HL Logic: The Study of Arguments PDF

Logic: the study of arguments Erasmus School for Philosophy Erasmus Institute for Philosophy and Economics (EIPE) Last Time + This Time This course................ Conceptual Analysis & Arguments. Arguments: example.......... Arguments and Logic.......... Logic is Fantastic............ Structure of...

Logic: the study of arguments Erasmus School for Philosophy Erasmus Institute for Philosophy and Economics (EIPE) Last Time + This Time This course................ Conceptual Analysis & Arguments. Arguments: example.......... Arguments and Logic.......... Logic is Fantastic............ Structure of this lecture............................................................................................................................................................................................................................................................................................ 2 3 4 5 6 7 8 1. Arguments and standard form What is logic?............. What is an argument?........ Propositions: some remarks.... What is an argument?........ Conversion to standard form (1). Conversion to standard form (2)..................................................................................................................................................................................................................................................................................... 9 10 11 12 13 14 15..... 16 17 18 19 20 21.......... 22 23 24 25 26 27 28 29 30 31 32...... 2. Inductive, abductive and deductive “Good” arguments............. A good inductive argument........ A good abductive argument....... A good deductive, valid argument... C.S. Pierce (1839-1914).......... 3. Valid (and sound) arguments Valid arguments........... Is Musician valid?.......... Is Musician sound?......... Valid, sound?............. Is Abortion valid?.......... Is Atheism valid?........... Atheism is valid........... Reasoning about Atheism..... Is Atheism sound?.......... Is Aliens valid?.................................................... arguments....................................................................................................................................... 4. Argument forms.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. 33 1 Argument forms............. The form of a poem.......... Logical form................ The atomic propositional form... Validity of form.............. Same valid form, different content. Valid form................. Modus Ponens.............. Modus Tollens.............. Modus Tollens and the UP...... Thee valid argument forms...... Validity beyond propositional form. The predicate form of Musician.. Logical forms and validity....... Validity beyond logical form..... Logic and valid forms.......... Wrapping up.......................................................................................................................................................................................................... 2................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Last Time + This Time 2 / 50 This course Philosophy is the systematic investigation into the foundational concepts and principles of any subject matter. Two methods of philosophical investigation: 1. Conceptual Analysis Lecture 1 2. Logic / Arguments. Lecture 2 All future lectures are then devoted, in one way or the other, to: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ The Utilitarian Principle (UP) ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ P1 P2 P3 P4 P5 Consequentialism Welfarism Weak Pareto Cardinal Comparability Transitional Equity 3 / 50 Conceptual Analysis & Arguments A conceptual analysis of X is a philosophical method which seeks to formulate, test and revise principles (via thought experiments) for X in order to increase our understanding of X. We discussed, amongst others, the following knowledge principle: (i) If A knows that p, then p is true. This relation between knowledge and truth seems to be self-evident: (i) follows immediately from reflecting on the knowledge concept or, as one may say, from “intuitive observation”. In other words, hardly any argument is needed for (i). Most principles, however, are not self-evident. (think of the UP) Philosophers typically present arguments to explain why you should, or should not, accept some principle or statement. 4 / 50 3 Arguments: example In How to argue for (and against) ethical veganism (2016), McPherson presents the following argument: P1 It is wrong to make animals suffer. P2 If it is wrong to make animals suffer, then it is wrong to kill animals. P3 If it is wrong to kill animals, then it is wrong to eat meat. C It is wrong to eat meat. With respect to this argument, McPherson remarks: This argument is valid. This means that the conclusion must be true if all of the [three] premises are true. I will defend each of these premises in turn. To defend the premises, McPherson then presents further arguments. McPherson’s accessible article is an excellent example of how philosophers argue for their conclusions and how they write down these arguments in an essay. Have a look at its structure: this will benefit your philosophical essay writing skills! 