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  How to Think About the Logical Connectives in Propositional Logic

My goal with this logic course has been to present just enough logic to be useful for critical thinking. That includes the formal logic needed to do argument analysis (which is very little, actually), but another benefit is the careful attention to meaning and language that the study of logic helps to cultivate. I think this has significant critical thinking benefits as well.

One of the downsides of giving a fairly superficial introduction to logical concepts is that you might miss out on the opportunity to correct misunderstandings about what logic is and how it works, which a more thorough and rigorous treatment could address.

One common set of misunderstandings surrounds the interpretation of the basic logical connectives — “and”, “or”, and “if-then” and “not-“. In the videos and course materials I’ve written basic statements of propositional logic like this:

  • not-A
  • A and B
  • A or B
  • If A then B

I’ve used the ordinary English word “and” to represent the operation of logical conjunction, the ordinary English word “or” to represent the operation of logical disjunction, and so on.

I did this because I wanted to make the material as accessible as possible. Some people’s brains just turn off when they see a symbol.

But it’s important to know that in using the English words for these logical connectives I’m deviating from the standard practice in symbolic logic. There are several different symbolic conventions, and textbook writers will pick one and stick with it:

  • not-A
    • ~ A
    • ¬ A
  • A and B
    • A & B
    • A • B
    • A ∧ B
  • A or B
    • A ∨ B
    • A | B
  • If A then B
    • A ⊃ B
    • A → B

So in our course I would symbolize a sentence like

“Either John will study hard tonight or he won’t go to the movies tomorrow”

as follows:

“S or not-M”

where S = “John will study hard tonight”

and M = “John will go to the movies tomorrow”.

But in a standard logic text the symbolization might look like this:

S ∨ ~M

Is this a superficial difference or an important difference? What’s the point of these symbolizations anyway? Is it just for convenience, or is there a principled reason for distinguishing between sentences of natural language and sentences of a formal language.

The answer is “no”, it’s not just for convenience, and “yes”, there is a principled reason for distinguishing between sentences of natural language and sentences of a formal language.

In this section I want to talk about why the use of symbols is standard practice, and what the symbols tell us about the nature of formal logic and its relationship to natural language.