Colette Sciberras, PhD (Dunelm) 2016 Please do not distribute without acknowledging the author!

Philosophy A/ Int. Level: Logic Notes and Readings Colette Sciberras, PhD (Dunelm) 2016 Please do not distribute without acknowledging the author!

PART 1: Readings from Copi: Introduction to Logic. Deduction and Induction Truth and Validity Fallacies PART 2: Propositional Logic 1: Introduction to Logic... 2 2: Elementary Propositions... 5 3: The Use of Elementary Assertions... 8 4: Complex Propositions I Negations and the Negator (NOT)... 10 5: Working out truth-tables I (Negator)... 13 6: Complex Propositions II Conjunctions and the Conjunctor (AND)... 15 7: Translations... 18 8: Working out Truth- Tables II (Negators and Conjunctors)... 21 9: The Adjunctor (OR)... 26 10: The Disjunctor... 29 11: The Subjunctor and Bi-Subjunctor... 31 12: Arguments and Implications... 36 PART 3: Formal Logic 13: Concepts from Formal Logic... 37 14: Logical Truth... 40 15: Implications... 41 16: Standard Implications (Modus Ponens and Modus Tollens)... 44 17: Equivalence... 48 18: Properties of Implications and Equivalences: (Reflexivity, Symmetry and Transitivity)... 49 19: Generalisation, Instantiation and Partial Replacement Rules... 55 From the old syllabus 20: Deriving Equivalent Formulae: De Morgan s and Double Negation Equivalence... 56 21: Deriving Valid Implications I: The Duality Principle... 59 22: Deriving Valid Implications II: Contraposition... 61 23: Deriving Valid Implications III: Transportation Rule... 64 24: A Complete System of Junctors... 66 Colette Sciberras, PhD (Dunelm) Logic, 2016 1

1: Introduction to Logic What is logic? Why do we need to study it in philosophy? Logic is a tool for checking whether our reasoning and arguments are valid, and for this reason it is essential for philosophy. Philosophers usually want to show that their view or theory (whether it s about physical reality, religion, ethics, language and so on) is the correct one - and they do this by providing arguments, which they and other philosophers can check, using logic. Having a good grasp of logic will help us to determine whether our reasoning, and that of others, is correct or not. Hurley (2008) 1 puts it this way; Logic may be defined as the science that evaluates arguments The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own (p. 1). However, before we can start to check whether arguments are valid or not, we need to build up our knowledge of a few more basic features of logic and of language in general. Content vs. Form An important distinction, which we should try to understand from the start, is that between content and form. Let s take a few examples: (1) Philosophy is an easy subject. (2) Socrates was a Greek philosopher. (3) I am really bored right now. (4) Giovanni Curmi Higher Secondary is the best school in Malta! If you study these sentences, you will see that they each share a similar form; in each case we are saying that something IS (or was) something else. Let us use a variable to represent these somethings for instance, in sentence (1), let A stand for Philosophy and let B stand for an easy subject. We can then represent sentence (1) in the following way: A is B In this way, we have brought out the form of sentence (1) without saying anything about the content; that is, I have shown the structure of my sentence, without specifying what it is about. The content of a sentence, therefore, refers to the ideas, or concepts, the people, things and so on that we are talking about. In our examples, these are philosophy, easy subjects, Socrates, Greek philosophers, me, being bored, our school, etc. The form of a sentence is its structure, in other words, it shows only how those concepts are related to each other, without specifying what the concepts are. We can denote the form of a sentence using symbols and variables, e.g. A is B. Colette Sciberras, PhD (Dunelm) Logic, 2016 2

The logic we study here is sometimes called formal logic, and this is because it is concerned ONLY with the form of arguments, and not with their content. That is, we are not interested in whether the argument is about Socrates, schools, or whatever; the content is irrelevant. Defining Variables In fact, sentences 1 4 can all be represented by A is b providing we define A and B each time to stand for the particular concepts we are talking about. For instance, Sentence (2) can be written as Sentence (3) can be written as Let A stand for Socrates Let b stand for a Greek Philosopher +Ab Let A stand for I Let b stand for really bored right now +Ab Importantly, for each exercise in logic, when we define a variable, for instance, by writing down, Let A stand for Socrates, that variable must always be used in the same way. That is to say, A must always represent Socrates and nothing else, and conversely, whenever the word Socrates appears in that exercise, we must always use the variable A to represent it. We can use the same variable to represent another word or phrase only if we have started a separate exercise. It is important to understand that a variable can stand for a single word, a phrase, or even an entire sentence too (later, we will refer to these sentences as propositions). For example, I can represent the whole sentence Socrates was a Greek philosopher by a single variable, say, a. The first step in logic, therefore, is to decide what variables we are going to use, and what they are going to stand for. We call this defining our variables. Using symbols for logical particles Our objective in logic, what we are trying to do, is to build a new sort of language, by removing the content from arguments and representing only their form. There are some words which are represented by variables, as we saw above, and the others, which are known as logical particles are represented by symbols. An useful analogy here is mathematics. Suppose I have two apples in my bag and another two in my fridge. I can represent this as: 2 + 2 = 4 Of course, this can also represent two people in class and two outside, or two days at the weekend and two mid-term holidays. Therefore, 2 and 4 can be compared to our variables, in that they can represent anything at all. The + and = signs, on the other hand, represent a sort of function, for instance + means that we add up whatever is to the left and to the right of the + sign. These signs are similar to our logical particles, and in fact logic too has such symbols, which represent certain functions. In sentences 1 4, the word IS relates the concept A to the concept B (by identifying them with each other.) The Colette Sciberras, PhD (Dunelm) Logic, 2016 3

