Logic & Proofs. Chapter 3 Content. Sentential Logic Semantics. Contents: Studying this chapter will enable you to:


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1 Sentential Logic Semantics Contents: TruthValue Assignments and TruthFunctions TruthValue Assignments TruthFunctions Introduction to the TruthLab TruthDefinition Logical Notions TruthTrees Studying this chapter will enable you to: 1. Explain what a truthvalue assignment is. 2. Give the truthconditions for the logical connectives. 3. Determine the truthvalue of a formula relative to a given truthvalue assignment. 4. Construct a truthtable for a given formula or argument. 5. Use truthtables to analyze arguments and formulae. 6. Explain what tautological, contingent, and contradictory formulae are. 7. Find a counterexample to an invalid argument, using a truthtable or truthtree. Chapter 3 Content
2 TruthValue Assignments and TruthFunctions TruthValue Assignments As we mentioned earlier, our primary interest in sentential logic has to do with the TRUTHVALUES of formulae. We have learned how the basic expressions of sentential logic combine, in accordance with the syntactic rules, to form compound formulae. The syntactic rules allow us to determine, for any particular expression, whether or not that expression constitutes a wellformed formula. Similarly, the semantic rules for sentential logic will allow us to determine the truthvalue of any formula, given that we know the truthvalues of all the atomic formulae involved. Great, but what if we don't know the truthvalues of all the atomic formulae? Well, we can still determine what the truthvalue of the formula will be for any possible assignment of truthvalues to atomic formulae. Such an assignment of truthvalues to atomic formulae is, appropriately enough, a truthvalue assignment. Before we get to the definition itself, there's just one thing we ought to mention explicitly: we make the assumption that there are just two truthvalues, truth and falsity. Since we will refer to these two truthvalues quite a bit, we abbreviate them as follows: t for truth and f for falsity. That said, here is the definition: Definition TruthValue Assignment A TRUTHVALUE ASSIGNMENT specifies a unique truthvalue (either t or f) for each atomic formula. So, how do we go about determining the truthvalue of any formula of sentential logic, relative to a given truthvalue assignment? Obviously, if our formula is atomic, all we need to do is to see what truthvalue it has been assigned in order to determine what its truthvalue is on that assignment. What about compound formulae? You may recall that we've mentioned that the logical connectives are all TRUTHFUNCTIONAL. This means that the truthvalue of a compound formula is a FUNCTION of the truthvalues of its parts a function determined by the formula's main connective. In order to see how this works, let us proceed to take a look at the semantics of the connectives. Page 1 of 23
3 TruthFunctions We have informally discussed the truthfunctions we take to correspond to the logical connectives. So far, we have relied on an intuitive understanding of the truthfunctions these connectives represent, but it is time to make explicit the truthfunctions at work. We will use TRUTHTABLES to represent them. Recall our first example of a conjunction from the previous chapter: John ran and Mary laughed. Now, this sentence is considered to be true just in case it is both true that John ran, and true that Mary laughed. If John didn't run, the conjunction is false, and similarly if Mary didn't laugh. This intuitive and informal understanding of the TRUTHCONDITIONS of conjunction, that a conjunction is true just in case both of its conjuncts are true, and false otherwise, is precisely what we want to capture. We can do this in a tabular form, by specifying the truthvalue of a conjunction for each possible combination of truthvalues t (true) and f (false) that the two conjuncts can take on. Here's the truthtable for this particular conjunction: After looking only briefly at the above truthtable, it should be pretty clear what is going on, but it can not hurt to go through it explicitly. In our truthtable, we have three columns: one for the conjunction itself, plus one for each of the two atomic formulae appearing in the conjunction. As for rows, this truthtable has four (not including the header row, where we specify the atomic formulae and the compound formula), just enough to include every possible combination of the truthvalues t and f that the two atomic formulae can have. The entries in the conjunction's column specify the truthvalue of the conjunction given that the conjuncts have the truthvalues indicated in the same row. Each of the rows in the truthtable thus represents a set of truthvalue assignments the set of truthvalue assignments where the atomic formulae listed in the truthtable are assigned the truthvalues specified on that row. The first row of the truthtable thus represents all truthvalue assignments where the atomic formulae J and M are both assigned the truthvalue t. This includes the truthvalue assignments where J, M, and, say, R are all assigned the truthvalue t as well as the truthvalue assignments where J and M are assigned t, but R is assigned f, and so on for every atomic formula not mentioned in the truthtable. Now, in the truthtable above, we listed some specific atomic formulae. The truthtable thus tells us what the truthvalue of the conjunction of those two particular atomic formulae will be on any truthvalue assignment. We would like, however, to have a way to represent the relationship between the truthvalue of a conjunction and the truthvalues Page 2 of 23
