Ella failed to drop the class. Ella dropped the class.

Propositional logic In many cases, a sentence is built up from one or more simpler sentences. To see what follows from such a complicated sentence, it is helpful to distinguish the simpler sentences from which it is built (its components), and how they are put together to form the more complicated sentence. We will use a single capital sentence to represent each of the simplest sentences. The connectors that apply to simple sentences to form compound (complex) sentence are represented by operators. An atomic sentence is one that does not contain any sentential operators. We will use four truth-functional operators. To say these are truth-functional means that the truth-value (truth or falsity) of the whole is completely determined by the truth-values of its component sentences and the operations applied to those components. 4 common truth-functional operators Symbol Sentence type Sentence form Read as negation A Not-A conjunction A B A and B disjunction A B A or B conditional A B If A then B Translating sentences from English into the language of propositional logic Use negation for any sentence that denies or negates a statement. 'Not' is the most common indicator that an English sentence should be translated as a negation. However, you should also use ' ' to translate any sentence that denies or negates a statement. Sometimes what we treat a sentence as denying or negating will depend on features of context, including our interests. If we are interested in whether or not Ned locked the door, we may not care about the differences between Ned failed to lock the door. Ned neglected to lock the door. Ned forgot to lock the door. We would then treat all of these as negating 'Ned locked the door.' However, if we are concerned with Ned's memory, we might want to point out the many things Ned has forgotten lately. In that context, we would treat the last one as negating 'Ned remembered to lock the door. More examples: This sentence might be seen as negating. John is not home. John is home. Ella failed to drop the class. Ella dropped the class. Not everyone was satisfied. Harry refuses to sign. Everyone was satisfied. Harry is going to sign. Harry is willing to sign. When a pair of descriptions that are opposites are used, it is best to see one of the pair as occurring in an atomic sentence, and treat sentences that contain its opposite as negations, as in these pairs: 1

i. Ned remembered Harry's phone number. R Ned forgot Harry's phone number. R ii. Rita passed calculus. P Rita failed calculus. P iii. Calculus is easy E Calculus is difficult. E iv. The kitchen light is on. L The kitchen light is off. L Use ' ' to express anything that says both of two things, or a combination of things, is true. 'And' is not the only indicator of conjunction: Rent and car insurance are more expensive in California than in Oklahoma. Herbert moved to Atlanta, as did Martin. Logic is required for math majors as well as computer science majors. Karen's new job provides paid vacation in addition to health insurance. There are several words in English that express contrast or tension, besides expressing conjunction. The language of propositional logic cannot capture this. We treat these the same way we treat 'and', recognizing that we lose some of the nuances of English as a result: Harry is passing French, but failing calculus. Martha likes George, though Jean doesn't. 'Sometimes 'while' is used this way, but sometimes used to express a time relationship. A time relationship cannot be represented by a truth-functional operator. conjunction: Harry is passing French, while he's failing calculus. atomic: Rita babysat while Harry went to class. Not every use of the word 'and' indicates a conjunction. 1. Some sentences with 'and' are really conditional (If then ) statements: Hit your brother again and you'll be sorry. 2. Some sentences with 'and' between two names or terms express a relationship rather than a conjunction. Contrast Seth and Jack are the same height. Seth and Jack are classmates. with Seth and Jack are 6 feet tall. The last sentence tells us something about Seth, which is independent of the information we are given about Jack. We are attributing a characteristic to each of them, and it is the same characteristic in both cases. We can see it as a combination of one statement about Seth, and another about Jack. In contrast, the first sentence doesn't make any statement about Seth alone, independently of Jack, or Jack alone, independently of Seth. Rather, it makes one statement, describing them as having a certain kind of (height) relationship. The second sentence is also like this. In English, we use 'or' in two different ways. The inclusive 'or' is used to say at least one of two things is true, including the possibility that both are. The exclusive 'or' excludes 2

