brain (and the cosmos) and their absolute normative governance. Thus they are transcendent. John Frame observes: “People may very well interpret the expression “law of thought” by analogy with the “law of nature” and then have in their mind features of thinking as mental occurrence. A law of thought in this sense would be a psychological law. … That would be a misunderstanding regarding the task of logic, for truth has not been given its proper place.”
The True God Exists
Unless I believe in God, I cannot believe in thought (C.S. Lewis).
Change is the condition of life. … But the unchangeableness of God is the negation of all imperfection, it is the negation of all dependence on circumstances, it is the negation of all possibility of decay or exhaustion, it is the negation of all caprice. It is the assurance that His is an underived, self-dependent being, and that with Him is the fountain of light; it is the assurance that, raised above the limits of time and the succession of events—that we might have a rock on which to build and never be confounded (Charles Spurgeon).
God has the ontological heft to account for everything. God, as the One who provides the a priori truth conditions for all things, has the ontic capacity to account for immutable universals (laws of logic, moral law, etc.). Mutable and non-universal entities are devoid of the sufficient attributes that are required, so they are ontologically undersupplied to account for the laws of logic. These laws are invariant universals and are required for communication and knowledge.
Come let us reason together (God: Isaiah 1:17).
God furnishes all the a priori essentials; the necessary epistemic equipment utilized in all thoughts and achievements. God has the ontic attributes of omniscience, immutability, and omnipotence (He has universal reach) enabling Him to be the ground for the universal and immutable laws of truth and ethical necessities (moral law) that are utilized in all thought and action. Any position that rejects the true God as the epistemic (knowledge) base not only leaves an unnerving fissure, but hopelessly fails. Consequently, whatever evidence one discovers must be discerned and processed with the rational implements that arise from Christian theism and the worldview that streams from the true God.
The true God is the primordial requirement for all knowledge, proof, evidence, and logic. He is the a priori verity condition for the intelligibility of reality. The immaterial, transcendent, and immutable God supplies the indispensable pre-environment for the use of immaterial, transcendent, universal, and immutable laws of logic (law of identity: A = A; law of non-contradiction: A~~A). Atheistic thought cannot furnish the necessary a priori truth conditions for the immutable universal laws of logic; therefore it results in futility because of its internal weakness. Non-theistic worldviews fall into absurdity inasmuch as they are self-contradictory and lead to conclusions that controvert their own primary assumptions. Without God, ultimately, nothing can make sense.
Under Strict Materialism There is No Reason to Trust Human Reason
Van Til warns that “the only alternative to thinking of God as the ultimate source of unity in human experience as it is furnished by laws or universals is to think that the unity rests in a void. Every object of knowledge must, therefore, be thought of as being surrounded by ultimate irrationality.”
If human reason is only the product of matter and its supporting capacity then materialism itself cannot be true. Without God as the highest mind, the source for human reason, one undercuts the reason one can trust human reason.
The laws of logic are potent apologetic tools. However, the Great Logos, Jesus Christ, came to speak and provide the greatest Goodnews: Christ’s death and resurrection atones for the sins of His people. May the reader flee to Christ in faith and find forgiveness, acceptance, and pardon (Titus 3:4-7).
by Mike Robinson: Granbury, Texas. Minister and author.
1. Encyclopedia of Philosophy: http://www.iep.utm.edu/frege/
2. Dictionary of Philosophy: http://www.iep.utm.edu/frege/
3. Encyclopedia of Philosophy: http://www.iep.utm.edu/frege/
4. J.P. Moreland and William L. Craig: Philosophical Foundations for a Christian Worldview, p. 197.
5. Friedrich Ludwig Gottlob Frege: Basic Laws of Arithmetic, p. 13.
6. Laws of Logic: The Law of Non-contradiction (A~~A) and the Law of Identity (A=A). Also known as the laws of reason and the laws of thought.
7. Mario Livio: Is God a Mathematician? p. 186.
8. G. E. M Anscombe: Three philosophers.
11. Richard Mendelsohn: The Philosophy of Gottlob Frege, p. 2.
12. Ibid, p. 7.
13. David S. Oderberg: The Old New Logic: Essays on the Philosophy of Fred Sommers, p. 33.
14. Mendelsohn, p. 1.
15. Oderberg, p. 211.
17. Livio, p. 18).
18. Moreland and Craig: p. 198.
20. Frege: Basic Laws of Arithmetic, p. 10.
21. Stephen Hawking, Editor: God Created the Integers, p. 1077.
22. Livio, p. 46.
23. Ibid., p. 182.
24. Ibid., p. 184.
25. Cornelius Van Til: Survey of Christian Epistemology, p. 216.
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