4 of its conjuncts in a general way, rather than just specific instances of the relationship. We can do this with a truthtable by using variables in place of specific formulae for the conjuncts, as follows: Definition Characteristic Truthtable for Conjunction This truthtable, since it doesn't mention any specific formulae, we call the CHARACTERISTIC TRUTHTABLE for conjunction. Let us go on to take a look at the characteristic truthtables for the other connectives. Later on, we will see how to construct a truthtable for any formula of sentential logic. Here is the truthtable for disjunction: Definition Characteristic Truthtable for Disjunction The truthtable specifies that a disjunction is true on any truthvalue assignment where either one or both of the disjuncts is true, and false just in case both of the disjuncts are false. Recall our example disjunction from the last chapter: John ran or Mary laughed. The truthtable tells us that this sentence will be false only if it is both false that John ran, and false that Mary laughed. If either of the two disjuncts is true, including the case where both disjuncts are true (as we decided that the symbol should reflect the inclusive understanding of disjunction), then the sentence as a whole is true as well. This brings us to another kind of argumentative step we can take, involving the elimination of disjunction. It's a little more complicated than the ones we have seen so far. The idea goes like this: if we know that (P Q), and we know that R follows from P, and that R also follows from Q, then R can be inferred from (P Q), since at least one of P or Q must be true, and either way, R must be true. The step looks like this: Page 3 of 23
5 (P1) (P Q) (P2) Assuming P, R (P3) Assuming Q, R (C) R We'll return to the notion of assumptions in the next chapter. For now, it is enough to grasp the intuitive notion underlying this argumentative step, elimination. Moving on to the last of our binary connectives, here is the truthtable for the conditional: Definition Characteristic Truthtable for the Conditional As you can see, a conditional is false just in case its antecedent is true, and its consequent false. If either the antecedent is false or the consequent true, then the conditional as a whole is true. The notion of assumption that we've just seen plays a role in argumentation allowing the introduction of a conditional: since a conditional is false only if its antecedent is true and its consequent is false, we may infer a conditional if we know that the consequent follows from the (assumption of) the antecedent. Here is introduction: (P1) Assuming P, Q (C) (P Q) Our final connective is negation. Since negation is a unary connective, we only have one component formula to worry about in the truthtable. The characteristic truthtable for negation thus has only two rows, since the single formula can only be either true or false. Here is the truthtable itself: Definition Characteristic Truthtable for Negation Now that we have seen the truthtables for each of the connectives, we should turn our attention to some mathematical, or rather metamathematical matters before we get any further. Page 4 of 23
6 See the online version for interactive material here! Page 5 of 23
7 Introduction to the TruthLab The TRUTHLAB is the application you will use to complete a variety of semantic exercises in this course. The TruthLab runs directly in your web browser, so it is important that the browser you are using meets the system requirements stated on the Test and Configure page (the link to this page can be found at the top of the course syllabus). Tip: If you haven't already verified that your browser meets the requirements stated on the Test and Configure page, and passes the tests provided there, please take a moment to do that now! If your browser is correctly configured, launching and running the TruthLab is as simple as clicking a link every problem that can be worked in the TruthLab will be followed by a link that says Click here to open this problem in the TruthLab, like the following: Chase truth up the parse tree of the formula ((L & A) & B) to determine its truthvalue when L is true, A is true, and B is true. See the online version to try this problem in the TruthLab! Clicking on such a link will launch the TruthLab. Once you click the link, a new browser window or frame will appear as the TruthLab applet begins downloading: After a moment or two required for Java to start up, if necessary, and the TruthLab applet to finish loading the TruthLab itself will open in a new Java applet window (not a regular browser window). Here is what the TruthLab looks like, running in Safari on a Mac (the menu bar and other details may look slightly different if you are using a different browser or operating system): Page 6 of 23
