LESSON 4 - Translating from Natural Language

Symbolic logic is most often taught as a way of examining arguments for deductive validity.  For instance, in the following argument the conclusion does follow from the premises even though nothing said here is actually true.

You have already seen something about symbolic logic just as a technique for working with symbols, regardless of what the symbols might represent.  Now we are going to learn more about how to express ordinary factual statements symbolically.  The key is that anything said to be factual is going to be either true or false, so this leaves out probabilities and it also leaves out value judgments.

Let's go back to something from the first lesson.  There we introduced the idea of working with variables and signals.  I'll repeat some of what we had there.

Let's keep it simple by talking only about the propositional variables (the letters in the alphabet from P to U) and these six signals expressing our key logical relationships:
        A (think of the "a" in "and") to express conjunction (and, although, but)
        B to express a biconditional or equivalence relationship (if and only if)
        C to express implication or a conditional relationship (if...then, ... only if ...)
        M to express the opposite of equivalence, which is a mutually exclusive relationship (either...or but not both)
        N to express negation (not)
        O (think of the "o" in "or") to express disjunction (or, unless)

PQA, then, represented the conjunction of the ideas represented by P and Q.
PQO is their disjunction (setting up alternatives where both could happen).
PQC sets up a conditional relationship.  See more about what this means by going to the SLIPS page on conditionals.

Let's set up a code for some of these letters that we use as variables.  (To keep things simple we'll also use words that match the letters.)
       P: Peter plays.
        Q: Quinton quits.
        R: Roger refuses.
        S: Sam shrugs.
PQA then expresses the thought that two things happen: Peter plays but Quinton quits.  It would also "translate" Peter plays although Quinton quits.
PQO could mean either Peter plays or Quenton quits.  This is the same as Peter plays unless Quinton quits.
PQC tells us that if Peter plays Quinton quits.  Also, Peter plays only if Quinton quits.
QPC turns things around: if Quinton quits then Peter plays or Quinton quits only if Peter plays.
PQARC tells us that Roger refuses if Peter plays but Quinton quits.  Note here that the English word order and the order of the two parts of the conditional do not always match, and this will be one of the biggest problems for the beginning student.
PQARSOC tells us that if Peter plays but Quinton quits then either Roger refuses or Sam shrugs.

Up to now we have been talking about truth values as colors (red or blue), but for working with sentences we will (finally) talk about them as either true (1) or false (0).  Any wff that is not just itself a variable is true or false depending on the truth values of the variables that are there.  To see all the possibilities for any wff we need to double the number of lines for each new variable we introduce.  For the last wff above the result would look like this:
 

What this table tells us is that the sentence will be false only under these conditions: Peter plays but Quinton quits and at the same time Roger doesn't refuse and Sam doesn't shrug.

Doing long truth tables by hand is rather tedious.  You can see the results very quickly by using the program "PLN2-colors" available from the files in the YahooGroups community we have for symbolic logic.  You need to be working with a PC and Windows and the Microsoft Explorer browser for this, but by going to the website you can download the program to your own computer and try it out at your convenience.

The main reason we think about all the possibilities for a sentence is that this is one way we can see whether an argument using one or more sentences as premises and another sentence as a conclusion is deductively valid, meaning that it is not going to happen that the premises could all be true while the conclusion is false.

Let's look at two examples.
(1) If Peter pays then Quinton quits, but Quinton is not quitting, so Peter must not have paid.
(2) If Peter pays then Quinton quits, and Quinton is quitting, so Peter must have paid.

We'll symbolize each (the differences in tense will not ordinarily matter) and look at truth tables that will show us what could happen.  To make things easier we will set up the variables themselves in separate columns.
 

What the tables are telling us is that it is possible in the second argument for the premises to be true and at the same time have the conclusion be false (what we will call a "bad line").  This goes against the definition of a valid argument, so the second argument is not deductively valid: Quinton might quit even though Peter doesn't pay.

We'll be testing arguments with truth tables later on, but for right now the task is to develop more facility in "translating" sentences.  In the assignment to go with this lesson you will have a code for the variables and your task will be to symbolize correctly the sentences that you are given.  The letters will not, however, match the words used (for instance, P will be the idea that logic is easy).

Go on to assignment 4.