forall x York Edition P.D. Magnus University at Albany, State University of New York Modified for Cambridge by: Tim Button University of Cambridge Further modified for York by: Robert Trueman University of York P.D. Magnus would like to thank the people who made this project possible. Notable among these are Cristyn Magnus, who read many early drafts; Aaron Schiller, who was an early adopter and provided considerable, helpful feedback; and Bin Kang, Craig Erb, Nathan Carter, Wes McMichael, Selva Samuel, Dave Krueger, Brandon Lee, and the students of Introduction to Logic, who detected various errors in previous versions of the book. Tim Button would like to thank P.D. Magnus for his extraordinary act of generosity, in making forall x available to everyone. Thanks also to Alfredo Manfredini B ̈ ohm, Sam Brain, Felicity Davies, Emily Dyson, Phoebe Hill, Richard Jennings, Justin Niven, and Igor Stojanovic for noticing errata in earlier versions. Robert Trueman would also like to thank P.D. Magnus, and Tim Button as well. c © 2005–2020 by P.D. Magnus, Tim Button and Robert Trueman. Some rights reserved. This book is based upon P.D. Magnus’s forall x (version 1.29), available at fecundity.com/logic, which was released under a Creative Commons license (Attribution-ShareAlike 3.0). You are free to copy this book, to distribute it, to display it, and to make derivative works, under the following conditions: (a) Attribution. You must give the original author credit. (b) Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under a license identical to this one. — For any reuse or distribution, you must make clear to others the license terms of this work. Any of these conditions can be waived if you get permission from the copyright holder. Your fair use and other rights are in no way affected by the above. — This is a human-readable summary of the full license, which is available on-line at http://creativecommons.org/licenses/by-sa/3.0/ In accordance with this license, Tim Button made changes to P.D. Magnus’s original text, and added new material, and he offered forall x :Cambridge 2014-15 under the same Creative Commons license. The most recent version is available at http://www.nottub.com/forallx.shtml. Also in accordance with this license, Robert Trueman has further modi- fied Tim Button’s text, and offers forall x :York under the same Creative Commons license. The L A TEX source code is available on request from: rob.trueman@york.ac.uk Typesetting was carried out entirely in L A TEX2 ε The style for typesetting proofs is based on fitch.sty (v0.4) by Peter Selinger, University of Ottawa. Contents 1 Introduction 1 1 Intermediate Logic at York 2 2 Valid arguments 3 3 Other logical notions 5 2 Truth-functional logic 8 4 What is truth-functional logic? 9 5 Atomic sentences 10 6 Connectives 12 7 Sentences of TFL 19 8 Use and mention 25 3 Truth tables 29 9 Characteristic truth tables 30 10 Truth-functional connectives 31 11 Using truth tables 34 12 Semantic concepts 41 4 Natural deduction for TFL 48 13 The very idea of natural deduction 49 14 Basic rules for TFL 51 15 Additional rules for TFL 73 iii Contents iv 16 Proof strategies 78 17 Derived rules 79 18 Proof-theoretic concepts 84 5 First-order logic 88 19 FOL: the basics 89 20 Identity 103 21 Definite descriptions 108 22 Sentences of FOL 114 6 Interpretations 121 23 Extensionality 122 24 Truth in FOL 128 25 Semantic concepts 136 26 Using interpretations 138 27 Reasoning about all interpretations 148 7 Natural deduction for FOL 152 28 Basic rules for FOL 153 29 Conversion of quantifiers 166 30 Derived rules 169 31 Rules for identity 171 32 Proof-theoretic concepts and semantic concepts 175 Appendices 179 33 Symbolic notation 179 34 Alternative proof systems 182 35 Quick reference 187 Chapter 1 Introduction 1 Intermediate Logic at York 1 This edition of forall x is intended to be used as the textbook for the Inter- mediate Logic module at York. If you are taking this module, then you have already taken Reason & Argument . As a result, you will already be familiar with the key logical ideas, like validity and inconsistency . You will also already know how to formalise arguments and how to use truth-tables. So we won’t spend too long on any of those things — just a few short chapters to act as quick refreshers. The aim in this module is to build on what you learnt in Reason & Ar- gument One of the most important things you will learn is how to present formal proofs . This will give you a whole new way of demonstrating that an argument is valid. You will first learn how to present proofs in truth-functional logic (ch. 4), and then later you will learn how to present proofs in first-order logic (ch. 7). As well as learning how to prove things, you will also learn how to construct interpretations for first-order languages (ch. 6). These interpretations are a bit like truth-tables, but for first-order logic instead of truth-functional logic. These interpretations will let you demonstrate, in a rigorous way, that an argument written in first-order logic is invalid But that’s enough preamble. Let’s get down to work, starting with those quick refresher chapters. 