8 Note: The TruthLab is a moderately sized Java applet, so it may take a little while to download, particularly the first time. It should normally not take more than approximately 5 to 30 seconds to load when using a reliable, highspeed internet connection. If you are using dialup or have an otherwise slower or unreliable connection, the TruthLab may take significantly longer to load. It may also be sluggish in responding to clicks/ commands in such circumstances, as it communicates frequently with the OLI servers. For this reason, we strongly recommend that working in the TruthLab be avoided when using an unreliable or nonbroadband internet connection if at all possible. Please take a moment now to verify that you are able to launch the TruthLab by clicking the link following the problem above. Don't worry about finishing the problem but do take a moment to explore the interface! Page 7 of 23
9 Note: If you have any difficulty launching the TruthLab, get help right away! If you are in an inclass session, tell your instructor. If you are working outside of a class meeting, you can contact OLI technical help by clicking on the help link on any course page. If your instructor has provided instructions on what you are expected to do in the event of technical issues, be sure to follow those instructions. In addition to using the TruthLab, however, we do recommend that you take the time to do a few problems (e.g., those involving truthtable and truthtree construction) the old fashioned way with pencil and paper as well. Take a few moments to explore the Lab, then once you are ready, on to the truthdefinition! Page 8 of 23
10 TruthDefinition We promised earlier on that parse trees were going to come in handy when it comes to semantics. Indeed, they do. They offer a handy way to exploit the truthfunctional character of the logical connectives in order to determine the truthvalue of any formula on a particular assignment. To put it succinctly, we can "chase truth" up the parse tree of a given formula, matching rows of the truthtables to applications of syntactic rules in the tree. Let us look at an example: Movie Chasing Truth Text/printable version available as a separate PDF! That is really all there is to it: just replace the atomic formulae in the parse tree with the truthvalues assigned to each on the given assignment. For each application of a syntactic rule in the tree (from the bottom up), find the corresponding row in the connective's truthtable, and replace the subformula with the resulting truthvalue. Once you get to the top, you have the truthvalue for the formula as a whole. Page 9 of 23 See the online version for interactive material here! Of course, if you are interested in more than just a single truthvalue assignment, say if you wanted to determine whether a formula is true on every truthvalue assignment, things start getting a bit messy: for our example ((P & Q) R), you need to draw eight separate parse trees one for each possible truthvalue assignment to the three atomic formulae (we will explain how we got this number in just a bit) in order to classify the formula. Describing this is painful enough, actually doing it would be horrible. Fortunately, we have already seen a means for avoiding such a state of affairs the truthtables we used to present the truthconditions for the connectives. We can also use truthtables to provide a neat and compact presentation of the truthvalues of a given formula on any truthvalue assignment. In order to construct a truthtable for an arbitrary formula of sentential logic, the first thing you need to do is to count the number of different sentential letters that occur as subformulae of the given formula. This will tell you how many rows you need in your truthtable. As we have seen, if there is only a single atomic formula, you will need two rows, and with two atomic formulae, you will need four rows. See if you can determine how many rows are needed for larger numbers of atomic formulae, then read on. Now that you have thought about it, you can see that every additional atomic formula involved is going to double the number of rows you'll need in your truthtable. Thus, for a formula containing three atomic formulae, you need a truthtable with eight rows, for four you need sixteen rows, five atomic formulae would require thirty two rows, and so on. In general, you need 2 n rows in a truthtable for a formula containing n different atomic formulae as subformulae.
