Presentation on theme: "Propositional Logic – The Basics (2) Truth-tables for Propositions."— Presentation transcript:
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. L ~ FL ~ F
How about this one? This is a class in introductory logic, which includes a study of informal fallacies. L F T F This is a class in introductory logic (T), which includes a study of informal fallacies (F). F
Propositional Logic and Truth The truth of a compound proposition is a function of: a.The truth value of its component, simple propositions, plus b.the way its operator(s) defines the relation between those simple propositions. p qp v q T FTFFT
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 value 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 This is a logic class and I am seated in SOCS Actually and currently true on a class day. - Possibly true, but not currently true on Monday, Wednesday or Friday.
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. A truth table describes all possible combinations of truth values for a statement. It will, in fact, even tell you if a statement could not possibly be true in any world.
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 - # of Lines pq For statement forms, there are only two symbols. Thus, these require lines numbering 2 2 power, or 4 lines. pq
Constructing Truth Tables – Distribution across all Symbols pq pq 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. TTFFTTFF TTFFTTFF
Constructing Truth Tables – Distribution across all Symbols pq pq 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. TFTF TFTF TFTF TFTF
Constructing Truth Tables – Operator Definitions pq pq 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. T T T FFFFFF F
Constructing Truth Tables - # of Lines ( pq )q Remember that you are counting each symbol, not how many times symbols appear. 2 symbols: 1 appearance of p and 2 appearances of q
Exercises - 1 Using the tables which define the operators, determine the values of this statement. ( M > P ) v ( P > M ) TTFFTTFF TFTFTFTF TTFFTTFF TFTFTFTF TFTTTFTT TTFTTTFT TTTTTTTT
Exercises – 2 Using the tables which define the operators, determine the values of this statement. [(Q>P)( ~Q>R)]~(PvR) TTTTFFFFTTTTFFFF TTTTFFFFTTTTFFFF TTFFTTFFTTFFTTFF TTFFTTFFTTFFTTFF TFTFTFTFTFTFTFTF TTFFTTFFTTFFTTFF FFFFTTTTFFFFTTTT TFTFTFTFTFTFTFTF TTFFTTTTTTFFTTTT TTTTTFTFTTTTTFTF TTTFTTTFTTTFTTTF FFFTFFFTFFFTFFFT TTFFTFTFTTFFTFTF FFFFFFFFFFFFFFFF