the combination. The use of 'either or ' may suggest an exclusive 'or' but does not force it, and in some sentences 'or' alone may express an exclusive 'or': You'll get a potato or rice with your main dish. You need to rely on context to help you determine which is being expressed. We will read 'or' as the inclusive 'or', except where we have a particular reason to think the exclusive 'or' is intended. The ' ' expresses the inclusive 'or'. Although we do not have a symbol specifically for the exclusive 'or', we can express it with a more complicated sentence: Harry or Carla will be given the job. (H C) (H C) Sentences with 'neither nor' are best treated as denials of disjunctions. Neither Oscar nor Elaine will be promoted. (O E) Karen eats neither meat nor cheese. (M C) It may be tempting to think that the negation sign distributes to the parts, but it does not. Continuing to using 'M' for 'Karen eats meat' and 'C' for 'Karen eats cheese', here's what happens if we apply negation to each disjuncts separately: M C Karen doesn't eat meat or Karen doesn't eat cheese This would be true if she eats one but not the other (as well as if she doesn't eat either one). In contrast, (M C) Karen eats neither meat nor cheese will only be true if she doesn't eat meat and she also doesn't eat cheese. More complex sentences Use parentheses as you do in mathematics, to indicate which operation is performed first. Operations inside parentheses are performed before those outside. Ralph is studying calculus and either German or Spanish. C (G S) Either both Rene and Jerome are studying calculus or neither is. (R J) (R J) Larryis older than Wilma or Ethel, but not both, (W E) (W E) The truth functions expressed by these operators The introduction to these operators described them as truth-functional. The functions associated with ' ', ' ', and ' ' are as you would expect, based on how we are reading them: The rule for ' ' is that applying it to a true sentence yields a false sentence, and applying it to a false sentence yields a true sentence. The rule for ' ' is that when it joins two true sentences, the resulting sentence is true; in any other case the resulting sentence is false. The rule for ' ', since it expresses the inclusive 'or', is that when it joins two sentences, at least one of which is true, the resulting sentence is true. In the only other possible case, where both of the sentences it joins are false, the resulting sentence is false. 3

The rule for ' ' is less obvious. We use 'if then' in several different ways in English, and none of them are truth-functional. That is, the truth or falsity of conditional sentences in English depends on something besides the truth or falsity of the parts. As a result, our rule will not work exactly like any way we use 'if then' in English. However, it will capture one important feature of all the ways we use conditionals in English: The truth of a conditional and its 'if' part (its antecedent) requires that its 'then' part (its consequent) also be true. In other words, where the conditional is true and its antecedent is true, its consequent is also true. A conditional whose antecedent is true but whose consequent is false cannot be true. So our rule specifies that a conditional is false in the case where its antecedent if true and its consequent is false, but that it is true in every other case. Each of these rules can be represented by a table showing all the possible combinations of values for the component sentences, and the value of the compound formed by applied the operator: A A A B A B A B A B A B A B T F T T T T T T T T T F T T F F T F T T F F F T F F T T F T T F F F F F F F F T Truth Tables In a truth table, we apply the rules, working from innermost parentheses outward, to calculate the truth-value of a sentence or group of sentences in which we are interested, in every possible relevant situations. The relevant situations are all the possible situations, or combinations of ways truth=values could be assigned to the atomic sentences that are components of the sentence(s) of interest. The values of the atomic components are shown at the left (the reference columns). The sentence(s) we are interested in are shown to their right. The truth-value of each sentence in each possible situation is shown in the column under the last operator applied (the main operator). Each row across the table shows one possible situation (one way we could assign truth-values to those components). To show all the possible combinations, where there are n atomic components, the truth table must have 2 n rows. The table as a whole shows all the possible combinations or assignments, and shows the values of the sentences we are interested in on every possible assignment. In completing the table, values are shown for every sentence under its main operator. We fill in the values under all operators as we calculate them, though we only look at the column under each sentence's main operator to interpret the completed table. For atomic sentences (single letters), we need not repeat their values under the sentences that contain them. However, we need to show the value for every premise and conclusion in the appropriate column, even if the sentence is atomic. A truth table can be used to see whether an argument is deductively valid. Put the premises and conclusion across the top of a truth table, with the conclusion at the far right. Each row across will show the truth-values of the premises and conclusion in a single situation. A truth table that 4

has no row across where the premises are all true but the conclusion is false shows that the argument is deductively valid. If there is a row with that combination of values, we can only make a more qualified judgment that, at the level of detail shown, the argument is deductively invalid. The reason for this qualification is that some details that might make the argument valid (such as the role of 'all' or 'at least one' in the sentences) are not captured in this language. As a result, an argument that does not test as valid here might still be valid. We acknowledge this by saying that the argument is truth-functionally invalid. This means that the structure, as expressed by use of the truth-functional operators, does not determine the argument to be deductively valid. (Of course, if aspects of structure expressible by these operators determine that the argument is deductively valid, then even if we remove the limitation to those aspects, the argument is still deductively valid.) Example 1: It's not the case that both Carl and Dan were hired. Therefore Dan wasn't hired. Premise: (C D) Conclusion: D C D (C D) D Row 1 T T F T F Row 2 T F T F T Row 3 F T T F F Row 4 F F T F T 1 2 3 4 5 Here the argument has two atomic sentences as components of its premise and conclusion, so the table needs 4 rows to show the 4 possible situations (combinations of truth values). The premise is the negation of ' C D', so we first calculate the value of the conjunction (column 4), and then apply the ' ' rule to the result (column 3). Column 3 shows the value of the premise in all 4 possible situations, and column 5 shows the value of the conclusion in all 4 possible situations. There is at least one possible situation, shown on row 3, where the premises is true (row 3 column 3) yet the conclusion is false (row 3 column 5). So the argument is truth-functionally invalid. Example 2: If Elaine passed either math or philosophy, she completed her graduation requirements. In fact, she has not completed her graduation requirements. So she must have failed both math and philosophy. Translation: (M P) G G / M P M P G (M P) G G M P Row 1 T T T T T F F F F Row 2 T T F T F T F F F Row 3 T F T T T F F F T Row 4 T F F T F T F F T Row 5 F T T T T F T F F Row 6 F T F T F T T F F Row 7 F F T F T F T T T Row 8 F F F F T T T T T 1 2 3 4 5 6 7 8 9 5