5 / 50 Arguments and Logic The four branches of philosophy systematically investigate the foundational concepts and principles that are at stake when discussing: Metaphysics: what is the nature of reality? Epistemology: what can we know? Ethics: how should we act? Logic: how should we reason? So logic is both a branch and method of philosophy. One of the fundamental notions of logic is that of a valid argument. To be familiar with that notion is important for any subject. This lecture basically explains what a valid argument is. 6 / 50 4 Logic is Fantastic Once master the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real use to you in any subject you may take up. It will give you clearness of thought - the ability to see your way through a puzzle - the habit of arranging your ideas in an orderly and get-at-able form - and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Lewis Carroll (1832-1898) This lecture explains what a valid argument is and sketches how Symbolic Logic studies valid argument forms. 7 / 50 Structure of this lecture 1. Arguments and standard form. 2. Inductive, abductive and deductive arguments 3. Valid (and sound) arguments 4. Valid argument forms 8 / 50 5 1. Arguments and standard form 9 / 50 What is logic? Logic Logic is the study of arguments. What then, is an argument? An exchange of diverging or opposite views, typically a heated or angry one. “I’ve had an argument with my father” A reason or set of reasons given in support of an idea, action or theory. “There is a strong argument for submitting a formal appeal” “A good argument can be made for providing health insurance for all” Logic is concerned with the “supporting reasons sense” of arguments. 10 / 50 What is an argument? Example of an argument (Musician) All musicians can read music. John is a musician. Therefore, John reads music. Argument 1. A set of propositions. (“things that can be true of false”) 2. One of the propositions is called the conclusion, the other propositions are called the premisses. 3. The premisses are interpreted as offering reasons to believe or accept the conclusion. 11 / 50 6 Propositions: some remarks Propositions are those entities...... that... that... that... that can be true or false are expressed by declarative sentences. are the content of assertions. are the content of beliefs and other cognitive attitudes. Questions, Commands, Requests can’t be true or false. So they can’t figure as the premisses or conclusion of an argument. ‘Sneeuw is wit’ and ‘la neige est blanche’ express the same proposition. But then,... propositions are abstract entities? What is a proposition, i.e. what is its nature? That’s a heated debate in metaphysics that I won’t enter. For us, propositions are simply those things that can be true or false 12 / 50 What is an argument? Musician All musicians can read music. John is a musician. Therefore, John can read music. Musician is an argument in standard form. Argument in standard form First the premisses, then the conclusion. “Real life arguments” typically do not occur in standard form but, in order to assess them, it is convenient to convert them into standard form. 13 / 50 7 Conversion to standard form (1) Spiders Spiders have eight legs. So they cannot be insects because, if they were insects, they’d have six legs. Standard form of Spiders: P1 Spiders have eight legs. P2 If spiders were insects, they’d have six legs. C Spiders cannot be insects. To convert an argument into standard form, we rely on: Conclusion indicators: therefore, so, hence, thus, it follows that, as a result, consequently,... Premise indicators: because, since, if, from which it follows, for these reasons,... 14 / 50 Conversion to standard form (2) Cluedo Mrs. White must have committed the murder, because it was either her or Reverend Green, and if he’d done it, it would have been in the Kitchen, and it wasn’t. Standard form of Cluedo: P1 Either Mrs. White or Rev. Green must have committed the murder. P2 If Rev. Green had done it, it would have been in the kitchen. P3 The murder has not been committed in the kitchen. C Mrs. White must have committed the murder. 15 / 50 8 2. Inductive, abductive and deductive arguments 16 / 50 “Good” arguments Argument A set of premisses and a conclusion (which are propositions) with the premisses interpreted as offering reasons to believe or accept the conclusion. In the study of logic, we are interested in relationship between propositions, and the way in which some propositions can act as reasons for other propositions. (Restall, Logic (2006)) We are interested in “good” arguments, i.e. arguments in which the premisses provide a good reason to believe or accept the conclusion. I.e. arguments in which the conclusion “follows from” the premisses. 17 / 50 A good inductive argument Louise 96 % of the Flemish college students speak both Dutch and French. Louise is a Flemish college student. Hence, Louise speaks both Dutch and French. Louise’s premisses provide a (pretty) good reason to accept its conclusion. Louise is a good inductive argument: The conclusion “follows from” the premisses on the basis of frequencies / statistics / generalization Louise’s conclusion does not necessarily follow from its premisses. 