word is, therefore, is a logical particle (known as a copula), and next week, we shall see that in logic, is can be represented by the + sign. Logic is a system of rules So, in logic, we first define variables to stand for certain words or phrases, and then relate these variables to each other by using symbols. Just as in maths, we need to know how these symbols work. If we say that 2 + 2 = 0 we have clearly not understood what + means, that is, we haven t understood the rules of addition. Similarly, in logic we need to know the rules for our logical particles. Just as in Mathematics, 2+2 is always equal to 4, whether we are talking about apples, people, holidays or whatever, in logic, if we apply our rules correctly, we will always obtain the correct result. This is why Riolo refers to logic as an ortholanguage (which, roughly means, a good or correct language) (Riolo 2001, 4). Exercises: 1. Identify the form and logical particles in these sentences. Logic is not difficult. The day before yesterday was not a very good day for me. He is not the person I thought he was. Make sure you have understood the following: - The content versus the form of a sentence - How to define variables - What a logical particle is Colette Sciberras, PhD (Dunelm) Logic, 2016 4

2: Elementary Propositions Sentences, assertions, and propositions We all know what a sentence is; in written language, it is a group of words (the first of which has a capital letter) which has a full-stop, a question mark or an exclamation mark at the end. Here are some sample sentences: (1) Socrates was Greek. (assertion) (2) Do you study philosophy? (question) (3) Open the door please. (request/ command) Logic is only concerned with sentences of type (1) that is, with assertions. How can we tell whether a sentence is an assertion or not? It s very easy ask yourself can this sentence be true or false? If the answer is yes, then the sentence is an assertion. Only sentence (1) is an assertion here the others cannot be true or false. In fact, sentence (2) is a question, whereas (3) is an imperative (a request or a command). To repeat then, in logic, we are only interested in those sentences which can be either true or false, and we call these assertions. Another commonly used term in logic is proposition. Propositions and assertions are more or less the same, and many authors use the words interchangeably. A proposition can be thought of as the meaning of an assertion. For example, the two assertions Socrates was Greek and Sokrate kien Grieg both express the same proposition. Riolo says that a proposition is the taking of a position or the commitment behind an assertion (2001, p7). In other words, we usually make an assertion because we believe that the proposition it expresses is true. However, at this stage you needn t worry too much about the difference between propositions and assertions the main thing to understand at this point is that propositions and assertions are those types of sentences which must be either true or false. Representing propositions symbolically As we have seen, we can represent the form of sentence symbolically, by using variables to represent the content, and symbols to represent the logical particles. It is important to remember that there are several ways of representing the same proposition. For example, the proposition expressed as (1) above, might be represented as follows: (i) Let A stand for Socrates Let B stand for Greek +Ab Alternatively, we could use a single variable to represent the entire proposition: (ii) Let a stand for Socrates was Greek a Colette Sciberras, PhD (Dunelm) Logic, 2016 5

Elementary and complex propositions The first thing we must understand is how to distinguish elementary propositions from complex ones. An elementary proposition is one which contains a subject, an object (sometimes called a complement) and some form of the verb to be (i.e. past, present or future, singular or plural, first, second or third person). An elementary proposition must contain nothing else. Most of our examples, so far, have been elementary propositions. Consider the following: (1) Philosophy is an easy subject. (2) Socrates was a Greek philosopher. (3) I am really bored right now. (4) Giovanni Curmi Higher Secondary is the best school in Malta. All of these contain a subject, a predicate, the verb to be, and nothing else. Even though sentence (4) seems a bit longer and more complex than the others, careful examination will show that it is actually an elementary proposition. The subject is Giovanni Curmi Higher Secondary, the predicate is the best school in Malta and of course there is the verb to be ( is ). In logic, we call the subject a nominator. The nominator is whatever it is we are talking about, or referring to. We can usually point to something and say This or we can use its name such as Philosophy as in example (1). The predicator is what we want to say about our nominator, in this case, that it is an easy subject. The verb to be is called the copula (plural copulae). We can have either a positive copula ( is, are, or was ) or a negative one ( isn t, aren t, weren t ). With our copula we either affirm or negate the predicator of the nominator. Compare elementary propositions with the following: (1) He is either studying or else he is on the phone. (2) Socrates was not Indian. (3) I am really bored and I am also tired now. (4) If GCHSS is the best school in Malta, then I want to go there! Some of these sentences contain more than one copula and more than one predicator, i.e. they contain more than one elementary proposition. They also contain other logical particles the word or in (1), not in (2), and in (3), and the words if then in (4). These are logical particles called junctors which we shall be meeting soon. Any proposition which contains a junctor is a complex proposition. Colette Sciberras, PhD (Dunelm) Logic, 2016 6