Three atomic sentences are used in this argument, so the table needs 3 reference columns (columns 1 3) and 2 3 (8) rows. There is no row across where the premises columns (5 and 6) have the value true and the conclusion column (column 8) has the value false, so the argument is deductively valid. There is a method for filling in the reference columns in order to guarantee that every possible combination of truth values will show up once and only once. For the atomic component at the extreme right, we alternate 'T' and 'F' all the way down the table. Each time we move left from one to the next, we double the number of consecutive T's and F's (1 and 1, then 2 and 2, then 4 and 4). The column at the far left should have 'T' for the top half and 'F' for the bottom half of the table. Example 3: Harold had to pass the vision test, the written test, and the road test to get his driver's license. He didn't his driver's license, even though he passed the vision test. So he must have failed the road test. Translation: ((V W) R) L L V / R V W R L ((V W) R) L L V R Row 1 T T T T F T T T F F F F Row 2 T T T F F T T T T T T F Row 3 T T F T T T F F F F F T Row 4 T T F F T T F T T T T T Row 5 T F T T T F F F F F F F Row 6 T F T F T F F T T T T F Row 7 T F F T T F F F F F F T Row 8 T F F F T F F T T T T T Row 9 F T T T T F F F F F F F Row 10 F T T F T F F T T T F F Row 11 F T F T T F F F F F F T Row 12 F T F F T F F T T T F T Row 13 F F T T T F F F F F F F Row 14 F F T F T F F T T T F F Row 15 F F F T T F F F F F F T Row 16 F F F F T F F T T T F T 1 2 3 4 5 6 7 8 9 10 11 12 Looking across row 2, we find 'T' in the columns for the w premises, but 'F' in the column for the conclusion, showing that the argument is truth-functionally invalid. (We could use line 6 instead of line 2 to establish this claim.) The Shorter Truth Table Technique In example 3, row 2 (or row 6) alone is sufficient to show that the argument is truth-functionally invalid. The shorter truth table technique gives us a way to find a combination of values that would establish this result, without having to write out the entire table. Since the number of rows in a complete table doubles with each additional atomic component, such a technique can save us a lot of writing. 6

In constructing a truth table, we write out every possible combination or assignment of truth values for the atomic sentences. We proceed in calculating the values from simpler to more complex sentences until we get the values for our premises and conclusion on every one of those assignments. Finally, we look through the results to see if the combination of values for the premises and conclusion that we are interested in (true premises with a false conclusion) appears on any row (assignment). The shorter truth table technique, can be seen as working in reverse. We start with the target values of interest to us (true premises and false conclusion), and work down toward the values of simpler sentences that would produce those results. By the time we get down to the atomic sentences, either we find an assignment for the atomic sentences that yields the target values, or we find that we are forced to a contradiction, showing no such assignment is possible. If no such assignment is possible, the argument is deductively valid. If we find an assignment where the premises are true yet the conclusion is false, that shows that the argument is truth-functionally invalid (although it might still be deductively valid), To apply this technique, we write the premises and conclusion with target values (Stage 1 below). Next, we begin assigning values to simpler immediate components, starting where there is only one way to get the target value. Only when no values can be determined do we try out values where we have options. Example 1, revisited: This is a simple case, where the target values determine all values: Stage 1: (C D) D T F The only way the premise can be true is for the conjunction to be false. The only way for the conclusion to be false is for D to be true: Stage 2: (C D) D T F FT We copy this value for D from the conclusion to the premise: Stage 3: (C D) D T F T FT The only way for the conjunction in the premise to be false, given that we have already assigned D the value true, is for C to be assigned the value false: Stage 4: (C D) D T F F T FT We now have assigned values to all the atomic components to yield the value true for the premise and false for the conclusion: C = F, D = T. Example 2, revisited: Again, we assign T to each premise and F to each conclusion: Stage 1: (M P) G G M P T T F The only value specifically determined by these target values is the value for G: Stage 2: (M P) G G M P T TF F 7