11 There are many different ways to go about constructing a truthtable. We will demonstrate one method in a moment, but it doesn't really matter how you go about constructing it, so long as you end up with the right thing. What you end up with for a given formula should include the following: For each subformula in the parse tree (including the formula itself), you should have a corresponding column in your truthtable; for each possible truthvalue assignment to the atomic subformulae, you should have a corresponding row in your truthtable, along with a header row where the subformulae are specified. The columns are usually ordered by traversing the parse tree from bottom to top, and from left to right, so you start out at the left of the table with atomic formulae, and work your way to the formula as a whole in the rightmost column. At this point, we should probably look at an example, because it really is much more straightforward than it may sound: Movie Truthtables I Text/printable version available as a separate PDF! As this example demonstrates, constructing a truthtable is pretty simple. Once complete, it shows us the truthvalue of our formula ((P & Q) R) on every possible truthvalue assignment. If we want to find its truthvalue on a particular assignment, we just have to find the row where the truthvalues assigned to P, Q, and R match the assignment in question, and the above truthtable will tell us whether our formula is true or false on that assignment. Let us take a look at one more example, just to make sure we have the hang of it all. This time, we construct the truthtable for the formula ( P Q): Movie Truthtables II Text/printable version available as a separate PDF! We hope to have made it evident that all we need to determine the truthvalue of any formula on a given truthvalue assignment is the truthvalue assignment itself, thanks to the truthfunctional nature of the connectives. See the online version for interactive material here! No matter how complicated a formula, we can always determine its parse tree and "chase truth up the tree of grammar". This insight can be given a precise mathematical formulation (and proof, for that matter): given a truthvalue assignment σ for atomic formulae, there is a (unique) truthvalue assignment σ * for all formulae, extending σ and assigning the correct truthvalues to Page 10 of 23
12 complex formulae. We can formulate this in terms of a definition of truth and falsity on σ *. Here is the definition: Definition Truth and Falsity Relative to a TruthValue Assignment 1. If φ is an atomic formula (sentential letter) of sentential logic, then φ is true on σ * just in case σ assigns the value t to φ, and false otherwise. 2. If φ is a formula of the form ψ, then φ is true on σ * just in case ψ is false on σ *, and false otherwise. 3. If φ is a formula of the form (ψ & ρ), then φ is true on σ * just in case both ψ and ρ are true on σ *, and false otherwise. 4. If φ is a formula of the form (ψ ρ), then φ is true on σ * just in case either ψ is true on σ * or ρ is true on σ *, and false otherwise. 5. If φ is a formula of the form (ψ ρ), then φ is true on σ * just in case either ψ is false on σ * or ρ is true on σ *, and false otherwise. If this definition seems simple, good! All it does is mathematically characterize the truthconditions we have presented informally by means of the characteristic truthtables of the connectives. It gives us everything we need to determine the truthvalue of any formula on any truthvalue assignment, and that was captured quite efficiently by the truthtables. Now that we have seen how we can use truthtables to determine the truthvalues of formulae on particular assignments, let us move on and look at some central logical notions from a semantic perspective. Page 11 of 23
13 Logical Notions We are going to put the definition of truth to use in order to characterize some central logical notions. For example, a formula is to be considered LOGICALLY TRUE if and only if it is true "independent of matters of fact", where the "facts" are given by a truthvalue assignment. This gives us the following definition: Definition Tautology A formula is called LOGICALLY TRUE or a TAUTOLOGY just in case it is true on every truthvalue assignment. If, on the other hand, a formula is always false, it is called you guessed it, CONTRADICTORY: Definition Contradictory Formula A formula is called a CONTRADICTORY just in case it is false on every truthvalue assignment. Finally, we have everything in between: Definition Contingent Formula A formula is called a CONTINGENT just in case it is true on some truthvalue assignments, and false on others. We have already seen quite a few contingent formulae, including all the atomic formulae, so we won't worry about any examples there. As an example of a tautology, on the other hand, consider the formula (P P). This formula has only one atomic subformula, P, so it's pretty easy to tell that it will be true whether P is assigned t on a given assignment (because that will make the lefthand disjunct true) or f (since that will make the righthand disjunct true). You have probably realized that the negation of any tautology is going to be false on every truthvalue assignment, and hence a contradictory. Similarly, the negation of a contradictory is a tautology. The negation of a contingent, however, will also be contingent. We will see in just a bit how useful these classifications for formulae can be when we consider inferences and arguments from a semantic perspective. As you will recall from the introduction, we consider an argument to be a good argument just in case its premises are all true, and those premises furthermore support the conclusion. The kind of support we are most interested in is, of course, validity, which we can now characterize in terms of truthvalue assignments: Page 12 of 23