Exercises: 1. Which of these sentences are assertions? 1. I like logic. 2. Did you learn anything new today? 3. She didn t study and that s why she failed. 4. Go home and do some work now! 5. Hamlet asked the question to be or not to be? 2. Are these assertions complex or elementary? 1. That girl is my cousin. 2. The girl I saw yesterday is my cousin s Polish babysitter. 3. The girl I saw yesterday was wearing a blue top and she had short, spiky hair. Make sure you have understood the following: - Assertions/ propositions are sentences that must be true or false - Elementary propositions vs. complex ones - Nominators, predicators and copulae Colette Sciberras, PhD (Dunelm) Logic, 2016 7

3: The Use of Elementary Assertions How can an elementary assertion be used? In other words, what sort of action might we take, having received and accepted a simple proposition? Riolo discusses this at some length and introduces the terms virtual bi-location, and virtual bi-temporation (2001, 11-13). We all understand what virtual means these days; a virtual reality, say, is a kind of experience in which it feels, or seems as if you re in different circumstances, even though you re not. Second Life, a virtual reality game, is apparently so realistic that some people become more involved in it than in their actual lives. You can spend virtual money there, which, again, although not real money, functions as though it were. You can even send or receive virtual gifts via Facebook, and again, even though they are not really gifts, getting one makes you feel as though you really received a present (or so I m told!) The prefix bi- means two; location has to do with place and temporation to do with time. Therefore, virtual bi-location means something like it seems like I am in two places at once, while virtual bi-temporation would be the impression of being present at two different times. Virtual bi-location Suppose you are at school, sitting in class, and the lesson has just started. Your friend is not there and you are worried that she may not know the classroom number. So you text her a message: The logic lesson is in room 7. (Please remember that the use of mobile phones is prohibited in class!) Here you are the speaker (let s call you person A) and you make the assertion the logic lesson is in room 7. You are witnessing the event, i.e. the lesson in room 7, and communicate it to your friend (call her person B). She receives your assertion when she opens and reads your text. If she understands what you say, and believes you, your friend accepts the assertion. Importantly, in the case of virtual bi-location, the event and the assertion occur at the same time. Person A tells person B about something that is happening now. Person A is witnessing the event, whereas person B is not. How can person A s assertion be useful to person B? Your friend, who was waiting in room 9 all this time, now has information about what is happening in another place, i.e. in room 7. Even though she is not in room 7, it s as if she were there. That is, thanks to your message, she is virtually present in the right classroom, witnessing the start of the lesson, but she is actually in another place. Once she has received and accepted your assertion, that is, it is as if she were in two places at once. Moreover, your friend can now act upon this information. In this case, she can leave room 9 and go to room 7. Therefore, one way in which an assertion can be useful to someone who receives and accepts it, is through virtual bi-location. For that person, it is as if she were in two places at once; she is actually present in one place and virtually present in another, witnessing (virtually) the event taking place there, about which she has received the assertion. Moreover, that person can then decide to act upon the information she has received. Virtual bi-temporation Virtual bi-temporation is similar to virtual bi-location, but concerns time, rather than place. For the person who accepts and receives the assertion, it is as though they were present in two different Colette Sciberras, PhD (Dunelm) Logic, 2016 8

times. Of course, that person is actually in the present, but he or she can also be virtually in the past or in the future. Therefore, there are two types of virtual bi-temporation: Present and Past: In this case, A tells B about an event which happened before, and so for B it is as though he was there at the time of the event, as well as being in the present time. For example, you tell your friend that yesterday s logic lesson was really interesting and as a result, for your friend it is as if he is both in the present, listening to you speak, and also in the past, witnessing yesterday s lesson. He can then decide to act on the information (assuming that he accepts your assertion, i.e. he believes you) and he decides to go for the next lesson, because he believes it will be interesting. Another example of this occurs when I recall something I witnessed earlier, in which case the maker and receiver of the assertion are the same person - me. Otherwise, I might read about an event that occurred in the past, say in a newspaper or history book. In this case, the author is A, who makes the assertion, while I, who receive the assertion, am person B. Present and Future: As above, but this time, A makes an assertion about the future. The most common cases of virtual bi-temporation involving present and future are those of promising, and forecasting. For example, I make the assertion there will be a test next week. This is an example of a forecast. You receive that assertion if you are listening and hear what I say, and you accept it if you believe me, and you don t think that I am bluffing. Again, for you, it is as though you are both in the present, listening to my assertion, and in the future, sitting for your test. Based on the information you receive, you will (hopefully) go home and do some revision. It is important, when discussing the use of assertions, that the receiver takes some action with respect to it. Thus, if someone tells me, for instance, that share prices are going down, I will decide to sell mine quickly (or else to buy some). If I hear on the weather forecast that it is going to rain tomorrow, I will decide to take an umbrella! Exercise: In not more than 10 lines, explain how an elementary assertion might be useful to someone who receives and accepts it. Make sure you have understood the following: - Virtual bi-location - Virtual bi-temporation - The use of elementary assertions Colette Sciberras, PhD (Dunelm) Logic, 2016 9

