What is Symbolic Logic?

What is symbolic logic?

This is a math course—that’s why it satisfies the Quantificational requirement—but it is a significantly different kind of math course than you are probably used to. It is also a language course—but again, we will take an approach to language that differs quite a bit from the approach we take in literature courses or in foreign language courses.

The differences bring some challenges, but they may also bring something that changes your orientation towards math and towards language—if you already like math, or languages, or both this course will give you a slightly different focus in exercising some of the same cognitive muscles; if you don’t already like math (which I didn’t when I took my first course in logic) but you do like languages (which I did), it might exercise these cognitive muscles in ways that will make learning and doing math slightly more interesting to you. If you don’t like either math or languages, and you don’t want to exercise those cognitive muscles, well,… there is only so much I can do.

So, what is logic?

“Logic is a science that aims to identify principles for good and bad reasoning” (Agler, p.1)

Alright, so what is symbolic logic?

“Symbolic Logic is a branch of logic that represents how we ought to reason by using a formal language consisting of abstract symbols” (Agler, p. 1)

What is a formal language—a language that dresses in tails for dinner, that refers to people by their title, and knows which fork goes with which dish?

English is not a formal language—it is what we call a natural language. It consists of words, sounds, symbols, rules and conventions that have developed over time, out of other languages, in the concrete situations in which human beings have communicated with one another for a variety of purposes. It is constantly changing and evolving through use, through the addition of new words, through the loss of others, and it has a decidedly different look, feel, cadence, and tone, depending on where it is being used and by whom. Native speakers begin to learn it and use it before they have any idea that what they are doing is called ‘using a language’. We might say that its use is simply continuous with all the other forms of communication between persons that begin well before we develop our linguistic capacities.

Reasoning in English can be an extremely complicated affair. I have to know what all these different words mean, recognizing that they get used slightly differently (or radically differently) depending on things like context. Grammatical rules can help, but there are times when it isn’t wrong to break some general rule of grammar, or even when purposely breaking the rule allows us to communicate something different from what we would communicate through following it—‘what ya gettin’ into?’ means something different from ‘into what are you getting?’ ‘Dunnit?’ Knowing when and where you can and cannot draw certain conclusions from what is said in English is a messy task.

Computer languages (programming languages like C ++, Python, and AppleScript, and markup languages like HTML, XML, and LaTex) are formal languages (or artificial languages). They have precisely defined rules of grammar (or syntax) and the meaning of the symbols used (or semantics) is clear and unambiguous. This makes them much easier to use for certain purposes—like giving commands; telling a computer exactly what to output, given any specific input.

When we adopt a formal language for the sake of reasoning, and for the sake of identifying and codifying patterns of reasoning that are paradigmatically good (or paradigmatically bad) we are taking a formal approach to logic; we are doing symbolic logic.

(This allows us to move around in pretty rarified air—and to deal very abstractly with the business of thinking and reasoning about how we ought to think and reason. There are different schools of thought concerning informal vs. Formal approaches to logic (Just as there are different schools of thought concerning whether athletes, writers, chefs, or musicians are better for relying on natural talent and practice, or whether they are better for subjecting themselves to very specific, regimented training programs that are devised according to some theory.) We aren’t going to get into any specifics concerning these differences, but I wanted to signal them because I think they are really interesting… oh, and because they might bear on our thinking about things like whether computers are (or could be) intelligent, or whether intelligence is something that has some necessary connection to the natural history and biology of a particular species.)

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