14 Definition Validity An argument is VALID just in case any truthvalue assignment that makes all the premises true also makes the conclusion true. With the concept of validity defined, we can now characterize an important relationship in terms of validity: Definition Logical Consequence If an argument with premises φ 1,...,φ n and conclusion ψ is valid, then ψ is a LOGICAL CONSEQUENCE of φ 1,...,φ n. With a precise characterization of validity, we can also provide a precise characterization of invalidity: Definition Invalidity An argument is INVALID just in case it is not valid, i.e., if there is some truthvalue assignment that makes the premises true, but the conclusion false. Definition Counterexample A truthvalue assignment that makes the premises of an argument true and its conclusion false is called a COUNTEREXAMPLE to the argument. Now that we know how to use truthtables to determine the truthvalue of a formula on any truthvalue assignment, we can apply this technique to the premises and conclusion of an argument, symbolized as formulae of sentential logic, in order to determine whether or not an argument is valid. We just need to be careful to ensure that we take into consideration all the right truthvalue assignments. Consider the following conversation: It's raining outside. If it's raining outside, then Kant won't stop at the grocery store to buy ice cream. Page 13 of 23 If Kant doesn't stop at the grocery store to buy ice cream, then we will have to be content with cookies for dessert.
15 Alas, we will thus have to be content with cookies for dessert. Interpreting this as an argument, we can symbolize it as follows: R (R K) ( K S) S We can make a truthtable for all the premises plus the conclusion by first noting the atomic formulae that occur as subformulae of any premise or of the conclusion, then setting up the truthtable for all such atomic formulae. We include a column for each of the premises and also the conclusion. Here's what the resulting table looks like: In order to determine whether the argument is valid or not, we now look for any rows in this truthtable where all the premises are true. In this case, there is only one such row the third one since the conclusion is true on this row, the argument is valid. See the online version for interactive material here! Despite the fact that we have just introduced it as such, we don't actually need to include the concept of validity as an independent notion: we can instead define validity in terms of tautologies. How? Well, since being tautologous is a property of formulae, and validity is a property of arguments, we need a way to "transform" an argument into a formula a formula that will be a tautology if the argument is valid, and not if it is invalid. If you think about this for a minute, the following definition should come as no surprise: Definition Conditional Analogue The CONDITIONAL ANALOGUE of an argument with premises φ 1,...,φ n and conclusion χ is the formula ((φ 1 & (... & φ n )) χ). Given this definition, we can now restate our above claim precisely: we do so by formulating it as a metamathematical proposition. Page 14 of 23
16 Proposition An argument with premises φ 1,...,φ n and conclusion χ is valid if and only if its conditional analogue ((φ 1 & (... & φ n )) χ) is a tautology. Proof: Before presenting the proof let us make a few remarks. First of all, note that this proposition is metamathematical, which means simply that it is a claim about the formal system of sentential logic, as contrasted with a claim expressed in the language of sentential logic. This reflective concern with the properties of logics and logical systems is deeply characteristic of modern work in logic. Secondly, the proof of a statement such as this involving the phrase if and only if (called BICONDITIONALS, and we will return to discuss these in more detail in a later chapter) divides into two parts: the if part, and the only if part. Here we will establish that if an argument is valid, its conditional analogue is a tautology. We leave the other part of the proof as an exercise for the reader. On to the proof: Assume that the argument with premises φ 1,...,φ n and conclusion χ is valid. Assume also, in order to obtain a contradiction, that its conditional analogue ((φ 1 & (... & φ n )) χ) is not a tautology. That means there is a truthvalue assignment σ on which the conditional analogue is false. Using the appropriate clauses of the truthdefinition, then premise φ i, for each i from 1 through n, is true on σ, and conclusion χ is false on σ. Thus σ is a counterexample to the argument, contradicting the assumption that the argument is valid. See the online version for interactive material here! Since we have proved our claim, we could simply define an argument to be valid just in case its conditional analogue is a tautology: these two notions are equivalent, so it doesn't really make a difference whether or not we include the concept of validity (for an argument) as a basic notion we can always define it in terms of tautologies instead. Truthtables are all we need to determine the truthvalue of a formula on any truthvalue assignment, and they are all we need to find a truthvalue assignment that gives a particular formula a given truthvalue, and in particular, all we need in order to find a counterexample to an argument. It might be nice, however, to have a way of working backwards, if you will, from the desired truthvalue for the formula to a truthvalue assignment on which the formula has that desired truthvalue and hence a more efficient and direct technique for finding counterexamples. Let's head on to the next section and look at a technique that will allow us to do just these things. Page 15 of 23
