Table of Contents
- Historical Philosophy
- Philosophical Ambiguity
- Formal Languages
- Formal Systems
- The Advantages of Formal Systems
- An Example
- Formal is Better
- End Notes
The essence of philosophy is speculation: the process of asking a question and then, through contemplation, seeking an answer. If primitive religion can be called philosophy, in that it provided unscientific answers to the epic questions about life, death, and the cosmos, then philosophy is a practice almost as old as homo sapiens itself. Of course, much of historical philosophy has yielded to more scientific or rigorous areas of study that do not rely solely on speculation. For example, ancient creation myths and cosmologies have given way to modern cosmology, a scientific domain that has made remarkable progress in the last century1. What was once called natural philosophy is now the science of modern physics. Speculations about the origins of life have yielded to modern evolutionary zoology and biology. Contemplations about the human thought process have produced the wholly rigorous program of modern logic. Even the mind-body problem in philosophy has started to yield to the relentless progress of science, in this case from the field of cognitive psychology.
Of course, subjects still remain under the domain of philosophy. Any statement that is believed to be true but that cannot be proven through logic or justified via the scientific method can be considered part of philosophy2. Subjects that deal with these type of statements constitute the bulk of philosophy, but it can also make contributions to sciences that are still in their infancy, such as cognitive psychology, by providing valuable ideas through speculation. Many of the subjects of the first kind are very important3, some because they deal with the very foundations on which science rests, and others because of their practical significance, such as with ethics.
Unlike science and mathematics, the tools of the enterprise of philosophy have changed little since the dawn of time. The language of philosophy today is the same as it always has been, natural language in the style of literary prose. Unfortunately, natural languages are notoriously ambiguous, not the least because they are inherently metaphorical and ostensive. Consider for example, the word “suitcase”. Part of our understanding of the concept of suitcase is metaphorical: such a thing is a case for suits. But our understanding of suitcases is more precise than this for ostensive reasons: we have been shown examples of suitcases so we should recognize one when we see one. However, there is no rigorous definition, no rule, that we can apply to a particular thing to say definitely either yes, it is a suitcase, or no it is not.
This inherent ambiguity in natural language is exacerbated in philosophy, because philosophy deals with difficult concepts that require great mental acuity to grasp. How can one philosopher understand precisely what another is saying if the language they use is inherently ambiguous? If we cannot be certain about the meaning of the word “suitcase”, how can we even come close to preciseness about subtle concepts like “normativity” or “intensionality”? Modern logic has given us the tools to express ideas, concepts, and whole systems of these (i.e., theories) with a precision much greater than natural language. It seems obvious that philosophy would gain immensely by adopting these tools.
Let us speculate about why philosophy does not use rigorous, formal language to express even its most important ideas and arguments. For one, it is hard to break with any tradition, and the practice of using natural language and literary prose in philosophy is no exception. This trend is perpetuated by the fact that modern curricula in philosophy typically retain a heavy emphasis on prose and language, with less of an emphasis on logic4. Many philosophers may feel that their skills with natural language are much stronger than those with formal logic. However, there are clear counterexamples to this hypothesis, the most stark example being the American philosopher Willard Van Orman Quine. Quine was considered one of the foremost experts on modern logic and set theory, having written several books on these subjects5 and having even developed two set theories of his own6. Yet, Quine is most famous for his traditional philosophical writings, which embrace wholeheartedly the traditional ambiguity of historical philosophy.
This is not to say that natural language must be eliminated from philosophy altogether. Natural language should continue to constitute the bulk of philosophical writing, as it does even in mathematics and the hard sciences. Where philosophy would gain by using the formal language of logic is in its definitions and proofs. Many philosophical publications contain so-called proofs, but being expressed in natural language, they contain ambiguities, and therefore cannot be considered sufficiently rigorous; they are not really proofs in the modern mathematical sense. The remainder of this essay will give a very brief introduction to formal languages and methods of rigorous proof, followed by an example of how a traditional philosophical argument can be made rigorous. This example will illustrate how to use rigorous methods and what can be learned by using them.
The fundamentals of modern mathematical logic are taught by mathematics and philosophy departments at major universities, and are documented in many textbooks on logic7. I will give a brief summary here for the purposes of discussion below. A formal language consists of the following types of symbols:
- Sentential logic operators, such as ¬ (not), & (and), || (or), ⇒ (implies), ⇔(if and only if), etc. It has been demonstrated that one such symbol is all that is required, although this makes the language nearly unreadable for humans. Usually two or three such symbols are used, and others are defined from these symbols for convenience. For example A ⇔ B is defined as (A ⇒ B) & (B ⇒ A).