2 Valid arguments 2 Logic is in the business of evaluating arguments; sorting the good arguments from the bad. By a ‘good’ argument, we mean a valid argument. An argument is valid if and only if it is impossible for all of the premises to be true and the conclusion false. Here is an example of a valid argument: Sharon studies archaeology. If Sharon studies archaeology, then she tells Rob a lot about old pots. So: Sharon tells Rob a lot about old pots. This argument is valid: if the premises are true, then the conclusion must be true as well. In fact, this argument is not just valid, it is sound as well. An argument is sound if and only if it is valid and all of its premises are true. Sharon (my wife) studies archaeology, and she does tell me a lot ( a lot! ) about old pots. Sound arguments are even better than valid arguments. If all we know is that an argument is valid, then we just know that the conclusion is true if the premises are ; but if we know that an argument is not just valid but sound as well, then we know that the conclusion is true, full stop . However, as logicians, it isn’t really our business to figure out if an argument is sound or not. Consider this argument: All mammals have hearts. Anything which has a heart has kidneys. So all mammals have kidneys. 3 2. Valid arguments 4 Logic can tell us whether this argument is valid or not. (It is.) But logic alone cannot tell us whether it is sound: if you want to figure out whether the premises are true, you need to go and do some biology. So logic is just interested in sorting the valid arguments from the invalid ones. Practice exercises At the end of some sections, there are problems that review and explore the material covered in the chapter. There is no substitute for actually working through some problems, because logic is more about a way of thinking than it is about memorising facts. A. Could there be: 1. A valid argument that has one false premise and one true premise? 2. A valid argument that has only false premises? 3. A valid argument with only false premises and a false conclusion? 4. A sound argument with a false conclusion? 5. An invalid argument that can be made valid by the addition of a new premise? 6. A valid argument that can be made invalid by the addition of a new premise? In each case: if so, give an example; if not, explain why not. Other logical notions 3 In § 2, we introduced the idea of a valid argument. In this section we will introduce some more ideas that are important in logic. 3.1 Truth values An arguments is made up out of some premises and a conclusion. Note that many kinds of English sentence cannot be used to express premises or conclu- sions. For example: • Questions , e.g. ‘Are you feeling sleepy?’ • Imperatives , e.g. ‘Wake up!’ • Exclamations , e.g. ‘Ouch!’ The common feature of these three kinds of sentence is that they are not assertoric : they cannot be true or false. It does not even make sense to ask whether a question is true (it only makes sense to ask whether the answer to a question is true). The general point is that the premises and conclusion of an argument must be capable of having a truth value . And the two truth values that concern us are just True and False. 3.2 Consistency Consider these two sentences: B1. Either Jane is 6ft tall, or Jane is 5ft11in tall. B2. Jane is not 6ft tall, and Jane is not 5ft11in tall. Logic cannot tell us which, if either, of these sentences is true. Yet we can say that if the first sentence (B1) is true, then the second sentence (B2) must 5 3. Other logical notions 6 be false. And if B2 is true, then B1 must be false. It is impossible that both sentences are true together. These sentences are inconsistent with each other. And this motivates the following definition: Sentences are jointly consistent if and only if it is possible for them all to be true together. Conversely, B1 and B2 are jointly inconsistent We can ask about the consistency of any number of sentences. For example, consider the following four sentences: G1. There are at least four giraffes at the wild animal park. G2. There are exactly seven gorillas at the wild animal park. G3. There are not more than two martians at the wild animal park. G4. Every giraffe at the wild animal park is a martian. G1 and G4 together entail that there are at least four martian giraffes at the park. This conflicts with G3, which implies that there are no more than two martian giraffes there. So the sentences G1–G4 are jointly inconsistent. They cannot all be true together. (Note that the sentences G1, G3 and G4 are jointly inconsistent all by themselves. But if sentences are already jointly inconsistent, adding an extra sentence to the mix will not make them consistent!) 