Phil. 4

The relevance of symbolic logic to other subjects






Let's consider why symbolic logic is of special interest to the philosopher. Applying the formal techniques of logic to a vague philosophical argument can help to clearly display the controversial parts of the argument. Symbolic statements are free of vagueness and ambiguity. For example, one philosopher claims that from the premise "God is loving and all-powerful" she can deduce the sentence "There shouldn't be earthquakes or murder or any other evil in the world." Some philosophers initially are likely to agree that this is a valid deduction; others are likely to disagree. One reason for their disagreement is that it's so hard to tell just what the two sentences are really saying. But if the sentences are translated into symbolic logic, then the translated sentences will be precise. With precise sentences it is much clearer whether the conclusion does or doesn't follow from the premises. If the conclusion doesn't follow, then it will be clearer just what else must be assumed to make the conclusion follow. Then the philosophers can concentrate on discussing whether these additional assumptions are acceptable. Therefore, the use of symbolic logic can help (and has helped) direct the philosophers' discussions toward the crucial points in their disputes.

Some philosophers believe that symbolic logic can reveal the structure of all possible good inference, and so reveal the common skeletal structure that underlies all reasonable thought processes. Bertrand Russell, Ludwig Wittgenstein, and other 20th century philosophers have argued that there is an intimate connection among these three things: logic, our mind, and the deep structure of the physical world.   This issue is discussed in Phil. 154 (language), Phil. 176 (20th Century Anglo-American Philosophy), and Phil. 181 (metaphysics).

The symbolic analysis of our natural language can reveal exciting new information about the character of language itself. For example, can all the grammatical sentences of English, but none of the ungrammatical ones, be generated mechanically by using a small number of symbolic rules? Can all the meaningful sentences of English, but none of the nonsensical ones, be generated mechanically by using a small number of symbolic rules? The attempt to answer these questions is an active area of contemporary philosophical research begun by Noam Chomsky at M.I.T.  This topic is taken up in Phil. 154.

Logic also impacts philosophy in other ways. Consider this seemingly good inference that has, unfortunately, an unacceptable conclusion. "Because 9 is the number of planets in our solar system, and because it is logically necessary that 9 is greater than 5, it follows by substitution that it is logically necessary that the number of planets in our solar system is greater than 5." This conclusion is not correct because the solar system might have contained fewer planets if it had evolved differently. This paradox about substitution is an unsolved problem in philosophy. 

Finally, symbolic logic is a very useful tool for clarifying the philosophically important concepts of meaning, truth, and proof. You will learn how to clarify proofs in Phil. 60, but attention to truth will have to wait for Phil. 160 (the sequel course to Phil. 60), and attention to meaning is given most attention in Phil. 154.



Now let's consider why symbolic logic is of special interest to the computer scientist. The short answer is that computer science is just logic implemented in electrical engineering.

One area of computer science is A.I. or artificial intelligence. An A.I. process is a process by which a computer or robot is able to perform tasks which, when they are performed by humans, require intelligence. For example, A.I. researchers hope to build a machine that can read an article written in Chinese and produce a summary of it in English. Researchers generally believe that making progress on this task of getting a computer to use English intelligently will require a massive introduction into the computer of knowledge about the world outside the computer. How are the researchers going to give all this knowledge to the computer so that it is available in a way that the computer can use it? Many A.I. researchers believe the key to success is to translate this knowledge into symbolic logic rather than into ordinary computer languages.

Here is a December 1999 quotation from a famous computer scientist, Hans Moravec of Carnegie Mellon University, in Scientific American magazine:

"Robot intelligence will surpass our own well before 2050. In that case, mass-produced, fully educated robot scientists working diligently, cheaply, rapidly and increasingly effectively will ensure that most of what science knows in 2050 will have been discovered by our artificial progeny."

Computers are logic machines in two senses: their electronic design follows basic principles of symbolic logic, and their programs are themselves based on principles of symbolic logic. More specifically, computer science is involved with symbolic logic in the following five ways:

(1) The first programming language evolved from the language of classical symbolic logic.

(2) The electrical engineer who designs digital computers creates the machines' gates and networks on its chips according to the principles of Sentential Logic, that is, Boolean Algebra.

(3) Symbolic logic is useful for simplifying complicated electrical circuits. The techniques of symbolic logic are used to create a simpler circuit that works the same as a more complicated and more expensive circuit.

(4) Symbolic logic is useful for analyzing the theoretical limits of ideal digital computers. Symbolic logic techniques can be used to establish what functions a computer can and cannot compute (in principle, that is, with no limits on the size of memory or the amount of time available). The techniques can be used to establish limitations of speed for certain kinds of calculations, and to establish whether a computer program will in principle correctly do what its programmer intends to have designed it to do.

(5) Symbolic logic techniques are used in automated reasoning programs. Automated reasoning programs can create the proofs of some statements, not simply check a proposed proof.



Symbolic logic is of special interest to the mathematician because predicate logic, augmented by some principles of set theory, is capable of expressing every mathematical statement without significant loss of its content. Thus the proofs and theorems of any field of mathematics can be translated into proofs and theorems of logic. When fields of mathematics are represented this way as a part of logic, the logician can more clearly see the extent of that field of mathematics and see its assumptions (such as its axioms). The automatic theorem-proving procedures of the logicians can be (and have been) applied to discover new theorems of mathematics that the mathematicians working alone never discovered. Also, after translating a mathematical theory into symbolic logic it is much easier to establish the answers to such questions as "Will this theory permit the deduction of a contradiction?" and "Could there be a machine which could always correctly answer whether a given statement is a theorem of this theory?"

The details of the ideas mentioned above about computers, philosophy, and mathematics are explored in detail in other courses, and you aren't expected in this course to know much about computers, philosophy, or mathematics. This course will be simply an introduction, giving the basics of symbolic logic plus an overview of how that logic can be applied. This course is a prerequisite to Philosophy 160, which continues the study of symbolic logic. Symbolic logic is a central topic in Math 161; and symbolic logic is studied further in several computer science courses at our university.