- Parentheses () to determine the precedence of logical operators
- Quantifier symbols, such as ∀ (“for all”) or ∃ (“there exists a”). Only one such symbol is required; usually it is ∀, with ∃x defined as ¬∀x¬.
- Object variables, or symbols that can denote any object. Often lower-case letters near the end of the alphabet such as x, y, z are used for this purpose.
- Object names, or symbols that denote a specific single object.
- Predicate symbols, or symbols that denote a specific attribute or relation.
- In modal logic, there are also modal operators, □ for necessity and ◊ for possibility. As with quantifier symbols, only one is required, usually □, with ◊ defined as ¬□¬ .
For convenience, formal languages often contain function symbols, too. A function is a special type of n-place relation (a relation involving n objects) in which knowing the value of a specific n-1 of the objects in the relation is enough to determine the nth value. For example, addition is a function, with the sum able to be determined from the two addends. Some formal languages also required a symbol for equality (=), although it can be defined in others. Typically the language will not contain predicate variables (and the quantifier symbols do not pertain to predicates). If this is true, the language will be one of first-order logic. If the language does contain predicate variables to which the quantifier symbols pertain, then it would be one of second-order logic. It has been proven, however, that one interpretation of second-order logic can be reduced to first-order logic by using sets in place of the predicate variables8.
Most if not all descriptive statements in natural language can be translated into a formal language. For example, if we wanted to say that “John is a big, fat liar”, we would need an object name for John (‘j’ for short), and predicate symbols for “big”, “fat”, and “liar”. If we adopt ‘B’, ‘F’, and ‘L’ for these, then “John is a big, fat liar” would be expressed as “Bj & Fj & Lj”. Or, to express the classic statement “all humans are mortal”, let H denote humans and M denote mortal, then this would be expressed as ∀x( Hx ⇒ Mx), where ∀x means “for all x”.
Formal logic is the tool for demonstrating that if one set of statements in a formal language is true, additional statements must also be true. There are two foundations to any formal system of logic:
- Primitive Predicates (or simply Primitives): These are predicates that are not defined. In natural language, definitions are circular, in that it is common practice to define words in natural language in terms of other words in the language. For example, the definitions of the words “be” and “exist” on Dictionary.com each refer to the other9. If formal systems this is not allowed. There must be a set of primitives that are not defined, and from which all other predicates are defined.
- Axioms (or Postulates): These are statements that are assumed to be true. As with definitions, formal logic does not allow circular implications. It may be true that we know that I am a mammal because I have hair, and we know that I have hair because I am a mammal, but in a formal system we can only use one of these to infer the other.
The primitives and axioms form the foundation of any formal system, from which the following can then be derived:
- Defined Predicates: These are predicate symbols that are defined from primitive predicates and other previously defined predicates. For example, we might define a mammal as that which is an animal and has hair (a poor definition that is for illustrative purposes only). In first-order logic, defined predicates can be thought of as mere conveniences, as the defined predicate symbol could always be replaced in any formula that used the symbol with the formula that is the definition of the symbol.
- Theorems: These are statements that are derived from the axioms and other theorems. The predicate calculus provides an explicit process by which these derivations may be constructed, which I will not detail here10.
This explicit foundation-and-edifice structure of formal logic is a powerful tool for ensuring clarity. Anyone who understands the meaning of the primitives should then understand the meaning of any statement in the system: there is no ambiguity in these statements beyond any ambiguity in the primitives themselves. For this reason it is imperative that primitives be selected wisely. The concepts embodied by primitives should be so simple and obvious that there is little or no ambiguity in their meaning. For example, if the primitive is for an attribute (a single-placed predicate), then given any object, it should be obvious to anyone whether or not the object has the attribute.
The meaning embodied in a formal system is also dependent on it’s definitions and axioms. Definitions are somewhat problematic because, as mentioned above, any defined predicate could be replaced by the formula used in its definition. Thus, in a way, definitions are superfluous. However, in philosophy the purpose of definitions is often not just for convenience, but to elucidate concepts associated with words in natural language in a formal and explicit manner. In this sense, despite agreeing on the meaning of primitives, different people might want to use different definitions for the same term. Strictly speaking, in such a case both definitions could be allowed (with different terms used for the defined predicates; e.g. normative-1 and normative-2), and doing so would not change the implications or meaning of the formal system. However, there would be contention regarding which of the definitions corresponded to the informal, natural language term that was being defined. Despite this contention, the process of using formal definitions is extremely helpful, as it makes unambiguously clear from whence the differences between the conceptions of the same natural language term derive.