17 TruthTrees We can approach the problem of finding a truthvalue assignment that assigns a particular truthvalue to a given formula (and hence counterexamples) in much the same fashion as we approach the problem of determining whether or not a given expression is a wellformed formula using trees. The big difference is that while we make use of the syntactic rules when constructing parse trees, we will use the definition of truth with respect to a truthvalue assignment in order to construct TRUTHTREES. There is one basic and very useful fact about the definition of truth: a formula is true just in case its negation is false, and vice versa. We can exploit this fact to ensure that we only need to worry about constructing trees for formulae that we want to be true (hence the term truthtree). If we want to find a truthvalue assignment that makes some formula false, all we need to do is to find a truthvalue assignment that makes its negation true. If our goal is to find a counterexample to a given argument, then, all we need to do is to find a truthvalue assignment that makes the premises and the negated conclusion of the argument true. Now, just as we break formulae down into their immediate subformulae in parse trees, we are going to break the truth of formulae down into the truth of their immediate and possibly negated subformulae in truthtrees. Branches serve a very distinctive purpose in truthtrees, though: each branch corresponds to a set of truthvalue assignments. Which truthvalue assignments? Those that make all the formulae on that branch true. We will make this more precise by introducing truthtree rules for each connective in turn. Starting with conjunction, we know that a conjunction (φ & ψ) is true just in case both φ and ψ are true. We thus know that any truthvalue assignment that makes (φ & ψ) true must make both of φ and ψ true as well. We have broken down the truthconditions for the original formula into truthconditions for two smaller pieces. Just like the subformulae in a parse tree, these two smaller pieces can be added to our truthtree. Unlike a parse tree, however, they do not end up on different branches. Since the truth of each subformula is guaranteed by the truth of the formula as a whole, both subformulae go on the same branch. Graphically, we can just write the two subformulae immediately below the original formula. It can also be very helpful to indicate in some fashion that you have already dealt with the original formula, such as putting a check mark next to it, or underlining it, or something similar. This way, you can keep track of which formulae in the tree have already been broken down, and which have not. (This becomes more important when analyzing arguments using truthtrees, but it is a good idea to get in the habit of checking them off right from the beginning.) Next, we know that a disjunction is true just in case either one of its disjuncts is true. Thus, for a disjunction (φ ψ), we have two sets of truthvalue assignments to consider: those where φ is true, and those where ψ is true. Since the truth of the disjunction can only guarantee the truth of one of these disjuncts, this tells us that the tree for the disjunction will branch, unlike the one for conjunction. The truthtree for disjunction will thus look much like the parse tree for the same formula, the only difference being that we will add in a check mark next to the formula we have just broken down. Page 16 of 23
18 On to consider our next binary connective, the conditional. We know that a conditional is true just in case either its antecedent is false or its consequent is true. As we mentioned earlier, however, the fact that a formula is true just in case its negation is false and vice versa allows us to reinterpret these truthconditions as follows: a conditional (φ ψ) is true just in case either the negation of its antecedent (i.e, φ) is true or its consequent (i.e, ψ) is true. If we compare this to the truthconditions for disjunction, it's easy to determine that the truthtree for a conditional must branch, and which formulae must be at the ends of those branches. Let us take a look at our truthtree rules so far: Definition TruthTree Construction Rules See the online version for interactive material here! How about our final connective, negation? We pointed out and exploited the fact that a formula is true just in case its negation is false and vice versa, but how are we going to use this fact to deal with negation in truthtrees? Consider for a moment the fact that each branch of a truthtree corresponds to a set of truthvalue assignments, namely, the set of truthvalue assignments that make all the formulae appearing on that branch true. This being the case, we thus know that if an atomic formula appears on a particular branch, then that formula must be