We copy this value for G over to the first premise Stage 3: (M P) G G M P T F TF F The first premise is a true conditional with a false consequent, so its antecedent must also be false: Stage 4: (M P) G G M P F T F TF F The only way the disjunction (M P) can be false is if both conjuncts are false: Stage 5: (M P) G G M P F F F T F TF F We copy those values under 'M' and 'P' in the conclusion< and then use them to build up the values for ' M' and ' P': Stage 6: (M P) G G M P F F F T F TF TF F TF However, given that both conjunctions are true, the conjunction comes must have the value true also, contradicting the original target value: Stage 7: (M P) G G M P F F F T F TF TF F TF T We were unable to assign values to the atomic components without a yielding a contradiction. This shows that there is no assignment, or possible situation, that will make the premises all true and the conclusion false. Thus the argument is deductively valid. Example 3 above is more complicated, because eventually we reach a point where either truth value would work. We can save some writing by showing the stages in one place. The numbers below indicate the stages in which they are entered. (1) Assign target values to premises and conclusion. (2) To assign T to the second premise we must assign T to both of its conjuncts Also, the only way to assign F to the conclusion is to assign F to R. (3) Having assigned L the value T, we must assign L the value F. (4) We copy the values determined thus far to the first premise. (5) In a true conditional with a true consequent, the antecedent can have either value. So we'll add one above and one below. (6) Next, we try out one of those possibilities. In this case, it won't matter which one we try. Choosing arbitrarily, let's continue to use the values below the sentences. For the negation to be false, the conjunct it negates must be true. (7) For ((V W) R) to be assigned the value T, (V W) must be assigned T. (8) For (V W) to be assigned the value T, W must be assigned T T ((V W) R) L L V R F T T T T T T T F T F T T F T 5 4 7 8 6 4 1 4 4 2 3 1 2 1 2 We now have assigned values to all the atomic components with no contradiction: V = T, W = T, R = T, L = F. This shows that it is possible to assign the values to 8

make the premises true and the conclusion false, so the argument is truthfunctionally invalid. Example 4: A B (A C) / (B C) The shortened table below illustrates a shortened table where one option leads to a contradiction, but the argument is still invalid. TO SHOW THAT AN ARGUMENT IS VALID, WE MUST SHOW THAT IT IS IMPOSSIBLE TO ASSIGN VALUES IN A WAY THAT WILL MAKE ALL PREMISES TRUE AND THE CONCLUSION FALSE. One effort that fails is not sufficient to show that this is impossible. We must also follow through any other alternatives to see whether a different one might yield that combination of truth values. This also shows the importance of strategic choices of the order for assigning values. A poor choice is made at step 6, just to illustrate finding a contradiction in an invalid argument. A better choice would give us a simpler shortened table. A) T B) P Q ~( P R) ~(Q R) C) F F F D) T T T F F F T F E) T T F F) T Additions in row A) 3 C) 6 7 8 D) 1 1 2 1 2 E) 3 4 5 5 F) 6 9 The numbers below indicate the order of insertion of values, as explained below: (1) We assign T to the premises, F to the conclusion (row D). (2) We add values determined by the initial values (row D). (3) At this point, no other values are determined, so we must begin to look at choices. The first premise can be T with either P or Q assigned T (row A or E) (4) We consider the possibility that P is T, using this in the second premise (row E). (5) For the 2nd premise to be T while P has the value F, R must have the value F (row E). (6) For the 1st premise, Q could be F or T with the values in row E, so we put these one above and one below row E on rows E and F. (This is a phony step, just to allow us to show a contradiction in a relatively simple argument that is valid. Really, we should notice that Q must be T for the conclusion.] (7) Consider giving Q the value F, putting this value in for Q across row C. (8) That forces the value F for the disjunction Q R in the conclusion (row C), contradicting the value we already had on in row D. (9) Consider a different alternative: Keep the values in rows D and E, but try the other 9

choice from step 6, shown in row F. Insert this value for Q in the conclusion. The shortened table above shows that if we assign the values P = T (as we did at step 4) Q = T (step 9) R = F (step 5) the premises in this argument all come out true, but the conclusion comes out false. The ability to get this combination of values shows that the argument is truth-functionally invalid. This example also shows that, where the premises and conclusion all have more than one way to get the target truth values (T for the premises, F for the conclusion), the shortened table is not short at all. If the first choice we make leads to a contradiction, we need to examine our alternatives. If additional choices arise later, the table becomes hard to follow. So this technique is not very useful for such arguments. The uses and limits of truth tables As mentioned earlier, an argument could be deductively valid even though it is truthfunctionally invalid. That would happen if there is something other than the propositional structures of the statements that is making the argument valid. For example, it could be that relationships between descriptions in the sentences guarantee that if the premises are true, then the conclusion must also be true. On the other hand, if an argument is truth-functionally valid, then it is deductively valid. The truth-functionally valid arguments are a special subgroup of the deductively valid arguments. Another such subgroup is the group of arguments that are valid because of their categorical structure. 10