18 / 50 9 A good abductive argument Fight Last week, Tim and Harry had a terrible fight that ended their friendship. But yesterday, they were seen together jogging. So, they must be friends again. Fight’s premisses provide a (pretty) good reason to accept its conclusion. Fight is a good abductive argument: The conclusion “follows from” the premisses on the basis of a plausible explanation. Fight’s conclusion does not necessarily follow from its premisses. 19 / 50 A good deductive, valid argument No Free Will If the world is deterministic, then humans have no free will. The world is deterministic. Therefore, humans have no free will. No Free Will’s premisses provide a good reason to accept its conclusion. No Free Will is a good deductive, i.e. valid argument. If the premisses are true, then the conclusion must be true as well. No Free Will’s conclusion necessarily follows from its premisses. 20 / 50 10 C.S. Pierce (1839-1914) Pierce introduced the term ‘abduction’ and the study of abductive arguments. He used the following example to explain abduction. Deductive beans All beans in the bag are white. These beans are from the bag. Hence, these beans are white. Inductive beans These beans are from the bag. These beans are white. Hence, all beans in the bag are white. Abductive beans All beans in the bag are white. These beans are white. Hence, these beans are from the bag. 21 / 50 3. Valid (and sound) arguments 22 / 50 Valid arguments A good deductive argument is called a valid argument: Valid argument (NB (1) and (2) are equivalent) (1) In every situation in which all premisses are true, the conclusion is true as well. (2) There’s no situation in which all premisses are true and the conclusion is false. (1) The conclusion necessarily follows from the premisses. (2) It’s impossible that all premisses are true while the conclusion is false. Quite a few authors would say that... Logic is the study of the (in-)validity of arguments. 23 / 50 11 Is Musician valid? Musician All musicians can read music. John is a musician. Therefore, John can read music. Yes, Musician cleary is valid: If its premisses are true, then its conclusion must be true as well. It is impossible that it’s premisses are true while its conclusion is false. 24 / 50 Is Musician sound? Musician All musicians can read music. John is a musician. Therefore, John can read music. Sound argument An argument is sound := (1) the argument is valid and (2) all its premisses are true. Jimi Hendrix did not read music so premise 1 is false! Musician is valid but not sound! 25 / 50 12 Valid, sound? Socrates Socrates died of poison or was killed in an accident. Socrates did not die of poison. Therefore, Socrates was killed in an accident Socrates is valid but not sound. Princess Diana Princess Diana was assassinated or was killed in an accident. Princess Diana was not assassinated. Therefore, Princess Diana was killed in an accident. Princess Diana is sound (and so, per definition, valid) 26 / 50 Is Abortion valid? Fred puts forward the following argument: Abortion P1 Anytime one ends the life of a person, it is murder P2 Abortion ends the life of a fetus C Therefore, abortion is murder. Abortion is, strictly speaking, invalid. P3 A fetus is a person. But adding P3 , which Fred will clearly accept, renders Abortion valid. Abortion is an enthymeme: an invalid argument with suppressed premisses that, when added, render the argument valid. Suppressed premise: implicitly accepted by a proponent of the argument. 27 / 50 13 Is Atheism valid? Atheism P1 If God exists, there is no evil in the world. P2 There is evil in the world. C God does not exist. We may show that Atheism is valid as follows: We suppose that its premisses are true and its conclusion false. We derive a contradiction from this supposition. We conclude that the supposition is impossible, hence the argument is valid. I.e., it’s impossible that its premisses are true while its conclusion false, i.e. the argument is valid. 28 / 50 Atheism is valid Atheism P1 If God exists, there is no evil in the world. P2 There is evil in the world. C God does not exist. Suppose that P1 and P2 are true and C false. 1. Then ⟨God does not exist⟩ is false C false 2. ⟨God exists⟩ is true 1, ‘not’ 3. ⟨There is no evil in the world⟩ is true 2, P1 true, ‘if then’ 4. ⟨There is evil in the world⟩ is false 3, ‘not’ 5. ⟨There is evil in the world⟩ is true P2 true Propositions can’t be true and false, so 4 and 5 contradict one another. Hence, the supposition is impossible: Atheism is valid. 