3.3 Necessity and contingency In assessing arguments for validity, we care about what would be true if the premises were true. But some sentences just must be true. Consider these sentences: 1. It is raining. 2. Either it is raining here, or it is not. 3. It is both raining here and not raining here. In order to know if sentence 1 is true, you would need to look outside or check the weather channel. It might be true; it might be false. Sentence 2 is different. You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or it is not. That is a necessary truth 3. Other logical notions 7 Equally, you do not need to check the weather to determine whether or not sentence 3 is true. It must be false, simply as a matter of logic. It might be raining here and not raining across town; it might be raining now but stop raining even as you finish this sentence; but it is impossible for it to be both raining and not raining in the same place and at the same time. So, whatever the world is like, it is not both raining here and not raining here. It is a necessary falsehood Something which is capable of being true or false, but which is neither necessarily true nor necessarily false, is contingent Practice exercises A. Look back at the sentences G1–G4 in this section (about giraffes, gorillas and martians in the wild animal park), and consider each of the following: 1. G2, G3, and G4 2. G1, G3, and G4 3. G1, G2, and G4 4. G1, G2, and G3 Which are jointly consistent? Which are jointly inconsistent? B. Could there be: 1. A valid argument, the conclusion of which is necessarily false? 2. An invalid argument, the conclusion of which is necessarily true? 3. Jointly consistent sentences, one of which is necessarily false? 4. Jointly inconsistent sentences, one of which is necessarily true? In each case: if so, give an example; if not, explain why not. Chapter 2 Truth-functional logic 8 What is truth-functional logic? 4 The aim of this chapter is to introduce you to truth-functional logic (TFL). Or actually, I should say that the aim is to re- introduce you. That’s because you have already studied TFL, but under a different name: it is what you called ‘propositional logic’ in the Reason & Argument module. Consider this argument from the last chapter: Sharon studies archaeology. If Sharon studies archaeology, then she tells Rob a lot about old pots. So: Sharon tells Rob a lot about old pots. And now compare it to this argument: It is raining outside. If it is raining outside, then Simon is miserable. So: Simon is miserable. Both arguments are valid, and there is a straightforward sense in which we can say that they share a common structure. We might express the structure thus: A If A, then C So: C This looks like an excellent argument form . Indeed, surely any argument with this form will be valid. This is not the only valid form, there are lots of others too. TFL is an artificial language which allows us to symbolise many arguments in such a way as to show that they have a valid form. 9 Atomic sentences 5 TFL starts with a collection of atomic sentences . These are the basic building blocks out of of which more complex sentences are built. Every capital letter counts as an atomic sentence: A, B, C, . . . , Z Annoyingly, though, there are only twenty-six letters of the alphabet, and there is no limit to the number of atomic sentences that we might want to consider. So we will also allow ourselves to make new atomic sentences by attaching numerical subscripts to capital letters, like this: A 1 , B 17 , Q 58 , W 254 , Z 3064 , . . . We shall use atomic sentences to represent, or symbolise , certain English sentences. To do this, we provide a symbolisation key , such as the following: A : Sharon studies archaeology C : Sharon tells Rob a lot about old pots In doing this, we are not fixing this symbolisation once and for all We are just saying that, for the time being, we shall think of the atomic sentence of TFL, ‘ A ’, as symbolising the English sentence ‘It is raining outside’, and the atomic sentence of TFL, ‘ C ’, as symbolising the English sentence ‘Sharon is miserable’. Later, when