The final source of contention in any formal system is the selection of axioms. Just as a well-selected primitive should have little or no ambiguity, so also a well-selected axiom should be quite obviously true. As the Wikipedia article on Axioms states: “an axiom is a premise so evident as to be accepted as true without controversy”11. This may seem like a requirement that may not be completely satisfiable in many cases, but I have found that in most cases more contention arises from the definitions that correspond to natural language terms as described above than from the selection of axioms. Note that ambiguity is not introduced by axioms: like any other statement in a formal system, the only ambiguity derives from ambiguity in the primitives.
Given a set of primitives and axioms, the predicate calculus can then be used to derive theorems in the system. These theorems are necessarily true given the truth of the axioms: there can be no challenge to the theorems themselves. Any challenge to the truth of a formal system must be directed at its axioms. The truth or falsehood of the entire system rests on their shoulders. Likewise, as stated above, there can be no challenge to the ambiguity of any statement, whether an axiom or theorem. Any challenge to the clarity or meaning of any statement in the system must be directed at the system’s primitives.
The Advantages of Formal Systems
There are numerous advantages of using formal systems for the purposes of philosophical discourse which I will describe in detail in a moment. First, let me state the one main disadvantage of using formal systems: doing so is extremely hard. Natural language is so-called because its use is natural, especially to those trained in or with a dispensation for literary prose. The use of natural language allows a fuzziness and ambiguity that is not allowed in formal languages. For this reason, it can be very difficult to translate what seems like a perfectly clear concept or statement in natural language into a formal system. Nonetheless, I believe that the benefits of doing so justify this effort.
The biggest advantage of formal systems, as I have mentioned above, is their limited ambiguity. In natural language, every term, every word, is a potential source of ambiguity. In formal languages, this ambiguity is reduced to the primitives of the language. Identifying and/or selecting these primitives can be extremely difficult, not the least because all of the other terms in the language must be definable from these primitives. However, not only is this process beneficial in that it reduces ambiguity, but it is also very helpful in the mental process of constructing a theory. Using formal languages forces the philosopher to carefully consider each term in the language of the theory, a process that is often done in only a cursory manner. I personally have found that being forced to carefully consider every term, either as a primitive or as a formal definition, provides huge benefits with respect to my own understanding of a theory and it’s implications.
Another advantage of formal systems is that attacks on the system can only be directed at the primitives and axioms. In normal philosophical discourse, any facet of a particular argument is open to such an attack, including the meaning of any of the terms, the validity of any of the statements, or the validity of the logic used to derive one statement from another. This is not true of formal systems. The validity of the logic cannot be questioned (unless there is a mistake in the application of the rules of the predicate calculus, which could be discovered by an automated process). The validity of theorems cannot be questioned given the validity of the axioms. The validity of definitions cannot be questioned as they are mere conveniences. One caveat I have mentioned above is that the definition of terms in the formal system that have specific meanings in natural language can be disputed, but these disputes affect only the translation from the formal system to natural language. The meaning of the formal system is unaffected by the outcome of these disputes.
As an example, let us attempt to translate St. Anselm’s ontological proof of the existence of God12 into a formal system. It goes with out saying that such translations are difficult, and that for any particular argument there is more than one way to render the prose into formal syntax. This is a function of the ambiguity of the original prose rather than a limitation on the expressiveness of formal languages. For St. Anselm’s proof, I will treat the notion that we can conceive something in our mind as representing the modal concept of possibility, and the statement that God must exist as saying that necessarily God exists. Given this, I see his system expressed as follows:
- Greater than (symbolized by >): In this case meaning a comparison of two things on some sort of scale of worth, holiness, or perfection.
- Greater than or equal to (symbolized by ≥): x≥y ≡ x>y || x=y, where ≡ means “is defined as”
- God, or strictly speaking the set of all things that is God: x∈God ≡ ∀y[ x≥y ] – i.e., God is the greatest of all things
- ◊∃x[ x∈God ] – This is the assumption that God’s existence is possible, or that we can conceive of Him in our minds
- ◊∃x[ x∈God ] ⇒ □∃x[ x∈God ] – This is the assumption that if we can conceive of God, then He must exist.
- ◊∃x[ x∈God ] – Postulate #1
- ◊∃x[ x∈God ] ⇒ □∃x[ x∈God ] – Postulate #2
- □∃x[ x∈God ] – (1), (2), and modus ponens (a deduction rule of the predicate calculus).
Line three of the proof states that necessarily there is a God. Given my translation of his argument as shown, the proof is indeed valid (steps 1-3 are logically valid). Therefore, any attack on Anselm’s proof as translated must be directed against the primitives and postulates. There are problems with both:
- The primitive “greater than” is an extremely poor primitive, in that it is not clear at all what is meant by it. Anselm expects us to understand that this means some sort of divine metric, some measurement of perfection. However, given any two objects, it is clear that different people will come up with a different answer as to which of the two objects is “greater”. The ambiguity in the primitive carries forward to ambiguity in every statement in this system.
