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A formal vocabulary

15 June, 2015 - 15:24

Imagine an offer where, for a mere $6.99, you can get: EE, (FF or CF or OB or HB) or CC and PH and BR and GR or WB and PJ. Some fine print clarifies for us that BR includes T (Whi, Whe, Ra, or Hb), FT, HM (Bb, Ba, or Ca), EM, B with CrCh, BB (GR from 6-11am). Unfortunately, it's not clear at all how the "and" and "or"s relate. Fundamentally, is "x and y or z" meant to be interpreted as "(x and y) or z", or as "x and (y or z)"? With some context, we might be able to divine what the author intended: the above ofer is the direct translation from the menu of a local diner 2 : 2 eggs, potatoes (french fries, cottage fries, O'Brien or hashed brown) or cottage cheese and peach half (grits before 11am) and choice of bread with gravy or whipped butter and premium jam. Bread choices include toast (white, wheat ,raisin or herb), hot four tortillas, homemade muffin (blueberry, banana or carrot), English muffin, bagel with cream cheese, homemade buttermilk biscuits. Grits available from 6:00am to 11:00am. (In a brazen display of understatement, this meal was called "Eggs Alone".) Even given context, this offer still isn't necessarily clear to everybody: can I get both french fries and a peach half? Happily, coffee is available before having to decipher the menu. In this example, parentheses would have clarified how we should interpret "and", "or". But before we discuss how to connect statements, we will consider the statements themselves.

Definition 2.1: proposition

A statement which can be either true or false.

Example

Your meal will include hashbrowns.

Definition 2.2: propositional variable

A variable that can either be true or false, representing whether a certain proposition is true or not.

Example

HB

We will often refer to "propositional variables" as just plain ol' "propositions", since our purpose in studying logic is to abstract away from individual statements and encapsulate them in a single variable, thereon only studying how to work with the variable.

For a proposition or propositional variable X, rather than write "X is true", it is more succinct to simply write "X". Likewise, "X is false" is indicated as "¬X".

ASIDE: Compare this with Boolean variables in a programming language. Rather than (x == true) or (x == false), it's idiomatic to instead write x or !x.

Observe that not all English sentences are propositions, since they aren't true/false issues. Which of the following do you think might qualify as propositions? If not, how might you phrase similar statements that are propositions?

  • "Crocodiles are smaller than Alligators."
  • "What time is it?"
  • "Pass the salt, please."
  • "Hopefully, the Rice Owls will win tomorrow's game."
  • "Mr. Burns is filthy rich."
  • "Fresca® is the bee's knees."