we are dealing with different sentences or different arguments, we can provide a new symbolisation key; as it might be: A : It is raining outside C : Simon is miserable But it is important to understand that whatever structure an English sentence might have is lost when it is symbolised by an atomic sentence of TFL. From 10 5. Atomic sentences 11 the point of view of TFL, an atomic sentence is just a letter. It can be used to build more complex sentences, but it cannot be taken apart. Connectives 6 TFL starts with atomic sentences, but it also builds bigger, more complex sentences out of those atoms. It does this by combining sentences with con- nectives You were introduced to all of these connectives in the Reason & Argument module, but we will quickly run through all of them now. 6.1 Negation The first connective is negation. In this module, we will use ‘ ¬ ’ as our negation symbol. In Reason & Argument , you used a different symbol, ‘ ∼ ’. Both symbols are completely standard, and the issue of which to use is just a matter of notational preference. You will not be penalised if you use ‘ ∼ ’ in the exam. ‘ ¬ ’ combines with a sentence, A , like this: 1. ¬ A Sentence 1 can be read as ‘it is not the case that A ’; 1 is known as the negation of A . So if we were using the atom ‘ P ’ to symbolise ‘Sharon studies archaeology’, ‘ ¬ P ’ would symbolise ‘It is not the case that Sharon studies archaeology’. In a way, it is a bit weird to call negation a connective You expect a connective to connect two sentences together, but ‘ ¬ ’ just attaches to one sentence at a time. Because of this, some logicians prefer to call ‘ ¬ ’ an operator 6.2 Conjunction The next connective is conjunction. In this module, we will use ‘ ∧ ’ as our conjunction symbol. In Reason & Argument , you used a different symbol, ‘&’. Again, both symbols are completely standard, and you will not be penalised if you use ‘&’ in the exam. 12 6. Connectives 13 ‘ ∧ ’ combines with two sentences, A and B , like this: 2. ( A ∧ B ) Sentence 2 can be read as ‘ A and B ’; 2 is known as the conjunction of A and B , and A and B are known as the conjuncts of 2. So if we were using the atom ‘ P ’ to symbolise ‘Sharon studies archaeology’, and ‘ Q ’ to symbolise ‘Sharon tells Rob a lot about old pots’, ‘( P ∧ Q )’ would symbolise ‘Sharon studies archaeology and Sharon tells Rob a lot about old pots’. You may have forgotten why we need to put brackets around a conjunc- tion. It’s so we can tell which sentences the ‘ ∧ ’ is connecting together. (More formally, it is so we can determine the scope of the conjunction. See § 7.2 for more details.) Compare these two different sentences: 3. I didn’t finish my work and go home on time. 4. I didn’t finish my work, and I went home on time. Sentence 3 can be paraphrased as ‘It is not the case that: I finished my work and I went home on time’. Using this symbolisation key: S 1 : I finished my work. S 2 : I went home on time. We would symbolise ‘I finished my work and I went home on time’ as ‘( S 1 ∧ S 2 )’. To symbolise sentence 3, then, we simply negate the whole sentence, thus: ‘ ¬ ( S 1 ∧ S 2 )’. To symbolise sentence 4, on the other hand, we first need to negate ‘ S 1 ’, and then conjoin it to ‘ S 2 ’, giving us ‘( ¬ S 1 ∧ S 2 )’. The brackets here are essential. If we didn’t have them, we would just write ‘ ¬ S 1 ∧ S 2 ’, and it wouldn’t be clear if we were trying to symbolise 3 or 4. 6.3 Disjunction We come now to disjunction. In this module, we will use ‘ ∨ ’ as our disjunction symbol, just as you did in Reason & Argument ‘ ∨ ’ combines with two sentences, A and B , like this: 5. ( A ∨ B ) Sentence 5 can be read as ‘ A or B ’; 5 is known as the disjunction of A and B , and A and B are known as the disjuncts of 5. So if we were using the atom 6. Connectives 14 ‘ P ’ to symbolise ‘Sharon studies archaeology’, and ‘ Q ’ to symbolise ‘Sharon tells Rob a lot about old pots’, ‘( P ∨ Q )’ would symbolise ‘Sharon studies archaeology or Sharon tells Rob a lot about old pots’. It is important to mention that ‘ ∨ ’ is an inclusive disjunction. That means that ( A ∨ B ) will be true if A is true, or B is true, or A and B are both true. There is another kind of disjunction, exclusive disjunction, which is true just when A is true or B is true, but not both. However, we do not have a special symbol for that kind of disjunction in TFL. That is not a big problem: if we ever need an exclusive disjunction, we can just write ‘(( A ∨ B ) ∧ ¬ ( A ∧ B ))’. 