Presentation on theme: "Propositional Logic – The Basics (2)"— Presentation transcript:
1Propositional Logic – The Basics (2) Truth-tables for Propositions
2Assigning Truth L ● ~ F 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.”“This is a class in introductory-level logic, which does not include a study of informal fallacies.”L ● ~ F
3How about this one? L ● F T F F “This is a class in introductory logic, which includes a study of informal fallacies.”“This is a class in introductory logic (T), which includes a study of informal fallacies (F).”L ● FT FF
4Propositional Logic and Truth The truth of a compound proposition is a function of:The truth value of it’s component, simple propositions, plusthe way its operator(s) defines the relation between those simple propositions.p ● q p v qT F T FFT
5Truth 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.
6Truth and the Actual World Some statements describe the actual world - the existing state of the world at “time x”; the way the world in fact is.“This is a logic class and I am seated in SOCS 203.”- Actually and currently true on a class day.- Possibly true, but not “currently” true on Monday, Wednesday or Friday.
7Truth and Possible Worlds Some statements describe possible worlds - particular states of the world at “time y”; a way the world could be..“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.
8Constructing 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 2n 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.
9Constructing Truth Tables - # of Lines For statement forms, there are only two symbols. Thus, these require lines numbering 22 power, or 4 lines.pq184.108.40.206.pq220.127.116.11.
10Constructing Truth Tables – Distribution across all Symbols 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.”p●q18.104.22.168.p≡q22.214.171.124.T T F FT T F F
11Constructing Truth Tables – Distribution across all Symbols 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.”Repeat for lines 3 & 4, inserting “T” and “F” respectively.p●q126.96.36.199.p≡q188.8.131.52.T T F FT T F FT FT FT FT F
12Constructing Truth Tables – Operator Definitions Thinking about the corresponding English expressions for each of the operators, determine which truth value should be assigned for each row in the table.p●q184.108.40.206.p≡q220.127.116.11.T T F FT T F FTT FT FTF FF F FT FT FT
13Constructing Truth Tables - # of Lines Remember that you are counting each symbol, not how many times symbols appear.( p≡q )●q18.104.22.168.2 symbols: 1 appearance of “p” and 2 appearances of “q”
14Exercises - 1 1. 2. 3. 4. ( M > P ) v ( P > M ) T T F F T F T T Using the tables which define the operators, determine the values of this statement.22.214.171.124.( M > P ) v ( P > M )T T F FT F T TT F T FT T T TT F T FT T F TT T F F
15Exercises – 2Using the tables which define the operators, determine the values of this statement.[(Q>P)●( ~QR)]~(PvR)126.96.36.199.188.8.131.52.T T T T F F F FT T F F T T T TT T F F T T F FT T F F T F T FF F F F T T T TT T T T F F F FT T T T T F T FT F T F T F T FF F F F F F F FF F F T F F F TT T F F T T F FT T T F T T T FT F T F T F T FT T F F T T F F