true on each of the corresponding truthvalue assignments, and hence that each such assignment assigns the value t to that atomic formula. Now consider again the fact that a formula is false just in case its negation is true if the negation of an atomic formula appears on the branch, then we know that the truthvalue assignments in question must make this negation true, and hence must assign the value f to the atomic formula. Great, so how does this help us figure out what to do with negation in truthtrees? Well, we know that as soon as we have broken down a formula to the point where all we have left is the negation of an atomic formula, we are done. This tells us everything we need to know about the truthvalue assignments in question, i.e., that they must assign the value f to that atomic formula. All that is left, then, is to worry about the negations of formulae other than atomic ones. In that light, there are going to be exactly four rules we need in order to deal with negation in the context of truthtrees: One for negated conjunctions; one for negated disjunctions; one for negated conditionals; finally, one for negated negations. Why not see if you can extrapolate the rules from the ones we have presented above and the truthconditions of the connectives: See the online version for interactive material here! Page 17 of 23
19 Here are all the truthtree rules we need: Definition TruthTree Construction Rules There is just one thing to note about these rules: they must be used to extend every branch of the tree on which the formula in question is located. In other words, if you are applying a rule to a formula that already has more than one branch growing out of it, you must extend each such branch before you can check off the formula. Just to make sure this is absolutely clear, here is an example: There are a few other things we ought to mention regarding truthtrees. First, though it might be intuitive, we should spell out how to read a truthvalue assignment from a truthtree. Each branch of the tree corresponds to a set of truthvalue assignments, namely, the truthvalue assignments that make all the formulae appearing on the branch true. Because we've applied the truthtree rules to them, we don't need to worry explicitly about any formulae that have been checked off which means that the only formulae we need to worry about once we are done constructing the tree is atomic formulae and the negations of atomic formulae. Given that, the first thing to do in determining the truthvalue assignment specified by the branch is to go along the branch, and make a note of every such atomic formula and negation of an atomic formula on that branch. The truthvalue assignments that correspond to the branch are all those that assign t to the atomic formulae appearing on the branch and f to atomic formulae whose negations appear on the branch. Page 18 of 23
20 Now would be a good time for an example. Consider the formula (P & (Q P)). We will try to find a truthvalue assignment that makes this formula false, so to do that we construct the truthtree for the formula's negation. Here is the completed tree: The only atomic formula or negation of an atomic formula appearing on the lefthand branch is P, which tells us that assigning f to P is enough to make (P & (Q P)) false. We can see that this is correct by looking at the truthtable for the formula all the rows where P is false make the formula false: Similarly, the atomic formulae and negations thereof on the righthand branch are P and Q, which tells us that if we assign t to P and f to Q, this will also make the formula's negation true, and hence the formula itself false. Here's the truthtable again to highlight this: It is worth pointing out that while we went to the trouble of completing the tree for (P & (Q P)), we really didn't have to. If you consider the tree after just one step, i.e., the application of the rule for negated conjunctions to the formula as a whole, note that the lefthand branch of the tree is already complete every formula on the branch is either atomic, the negation of an atomic formula, or checked off. We could have stopped at that point and read our truthvalue assignment from this branch, not even worrying about completing the righthand branch of the tree! There's one hitch to reading off assignments: what happens if both some formula and its negation appear on the same branch of a truthtree? Since no truthvalue assignment can make any formula both true and false, and so cannot make both a formula and its negation true, then we obviously cannot associate a truthvalue assignment with such a branch. Thus, if a formula and its negation ever do appear in the same branch, we CLOSE OFF that branch, marking it with an asterisk to indicate that the branch does not represent any possible truthvalue assignments. Such a branch is, unsurprisingly, referred to as Page 19 of 23