4: Complex Propositions I Negations and the Negator (NOT) You will remember that a simple or elementary proposition is one which is made up of a nominator, a predicator, and a copula. Now we are going to turn to complex propositions. Therefore, we shall be using a different type of notation instead of +Np or Np, as this is only useful for elementary propositions. Another important point is that from now on, wherever the word not occurs, we shall be treating that proposition as a complex one. Previously, we interpreted a sentence such as Socrates was not Italian as an elementary proposition, symbolized as Np. Now, we shall think of it as a complex proposition, and in fact, the word not is the first junctor we shall be using to make our first type of complex propositions the negation. A complex proposition can be defined as any proposition that is made up of at least one elementary proposition and at least one junctor. Therefore the sentence Socrates was not Italian is made up of the elementary proposition Socrates was Italian plus the junctor not. Here are some more examples: (1) If George Washington was beheaded, then George Washington is dead. This is made up of two elementary propositions, George Washington was beheaded and George Washington is dead which are combined with the junctor if then. (2) London is North of Paris and South of Edinburgh. Again, this contains two elementary propositions, London is north of Paris and London is south of Edinburgh joined together with the junctor and. There are several other junctors, which we will be encountering later on. This week, we shall only be dealing with our first junctor the negator (NOT). The negator can appear as the word not, or as the phrases it is not the case that, it is not true that and so on. Expressing negations symbolically Important! We use single letters as variables for elementary propositions when these form part of a complex proposition. That is, instead of writing +Np for Socrates was Italian we will express it as follows: Let a stand for Socrates was Italian a Riolo uses German letters, e.g. a. Do not let this worry you they are simply different systems. We represent the junctor NOT through the symbol Therefore, we can represent the complex proposition Socrates was not Italian as follows: Let a stand for Socrates was Italian a Colette Sciberras, PhD (Dunelm) Logic, 2016 10

Notice that when we define a variable it must stand for an elementary proposition. Therefore it is wrong to say Let a stand for Socrates was not Italian. Instead, we use the symbol to express the word not. The negation of a proposition is its counterpart We obtain the counterpart of a single elementary proposition by changing the copula from a positive to a negative or vice-versa. For example, the counterpart of Socrates is Greek is Socrates is not Greek. Therefore, to add the word not to a proposition, that is, to negate it, is to obtain its counterpart. This is true also of complex propositions. Whatever complex proposition the variable [a] stands for, if we negate it we are saying something like it is not the case that *a+ or it is not true that *a+ (symbolized as a). This will give us the counterpart of [a]. With elementary propositions, we added not to the copula. With complex ones, we put the negator in front of the entire proposition. For example: Complex proposition 1: If George Washington was beheaded, then George Washington is dead Counterpart: It is not true that [if George Washington was beheaded then George Washington is dead] Complex proposition 2: London is North of Paris and South of Edinburgh. Counterpart: It is not the case that [London is North of Paris and South of Edinburgh.] It should be obvious that if a proposition is true then its counterpart will be false, and that if a proposition is false, its counterpart will be true. From this fact, we can obtain the rules for the use of the negator, as we shall see below. Rules for the negator We use the symbol to show that we are permitted to move from that which is on the left, to that which is on the right. We can read it out as leads to or results in. E.g. a b stands for proposition [a] leads to proposition [b] The symbol shows that we are permitted to move in both directions. That which is on the right leads to that which is on the left and vice-versa. E.g. a b stands for proposition a leads to proposition b and proposition b leads to proposition a Rule 1 (R1): If a proposition [a] is true then its counterpart [ a] will be false If a proposition [a] is false, then its counterpart [ a] will be true We can express this symbolically as follows: (i) a is true a is false (ii) a is false a is true RI Colette Sciberras, PhD (Dunelm) Logic, 2016 11

Truth-table for Negator (NOT) Every junctor we introduce will have a truth-table as a summary of its rules. It is important to know how to write these rules, both symbolically (as above) and as a truth-table. The truth-table has two sets of double lines; one vertical and one horizontal. To the left of the vertical double-lines we write down the variables for all the elementary propositions which occur in our rules. With the negator, there is only one variable a and therefore we only need one column before the vertical double line. We draw the horizontal double-line underneath this variable: a Underneath our variable, we write down all the possible truth-values that the variable can have. Clearly, a single proposition [a] can only be true or false, and therefore, we will have two rows underneath our variable. We write T to represent true, and F for false: a T F On the right of our vertical double-line, we write down the complex proposition which occurs in our rules (here a) a T F a Below that, we put down the truth-values which are stated in our rules. We know from our rules that when a is true, a is false and that when a is false, a is true. Therefore, in the first row (where a is T) we put F under a, and in the second (where a is F) we put T. a T F a F T This is the truth-table for our first junctor, the negator (NOT). It is simply a summary of R1. Exercise: Study the rules and the truth-table for the negator until you know them by heart! Make sure you have understood the following: - A complex proposition is any proposition that contains at least one elementary proposition and at least one junctor - The negator is the junctor NOT and its symbol is - The negation of any proposition is its counterpart - Rules and truth- tables for the negator Colette Sciberras, PhD (Dunelm) Logic, 2016 12

