Propositional Logic – The Basics (2) Truth-tables for Propositions

Assigning Truth True or false? – “This is a class in introductory-level logic.” “This is a class in introductory-level logic, which does not include a study of informal fallacies.”

How about this one? “This is a class in introductory logic, which includes a study of informal fallacies.” Symbolize this statement, and determine if it is true or false.

Propositional Logic and Truth The truth of a compound proposition is a function of: a.The truth value of it’s component, simple propositions, and b.the way its operator(s) defines the relation between those simple propositions.

Truth Table Principles and Rules Truth tables enable you to determine the conditions under which you can accept a particular statement as true or false. Truth tables thus define operators; that is, they set out how each operator affects or changes the content of a statement.

Some statements describe the actual world - the existing state of the world at “time x”; the way the world in fact is. Truth and the Actual World Provide an example or two of such a statement. “Translate” your statement into symbolic form.

Some statements describe possible worlds - particular states of the world at “time y”; a way the world could be.. Truth and Possible Worlds “This is a history class and I am seated in SOCS 203.” Possibly true, but not currently true. Actually true, if you have a history class here and it is a history class day/time. Your example of a possibly but not actually true statement:

Constructing Truth Tables 1. Write your statement in symbolic form. 2. Determine the number of truth-value lines you must have to express all possible conditions under which your compound statement might or might not be true. Method: your table will represent 2 n power, where n = the number of propositions symbolized in the statement. 3. Distribute your truth-values across all required lines for each of the symbols (operators will come later). Method: Divide by halves as you move from left to right in assigning values.

Constructing Truth Tables – Distribution across all Symbols p●q p≡q Under “p,” divide the 4 lines by 2. In rows 1 & 2 (1/2 of 4 lines), enter “T.” In rows 3 & 4, (the other ½ of 4 lines), enter “F.”

Constructing Truth Tables – Distribution across all Symbols p●q p≡q Under “q,” divide the 2 “true” lines by 2. In row 1 (1/2 of 2 lines), enter “T.” In row 2, (the other ½ of 2 lines), enter “F.” TTFFTTFF TTFFTTFF Repeat for lines 3 & 4, inserting “T” and “F” respectively.

Constructing Truth Tables – Operator Definitions p●q p≡q TTFFTTFF TTFFTTFF TFTF TFTF TFTF TFTF Thinking about the corresponding English expressions for each of the operators, determine which truth value should be assigned for each row in the table.

Constructing Truth Tables - # of Lines For statement forms, there are only two symbols. Thus, these require lines numbering 2 2 power, or 4 lines. Remember, you are counting each symbol, not how many times symbols appear. ( p≡q )●q

Exercises - 1 Using the tables which define the operators, determine the value of this statement. ( M > P ) v ( P > M )

Exercises – 2 Using the tables which define the operators, determine the value of this statement. [(Q>P)●( ~Q>R)]●~(PvR) TTFFTTFFTTFFTTFF