21 CLOSED. A branch that is not closed is, equally unsurprisingly, referred to as OPEN. A tree is COMPLETED once every formula on any open branch is either an atomic formula, the negation of an atomic formula, or has been analyzed (i.e., broken down and checked off). Note that this means in particular that if all of the branches of a tree are closed, then that tree is completed. So, what does all this about closed and open branches and completed trees have to do with using truthtrees to analyze the logical character of formulae and arguments? It is, on the surface, rather simple: if a formula has a truthtree that contains only closed branches, then that formula isn't true on any truthvalue assignments, and hence is contradictory. We already know that if a formula is a tautology, its negation will be contradictory. Consequently, if a formula is a tautology, then its negation will have a closed truthtree. As an example, consider the truthtree for the negation of the tautology (P P): The only branch in the tree is closed, which means there are no truthvalue assignments that make (P P) true, and thus none that make (P P) false. (It's also worth noting that the formula and its negation that closed the branch in the first place are P and P, illustrating an important point: if any formula and its negation appear on a branch, the branch can be closed immediately. It need not be an atomic formula and its negation you can save yourself a lot of work if you remember to check for any formulaanditsnegation pairs after each rule application and close all branches as soon as possible.) As for arguments, just remember the connection between an argument and the logical character of its conditional analogue to see how to apply truthtrees to the analysis of arguments. These considerations are true, but don't establish the connection between the syntactic considerations and the semantic interpretation rigorously enough. We will come back to this quite important issue at the end of this section. See the online version for interactive material here! It can't hurt to summarize everything we discussed regarding truthtrees so far into a procedure for generating a completed truthtree. Here's that procedure: Definition Procedure for Generating TruthTrees 1. Start by writing down the formula for which you want to generate a truthtree. 2. Based on the syntactic form of the expression, apply the appropriate truthtree rule, putting a check mark next to the formula to indicate that it has been analyzed. Page 20 of 23
22 3. For each open branch, determine whether the branch contains both any formula φ and its negation φ. If any branch does contain a formula and its negation, mark the branch closed. If all branches in the tree are closed, you are done. Otherwise, continue to the next step. 4. If the only formulae on open branches that do not have check marks next to them are atomic formulae or negations of atomic formulae, you are done. Otherwise, continue to the next step. 5. Choose an unchecked formula on an open branch in the truthtree that is not atomic and not the negation of an atomic formula, and apply this procedure to that formula, starting with step 2. Let us use this procedure to construct a truthtree for the formula ( (A & B) & ( (A B) (A & B))): Movie TruthTrees Text/printable version available as a separate PDF! Our final example: consider the formula ((P Q) (P & Q)) and decide whether there are any truthvalue assignments that make this formula false by constructing the truthtree for its negation. Here is the completed tree: Page 21 of 23 See the online version for interactive material here! Since the formula ((P Q) (P & Q)) is a conditional, then it could be taken to be the conditional analogue of the following argument: (P Q) (P & Q) As such, the truthtree for the formula demonstrates that the argument is not valid: each open branch provides a counterexample. The first open branch contains Q and P, while the second contains P and Q. That tells us that two truthvalue assignments are counterexamples: the assignment where Q is assigned t and P is assigned f is the first,
23 and the assignment where P is assigned t and Q is assigned f is the other, as we can clearly see in the truthtable: When it comes to arguments, however, just as we can use either the individual premises and conclusion in a truthtable, rather than the conditional analogue, we can use the individual premises and conclusion in a truthtree in order to test for validity. The only difference is that in the truthtree, we list the premises and the negation of the conclusion; in essence, we consider the negation of the conditional analogue, but skip the first step in analyzing the negation; instead we try to obtain a counterexample to the argument directly. Let us consider the following argument as an example: (A B) (B C) (A C) The completed tree for this argument is shown below. Note that the tree is closed, which demonstrates that the conditional analogue is in fact a tautology, and the argument is valid: If you like, complete the truthtable yourself to verify what the closed tree says: there are no truthvalue assignments that make the premises true and the conclusion false. Page 22 of 23 See the online version for interactive material here! Before heading on to the exercises, let us indulge in a little bit of metamathematical reflection. We have specified a thoroughly syntactic, indeed quite mechanical procedure and interpreted the completed truthtree obtained via this procedure in a semantic way, supporting or refuting the claim that a particular formula is a tautology of that a given argument is valid. Why are we justified in interpreting the completed tree in this way, and can we be sure that the procedure always yields a completed tree in finitely many steps?
24 We will come back to these issues, and address them thoroughly in a later chapter devoted to metamathematical issues. For the time being, however, do not forget the questions, or their significance! Page 23 of 23
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