5: Working out truth-tables I (Negator) Truth-tables do not only summarize rules; we can also use them to discover whether a complex proposition is true or false, given the truth or falsity of its constituent elementary propositions. For example: Let a stand for Today is Thursday It is quite clear that if [a] is true then [ a] is false. We can also work out mentally that [ a] is true. ([ a] will mean something like It is not the case that today is not Thursday which simply means Today is Thursday ) But what about more complex propositions, like [ a]? Fortunately, we do not have to work this out mentally; there is a very easy, systematic way of doing this. Example 1: Work out the truth-table for a We start, as before, by putting the variables for all the elementary propositions which our complex proposition contains, to the left of our double-line, and writing the complex proposition to the right. As before, there is only one variable, namely, [a]: a a Under [a] we put all the possible truth-values for it (i.e. T or F): a a T F This time we are going to work out the values of [ a] rather than just copying them from the rules. We start by copying out the values of [a] under the variable [a] to the right of the double-line (i.e. under column 1): 1 a a T T F F Using the rules for the negator ( ) we can work out the values of the column to the near left of the last one we have worked out. That is, we first work out the values for [ a] under column 2. When [a] is T [ a] is F and when [a] is F, [ a] is T. Colette Sciberras, PhD (Dunelm) Logic, 2016 13

Rules for a is T a is F a is F a is T 2 1 a a T F T F T F Once we have worked out the negation of a column, we do not use that column any more. This is why I have now put the values under column 1 in grey. We now have the values for [ a], so we do not need the values of [a] any more, and therefore, we can ignore them from now on. The next step is to work out the values of [ a], i.e. again, the column to the near left of the last one we worked out (line 3). We do this by using the same rules for the negator: 3 2 1 a a T T F T F F T F Where we have F, its negation is T, and where we have T, its negation is F. Again, once I have this new set of values, I ignore the last one. We work out the final column in the same way: 4 3 2 1 a a T F T F T F T F T F * We put an asterisk below the last column we worked out to show that this is our final answer. What we have discovered in short is the following: Exercise: When [a] is true, [ a] is false. When [a] is false, [ a] is true. Riolo, p. 20, no. 1* (p. 19 of the first edition) Make sure you have understood the following: - Step-by step method for working out truth-tables Colette Sciberras, PhD (Dunelm) Logic, 2016 14

Complex Propositions II Conjunctions and the Conjunctor (AND) So far we have introduced our first junctor, the negator. We have set out the rules for its use, and summarized them as a truth-table. We have also seen how to work out the truth-tables for further complex propositions involving negation, e.g. [ a], [ a] etc. (see Riolo, p.20-21). This week, we shall introduce our second junctor, the conjunctor, which, in spoken English we read as and, and in logic we represent with the symbol. A complex proposition that involves a conjunctor is called a conjunction. Expressing conjunctions symbolically The conjunctor is a two-place junctor which means that it requires at least two elementary propositions. This is easy to understand if you know what the word and means. Consider the following proposition: (1) I like ice-cream and chocolate This is a complex proposition, although it might seem like an elementary one. It is made up, in fact, of two elementary propositions: (i) I like ice-cream (ii) I like chocolate When we combine these two, using the conjunctor, we get the above sentence (1). I like ice-cream AND (I like) chocolate All the junctors we will encounter from now on are two-place junctors. Only the negator is not. As we saw in the previous lesson, the negator can be added to a single elementary proposition to form a complex one, represented by, e.g. [ a]. The conjunctor, on the other hand, is used to join at least two elementary propositions, and therefore, we will be requiring more than one variable to symbolize a conjunction. This is how we represent sentence (1) symbolically: Rules for the Conjunctor Let a stand for I like chocolate Let b stand for I like ice-cream a b Let us think about proposition (1) above. Under which conditions is it true? Clearly, if it is true that I like both chocolate and ice-cream, then I have not told a lie, and my proposition is true. If, on the other hand, I don t like ice-cream, then my sentence I like ice-cream and chocolate is false. In the same way, the sentence is false if I don t happen to like chocolate. Finally, the sentence is also false if I like neither chocolate nor ice-cream. In other words, the conjunctor has the following rules: Colette Sciberras, PhD (Dunelm) Logic, 2016 15

(i) a is true, b is true a b is true (ii) a is false a b is false R2 (iii) b is false a b is false Truth-table for Conjunctor (AND) Recall that we begin our truth-table by writing down, to the left of the vertical double-line, all the variables that occur in our rules (representing the elementary propositions). With the negator, there was only one variable and therefore we only needed one column before the vertical double line. The conjunctor, as we said, is a two-place junctor, and that means that our rules are going to involve two elementary propositions (as seen above). Therefore, this time we need to put two columns before the vertical double-line; one for each variable that occurs in the rules (that is, [a] and [b]). We put our complex proposition, the conjunction [a b], to the right. a b a b We draw a single line between every individual proposition, whether elementary or complex. Therefore, I put a line between [a] and [b] because these are representing single elementary propositions. On the other hand, I left [a b] as a single column because this represents a single complex proposition. The next step, remember, is to write down all the truth-values for [a] and [b] to the left of the double-line. This was very easy in the case of the negator, as there was only one variable [a], which could only be either true or false. When there is more than one variable, it gets a little more complicated however, there is a systematic way of doing it which we will look at here. First of all, the important thing to remember is that we want to write down all the different combinations of truth-values that there can be for all the variables we have. When there are two variables, as in this case, we get four combinations: (1) [a] and [b] are both true, (2) [a] and [b] are both false, (3) [a] is true and [b] is false (4) [a] is false and [b] is true. Here is the conventional way of doing it, step-by-step: To calculate how many combinations there are, that is, how many rows you are going to need under your variables, take the number of variables (in this case 2) and calculate 2 to the power of that number, i.e. 2 2 = 4 This is how we know that there are four combinations (as seen above). Colette Sciberras, PhD (Dunelm) Logic, 2016 16