- The second postulate is problematic because in general possibility does not imply necessity. Anselm would want us to accept this postulate as an inherent property of his primitive “greater than”. However, to do so, we should have a better understanding of this primitive. In fact, to solve this problem and the previous one, “greater than” should be defined in terms of more basic (and thus better) primitives, and this postulate should emerge from that definition.
I should also point out that there are interpretive problems with this proof. For one, the definition of God may be inadequate. Simply “greater than” or equal to everything else does not imply any other properties. For example, it does not imply the properties attributed to the Christian God, such as omniscience, omnipresence, omnipotence, and omnibenevolence. Again, these properties might emerge from a more fundamental characterization of the primitive “greater than”. Another interpretive problem is that there is nothing in this system that prevents the existence of more than one god. Any statement that said this would have to be added as an additional postulate; this system certainly does not prove this.
These criticisms do not falsify Anselm’s proof. It is conceivable that the argument could be improved, essentially by defining “greater than” in more fundamental terms that represent better primitives, and which lead to some of the statements that are now taken as postulates (although I doubt that postulate #2, which is essential to the proof, could ever be satisfactorily supported). The point of this exercise is to demonstrate that by translating the argument into a formal system, we challenge hidden assumptions, especially about the terms used in the natural language “proof”, that would otherwise go unchallenged.
Formal is Better
It should be clear by now that using formal languages and proofs in philosophy would be greatly beneficial to the entire enterprise. The reasons for doing so can be summarized as follows:
- The philosopher is forced to carefully examine her system, leading her to a better understanding of her terms, assumptions, and methods.
- Much of the ambiguity that exists in natural language is removed by expression in a formal system.
- The ambiguity that does exist is derived solely from the ambiguity in the primitives of the system.
- Any fundamental assumptions made in the argument must be documented explicitly as axioms or postulates.
- Proofs executed using a formal calculus are easier to examine for possible logical mistakes than those expressed in natural language.
- The strictly enforced structure of a formal system allows criticisms of the system to be directed only at specific parts of the system; i.e., primitives for ambiguity, axioms for truth, and proofs for validity of the deductions.
- The explicit nature of the definitions of the system likewise make it clear where any possible interpretive problems exist with the system.
- The lack of ambiguity in the consequences of the system (given that the primitives are unambiguous), makes it clear exactly what is and is not claimed by the system.
As mentioned above, the one problem with formal arguments is that using them is very difficult. But given their many advantages, it would seem that they are well worth the price of this difficulty13.
- For a history of cosmology since the beginning of the twentieth century, see Tenn, J.S., “Cosmology Since 1900”, Sonoma State University, September 22, 2012, URL=<http://www.phys-astro.sonoma.edu/people/faculty/tenn/cosmologysince1900.html>.
- For more details on this see my essay on truth.
- In my opinion, three of the most important subjects that still fall under philosophy are the nature of universals, the validity of the scientific method, and the entire subject of ethics.
- See for example the requirements for a Ph.D. in philosophy from Princeton University, published online at http://www.princeton.edu/gradschool/about/catalog/fields/philosophy/#requirements.
- See for example, Quine’s remarkable summary of the subject of set theory, Set Theory and Its Logic (Harvard University Press, Cambridge, MA, 1963).
- Quine’s two set theories are New Foundations, and a variation on it called Mathematical Logic. See Quine, W.V.O., “New Foundations”, From a Logical Point of View (Harvard University Press, Cambridge, MA, 1953). and Mathematical Logic, Harvard University Press, Cambridge, MA, 1940.
- See for example Enderton, Herbert, A Mathematical Introduction to Logic, Academic Press, San Diego, CA, 1972.
- Ibid., pg. 268 (Section 4.1) and pg. 281 (Section 4.4).
- Dictionary.com defines “be” as “to exist or live”, and it defines “exist” as “to have actual being; be”. (“be”, Dictionary.com, URL=<http://dictionary.reference.com/browse/be?s=t>; “exist”, Dictionary.com, URL=<http://dictionary.reference.com/browse/exist?s=t>.)
- For a detailed description of one such deductive calculus, see Enderton, op. cit., pg. 101 (Section 2.4).
- “Axiom”, Wikipedia, URL=<http://en.wikipedia.org/wiki/Axiom>.
- “Ontological argument”, Wikipedia, URL=<http://en.wikipedia.org/wiki/Ontological_argument#Anselm>.
- To see a more significant example of a formal philosophical system, see my Theory of Ethics. Examples of rigorous proofs from such a theory can be found in my essay on Formal Applications.