6.4 Conditional The next connective on the list is the conditional. In this module, we will use ‘ → ’ as our conditional symbol. In Reason & Argument , you used a different symbol, ‘ ⊃ ’. Again, both symbols are completely standard, and you will not be penalised if you use ‘ ⊃ ’ in the exam. ‘ → ’ combines with two sentences, A and B , like this: 6. ( A → B ) Sentence 6 can be read as ‘If A , then B ’; A is the antecedent of 6, and B is the consequent . So if we were using the atom ‘ P ’ to symbolise ‘Sharon studies archaeology’, and ‘ Q ’ to symbolise ‘Sharon tells Rob a lot about old pots’, ‘( P → Q )’ would symbolise ‘If Sharon studies archaeology, then Sharon tells Rob a lot about old pots’. It is important to bear in mind that the connective ‘ → ’ only tells us that if the antecedent is true, then the consequent is true. It says nothing about a causal connection between two events (for example). In fact, some philosophers think that we lose a huge amount when we use ‘ → ’ to symbolise English conditionals. But that is something you already covered at length in the Reason & Argument module. 6.5 Biconditional The final connective is the biconditional. In this module, we will use ‘ ↔ ’ as our biconditional symbol. In Reason & Argument , you used a different symbol, ‘ ≡ ’. 6. Connectives 15 Again, both symbols are completely standard, and you will not be penalised if you use ‘ ≡ ’ in the exam. ‘ ↔ ’ combines with two sentences, A and B , like this: 7. ( A ↔ B ) Sentence 7 can be read as ‘ A if and only if B ’. So if we were using the atom ‘ P ’ to symbolise ‘Sharon studies archaeology’, and ‘ Q ’ to symbolise ‘Sharon tells Rob a lot about old pots’, ‘( P ↔ Q )’ would symbolise ‘Sharon studies archaeology if and only if Sharon tells Rob a lot about old pots’. Really, we don’t need a special symbol for the biconditional, just as we didn’t need a special symbol for the exclusive disjunction. We could just write ‘(( A → B ) ∧ ( B → A ))’ instead. But biconditionals come up a lot in logic, and so having a special symbol will come in handy. Biconditionals also come up a lot in philosophy, and so lots of philosophers abbreviate ‘if and only if’ as ‘iff’. We will follow this practice. So ‘if’ with only one ‘f’ is the English conditional, and ‘iff’ with two ‘f’s is the English biconditional. But a word of caution. Ordinary speakers of English often use ‘if’ when they really mean to use something more like ‘iff’. Perhaps your parents said something like this to you when you were a child: ‘If you don’t eat your greens, you won’t get any pudding!’ Suppose you ate your greens, but that your parents still refused to give you any pudding, on the grounds that they were only committed to the conditional (roughly ‘if you get pudding, then you will have eaten your greens’), rather than the biconditional (roughly, ‘you get pudding iff you eat your greens’). Well, a tantrum would rightly ensue. So, be aware of this when interpreting people; but in your own writing, make sure you use the biconditional iff you mean to. 6.6 Symbolising English sentences in TFL is hard! We’ve just run through all five connectives of TFL, and explained what En- glish connectives they are meant to symbolise. But sometimes, it isn’t always that easy to figure out how best to symbolise an English sentence in TFL. However, you have already been given lots of advice about this in your Reason & Argument module, and so we won’t go back over all of that here. If you 6. Connectives 16 want to check that your skills are still sharp, then here are some symbolisation exercises. Practice exercises A. Using the symbolisation key given, symbolise each English sentence in TFL. M : Those creatures are men in suits C : Those creatures are chimpanzees G : Those creatures are gorillas 1. Those creatures are not men in suits. 2. Those creatures are men in suits, or they are not. 3. Those creatures are either gorillas or chimpanzees. 4. Those creatures are neither gorillas nor chimpanzees. 5. If those creatures are chimpanzees, then they are neither gorillas nor men in suits. 6. Unless those creatures are men in suits, they are either chimpanzees or they are gorillas. B. Using the symbolisation key given, symbolise each English sentence in TFL. A : Mister Ace was murdered B : The butler did it C : The cook did it D : The Duchess is lying E : Mister Edge was murdered F : The murder weapon was a frying pan 1. Either Mister Ace or Mister Edge was murdered. 2. If Mister Ace was murdered, then the cook did it. 3. If Mister Edge was murdered, then the cook did not do it. 4. Either the butler did it, or the Duchess is lying. 5. The cook did it only if the Duchess is lying. 6. If the murder weapon was a frying pan, then the culprit must have been the cook. 7. If the murder weapon was not a frying pan, then the culprit was either the cook or the butler.