In the first column furthest to the left, that is, under [a], we write half that number as T and the other half as F. Half of four is two, and therefore, this gives us two T s and two F s: a b a B T T F F For the next column, we divide that number into two again. Half of two is one, and so this time, we are going to write one T and one F. We repeat this pattern until we have filled our four rows: a b a B T T T F F T F F Now we have all the different combinations of truth-values for [a] and [b], we can start writing in the values for our complex proposition [a b] to the right of the double-line. Our rules say that [a b] is only true when both [a] and [b] are true. If either one of [a] or [b] is false, then [a b] is false. From this, we can see that [a b] is clearly also false when both [a] and [b] are false. Therefore, our conjunctor rules can be summarized in the following truth-table: a b a B T T T T F F F T F F F F This is the truth-table for our second junctor, the conjunctor (AND). It is simply a summary of R2. Exercises: 1. Study the rules and truth-table for the conjunctor until you know them by heart! 2. What if there were three elementary propositions in a conjunction? How many different combinations of truth-values would there be, and how would we write them out? Make sure you have understood the following: - The conjunctor is the junctor AND and its symbol is - Writing conjunctions symbolically - Rules and truth-table for the conjunctor - How to work out the different combinations of truth-values for any number of variables Colette Sciberras, PhD (Dunelm) Logic, 2016 17

7: Translations Now that we have two junctors (AND, NOT), we are in a position to start doing some translating. It is important to understand from the start that in logic, translations can mean both: (i) From ordinary English sentences into logical symbols (ii) From logical symbols into ordinary English Translating from English into logical notation Let us see how to translate the sentence: (1) It is not the case that Anne is both reading and watching T.V. (from Riolo, p. 29) First of all, remember that since we are dealing with complex propositions, we use single letters as variables to represent their constituent elementary propositions. It is important that for every translation, or every single task, once we have defined a variable, it must always be used in exactly the same way. The first thing to do is to identify the elementary propositions in the sentence (or sentences) to be translated. Our example contains two, namely: Anne is reading Anne is watching T.V How do we know this? Remember an elementary proposition has a nominator, a predicator, a copula and nothing else. Anne is the nominator in both our elementary propositions, they both contain the copula is, and they each have a predicator - reading in the first and watching T.V. in the second. What about the other words or phrases in our sentence? As you will recall, it is not the case is one way of expressing the negator, the junctor NOT, whereas and is the conjunctor. We will see what the word both signifies below. If we leave out the negator for a while, we have the sentence (2) Anne is (both) reading and watching T.V. We can translate this as follows: Let a stand for Anne is reading Let b stand for Anne is watching T.V. a b (2) Anne is reading and watching T.V. The phrase it is not the case that negates everything that follows it, in other words, that Anne is both reading and watching T.V. Therefore, the translation of our complete sentence is as follows: Colette Sciberras, PhD (Dunelm) Logic, 2016 18

Let a stand for Anne is reading Let b stand for Anne is watching T.V. (a b) (1) It is not the case that Anne is both reading and watching T.V. Notice the use of brackets around [a b+. This is where the word both comes in. If I had left out the brackets in my translation, the meaning would have changed altogether: Let a stand for Anne is reading Let b stand for Anne is watching T.V. a b (3) Anne is not reading and Anne is watching T.V. Sentence (3) means something very different from sentence (1). Sentence (1) means it is not true that she is doing both; that is to say, she may either be reading or watching T.V. but not both. It does not tell us which of the two activities she is doing; only that she is not doing both, and in fact, she may be doing neither. Sentence (3) on the other hand, does tell us which she is doing; it specifies that she is watching T.V., and not reading. The word both, therefore, specifies that the negator, the phrase it is not the case that, applies to the conjunction of our elementary propositions. It is usually an indicator that we are going to need to use brackets somewhere. Compare with the following: (4) It is not true that it rained, but the streets still got wet Again, there are two elementary propositions. Let a stand for it rained Let b stand for the streets still got wet a b (4) It is not true that it rained, but the streets still got wet NOTE: the word but must be interpreted as a conjunctor, i.e. as another form of the word and. A final example: (5) It is not the case that it both rained and the streets did not get wet. Remember, when working with complex propositions we define our variables to stand for affirmative elementary propositions (i.e. with an affirmative copula). We will add the not in did not get wet later on, as the negator. Let a stand for it rained Let b stand for the streets got wet (a b) (5) It is not the case that it both rained and the streets did not get wet. Colette Sciberras, PhD (Dunelm) Logic, 2016 19

Translating from logical notation to English sentences This is just the reverse of what we have been doing above. This time we are provided with a proposition in logical symbols, and with the definition of the variables used, and our task is to translate these into ordinary English. For example: Translate the following proposition into English: (a b) Where a stands for today is Monday b stands for we have a logic lesson We do this step by step, starting from inside the brackets, and translating any negations of elementary propositions first: b We do not have a logic lesson We can then move onto the conjunction: a b Today is Monday and we do not have a logic lesson Finally, we add the negation outside the brackets: (a b) It is not both the case that today is Monday and (the case that) we do not have a logic lesson Exercises: 1. Riolo p. 29 ex. 5 2. Translate the following into English: i) a b ii) (a b) Where a stands for b stands for I like ice-cream I like chocolate Where a stands for b stands for she studies English she studies philosophy Make sure you have understood the following: - How to identify the elementary propositions in a complex one. - Variables are always defined to stand for affirmative elementary propositions. - How to translate sentences into logical symbols - How to translate from logical notation to English - How to use brackets Colette Sciberras, PhD (Dunelm) Logic, 2016 20

8: Working out Truth- Tables II (Negators and Conjunctors) Once we have translated a complex proposition, we are often asked to write down its truth-table, or to work out its truth-value. Let us look again at an example from lesson 7. Example 1: We translated this as follows: It is not the case that Anne is both reading and watching T.V. Let a stand for Anne is reading Let b stand for Anne is watching T.V. (a b) How do we go about working out the truth-table for such a proposition? Here, I shall set out the steps that we must follow. As usual, we put all the variables for the elementary propositions to the left of our double-line, and the complex proposition itself on the right. We know that there are two variables ([a] and [b]) and therefore there will be four rows of values (2 2 ). We enter these values under [a] and [b] respectively. a b (a b) T T T F F T F F The next step is to work out the values for the complex proposition itself. Importantly, when there are brackets we must always begin with whatever is inside them. In this case, we have a conjunction [a b]. Using the rules for the conjunctor, we work out what the values will be, and write them down, under the symbol for the conjunctor (i.e. ) The rules for AND tell us that [a b] is only true when both [a] and [b] are true, i.e. only in the first row. All the other combinations of values result in false: 1 a b (a b) T T T T F F F T F F F F NB: It is not necessary to write down numbers corresponding to your steps I have only done this here to make it easier for you to follow what is going on. Colette Sciberras, PhD (Dunelm) Logic, 2016 21

Finally, we work out the values for the negator outside the brackets, using the last set of values we have obtained (under column 1). We do this by applying the rules for the negator to the values in line 1, and write our answers under the negator sign. After we work out line 2, the values under 1 can be ignored: 2 1 a b (a b) T T F T T F T F F T T F F F T F * We put an asterisk under the last column we have worked to show that this is our final answer. What we have found out is the following: When [a] is true and [b] is true, our proposition, (a b) is false When [a] is true and [b] is false, (a b) is true When [a] is false and [b] is true, (a b) is true When [a] is false and [b] is false, (a b) is true In other words, the proposition it is not the case that Anne is both reading and watching T.V. is FALSE when Anne is reading is true and Anne is watching T.V. is also true. In every other case, the sentence is true. This makes sense, of course, since our sentence says, precisely, that it cannot be true that Anne is both reading and watching T.V. Example 2: Work out the truth-table for the following proposition: ( a b) c Since there are three variables here, this will give us eight rows (2 3 ). Notice the system of distributing T s and F s is the same; half T s and half F s in the first column (i.e. 4 and 4), half of that in the second (2 and 2), and half again in the last (alternating T s and F s). With such long propositions, it is helpful to write down the values of [a], [b] and [c] wherever they appear in the complex proposition: 1 2 3 a b c ( A b) c T T T T T T T T F T T F T F T T F T T F F T F F F T T F T T F T F F T F F F T F F T Once again, we F F F F F F start from the brackets. This time, note that one of the elementary propositions inside the brackets (i.e. a) has a negator next to Colette Sciberras, PhD (Dunelm) Logic, 2016 22

it. It is important, in such cases, to start from that negator. We apply the rules of to the values in line 1, giving us the following: 4 1 2 3 a b c ( A b) c T T T F T T T T T F F T T F T F T F T F T T F F F T F F F T T T F T T F T F T F T F F F T T F F T F F F T F F F Next we work out the conjunctor inside the brackets. Note that we apply the rules of to the values under 4 and 2 (not those under 1). This is because we have already worked out the values of [ a], and therefore we don t need those of [a] any more. Our conjunction is [ a b], whereas, if we used the values under 1 and 2, we would be working out [a b] instead: 4 1 5 2 3 a b c ( a b) c T T T F T F T T T T F F T F T F T F T F T F F T T F F F T F F F F T T T F T T T F T F T F T T F F F T T F F F T F F F T F F F F We now have the final values for everything which is contained inside the brackets (under line 5), and therefore the values under lines 1, 2, and 4 are no longer needed. There is a negator outside the brackets which belongs to those brackets alone. Therefore, we work that out next, applying the rules of to the values under 5. (If there had been another set of brackets, i.e. (( a b) b) we would have left that negator for the very end, as it would belong to the outer set of brackets) 6 4 1 5 2 3 a b c ( a b) c T T T T F T F T T T T F T F T F T F T F T T F T F F T T F F T F T F F F F T T F T F T T T F T F F T F T T F F F T T T F F F T Now we have F F F T T F F F F column 6. There are a couple of steps left. the values for ( a b) under Colette Sciberras, PhD (Dunelm) Logic, 2016 23

First, note the negator next to [c] in line 7. We need to work that out next, applying the rules of to the values under line 3. Then, line 3 is no longer needed. 6 4 1 5 2 7 3 a b c ( a b) c T T T T F T F T F T T T F T F T F T T F T F T T F T F F F T T F F T F T F F T F F T T F T F T T F T F T F F T F T T T F F F T T T F F F F T F F F T T F F F T F Finally, we work out the conjunctor in the middle, that joining ( a b) with c. We do this, by applying the rules for to the values in lines 6 and 7. Since this is our final answer, we mark it with an asterisk: 6 4 1 5 2 8 7 3 a b c ( a b) c T T T T F T F T F F T T T F T F T F T T T F T F T T F T F F F F T T F F T F T F F T T F F T T F T F T T F F T F T F F T F T T F T F F F T T T F F F F F T F F F T T F F F T T F * Colette Sciberras, PhD (Dunelm) Logic, 2016 24

Working out the truth -value of a proposition: Sometimes, we are asked to work out a truth-value instead of a truth-table. In this case, we are always given the truth-values of each elementary proposition. For example: Work out the truth-value of a ( b a) Given that: a is True b is False In this case, we do not have to write down all the different combinations; instead, we know this time, that a is T and b is F. For this reason, we will only have one line under our variables: a b a ( b a) T F We follow the usual steps to get the following answer: Answer: when a is True and b is False, the proposition a ( b a) is FALSE Exercises: Riolo pp. 23-24, ex. 2, 3, 4. 7 1 8 6 4 3 2 5 1 a b a ( b a) T F F T F T F T F F T * Make sure you have understood the following: - How to work out truth-tables with two types of junctors (the negator and conjunctor) - How to work out truth-tables with three variables. - How to work out truth-values Colette Sciberras, PhD (Dunelm) Logic, 2016 25

9: The Adjunctor (OR) The adjunctor as one way of expressing the word or in logic The word or can be interpreted in two different ways, as we shall see, and each of these is represented by a different junctor. This lesson and the next will examine these in detail. The first junctor we shall be looking at is the most common way of interpreting or, sometimes written as either or. It is called the adjunctor or the inclusive or and it is symbolized by. To take an example, suppose I am expecting two guests for dinner, Bob and Tom. I hear the door bell and think It s either Bob or Tom. Clearly, if I find Bob standing at the doorway, the sentence is true. If Tom is standing there, the sentence is still true. What if Bob and Tom arrived together? If we interpret either or as an adjunctor, then the sentence it s either Bob or Tom is still true, even though it s actually both Bob and Tom standing at my door. Another example: I am at a restaurant and ask for the desert menu. The waiter says, you can have ice-cream or cake. Clearly here, what he means is I can have ice-cream alone, cake alone, or I can have both ice-cream and cake. What these examples show is that with an adjunction, our rules and truth-table are going to show three cases where the complex proposition is true (see below). Translating Adjunctions To translate sentences with an adjunction, we use the usual method. First we need to identify the elementary propositions (making sure they are affirmative) and define variables to represent them. Do not confuse the symbol for the inclusive or ( ) with that for and ( ). Example 1: It s either Bob or Tom Let a stand for It s Bob Let b stand for It s Tom a b Example 2: You can have ice-cream or cake Let a stand for you can have ice-cream Let b stand for you can have cake a b Rules for the Adjunctor As we saw above, the adjunction suggests three possibilities. In each of these, the sentences in the examples above will be true: 1. Only the first elementary proposition is true ( It s Bob ; You can have ice-cream ) 2. Only the second elementary proposition is true ( It s Tom ; You can have cake ), Colette Sciberras, PhD (Dunelm) Logic, 2016 26

3. Both elementary propositions are true (i.e. both Bob and Tom are standing behind my door; The waiter means I can have both ice-cream and cake if I like). The only option which is ruled out, i.e. which will make the sentences false, is where neither of the elementary propositions is true (i.e. there is someone else at my door, not Bob or Tom; there is neither ice-cream nor cake available after all). This will give us the following rules: a is true a b is true b is true a b is true R3 a is false, b is false a b is false Truth-Table for Adjunctor We draw the truth-table for the adjunctor rules in the usual way. a b a b T T T T F T F T T F F F N.B. Many students confuse the truth-table for the conjunctor (and) with that of the adjunctor (inclusive or.) Note that the adjunctor has three T s and one F; the conjunctor has one T and three F s (see lesson 6). Working complex truth-tables with adjunctors The adjunctor follows the same rules as the conjunctor that is, we work any negators which are exactly next to the variables before an adjunctor. After working out negators, we work any junctor, i.e. an adjunctor or conjunctor, within brackets. Then we work out negators outside brackets, and finally any junctors left (conjunctors or adjunctors) outside brackets. It is important that, just like we did with conjunctors, after working out an adjunctor, we must ignore the values of those elementary propositions that it combines into a complex proposition. These rules remain the same for all the junctors we will introduce, and will not be repeated. Example 1: It is not the case that either Bob or Tom is at the door Let a stand for Bob is at the door Let b stand for Tom is at the door (a b) a b (a b) T T F T T F F T F T F T F F T F * Colette Sciberras, PhD (Dunelm) Logic, 2016 27