29 / 50 14 Reasoning about Atheism In order to establish that Atheism is valid, we only used: Our supposition that its premises are true and conclusion false. The meaning of the logical connectives, ‘not’, ‘if then’. The fact that propositions can’t be true and false. (Classical logic) Note: we did not use the content of the propositions, i.e. the fact that they’re about God and evil did not play a role. 30 / 50 Is Atheism sound? Atheism P1 If God exists, there is no evil in the world. P2 There is evil in the world. C God does not exist. Although Atheism is valid, it is not clear whether it is sound. Logic is concerned with validity, not with soundness. Atheism is valid: there’s no situation in which the premisses are true and conclusion false. 31 / 50 15 Is Aliens valid? Aliens P1 If the universe ends tomorrow, we’ll never know if there’s alien life. P2 The universe does not end tomorrow. C We will get to know if there’s alien life. Consider the following situation: The universe ends in 7 days. In the next 7 days, we do not find out whether there’s alien life. In this situation, the premisses of Aliens are true, its conclusion false. This situation is a counterexample to Aliens’s validity: Counterexample to an argument’s validity: a situation in which the argument’s premisses are true and its conclusion false. 32 / 50 4. Argument forms 33 / 50 Argument forms We’ve seen that Musician, Socrates and Atheism are valid. As we’ll see, these (concrete) arguments are valid because they instantiate (abstract) valid argument forms. What then, is an argument form? 34 / 50 16 The form of a poem Twinkle, twinkle, little star, How I wonder what you are. Up above the world so high, Like a diamond in the sky Upon a nice mid-spring day. Let’s take a look at nature’s way. Breathe the scent of nice fresh air. Feel the breeze within your hair. The two poems have the same rhyme form: AABB. This rhyme form is obtained of a poem by abstracting from everything but the “end-sound” of it sentences. Like poems, arguments have forms (called logical form) 35 / 50 Logical form The logical form of an argument is that which remains of it when one abstracts away from the specific content of the premises and the conclusion, that is, words naming things, their properties and relations, leaving only those elements that are common to discourse and reasoning about any subject matter, that is, words such as “all”, “and”, “not”, “some” and so forth. One can represent the logical form of an argument by replacing the specific content words with letters used as place-holders or variables. Internet Encyclopedia of Philosophy A particular type of logical form is an argument’s propositional form Another type of logical form is an argument’s predicate form Related to these argument forms, one distinguishes between: Propositional logic and Predicate logic. 36 / 50 17 The atomic propositional form Obtaining the atomic propositional form of an argument One obtains the atomic propositional form of an argument by replacing atomic propositions that occur in the argument with letters, using the same letter for atomic propositions that occur more than once. The No Mammal argument and its atomic propositional form If people are mammals, then they are not cold-blooded. People are cold-blooded. If p then not-q. q. So, people are not mammals. So, not-p. E.g. people are not cold-blooded is a complex proposition: obtained by applying the logical connective ‘not’ to people are cold-blooded. Similarly, if people are mammals, then they are not cold-blooded is complex: obtained by applying ‘if then’ to two propositions. 37 / 50 Validity of form The atomic propositional form of Atheism and No Mammal: P1 P2 C If p, then not-q q So, not-p This form explains why Atheism and No Mammal are valid: Suppose that P1 and P2 are true and C false. 1. not-p is false. C false 2. p is true. 1, ‘not’ 3. not-q is true 2, P1 true, ‘if then’ 4. q is false. 3, ‘not’ 5. q is true P2 true Propositions can’t be true and false, so 4 and 5 conflict So, the supposition is impossible: the argument form is valid. 38 / 50 18 Same valid form, different content No Determinism. If everything is determined, then people are not free. People are free. So, not everything is determined. Atheism. If God exists, there is no evil in the world. There is evil in the world. God does not exist. No Determinism and Atheism have the same form as No Mammal. It follows from the previous slide that all 3 arguments are valid. 39 / 50 Valid form Socrates and its valid atomic propositional form Socrates died of poison or was killed in an accident. Socrates did not die of poison. So, Socrates was killed in an accident. p or k not-p k Socrates is valid because it instantiates a valid form: For any propositions p and k (atomic or not) that we substitute in this form, the resulting argument is valid. The (propositional) form of Socrates is called Disjunctive Syllogism: α or β, not-α. So, β. where α and β are arbitrary (not necessarily atomic) propositions. Disjunctive Syllogism is the name for a valid (propositional) argument form. 40 / 50 19 Modus Ponens Another well-known valid (propositional) argument form is Modus Ponens. Modus Ponens: If α then β, α. So, β. The Sceptic’s Argument (BIV = Brain In a Vat) P1 If I don’t know that I am not a BIV, then I don’t know that I have hands. P2 If I do not know that I am not a BIV. C I do not know that I have hands. The Sceptic’s Argument has Modus Ponens form: α: I do not know that I am not a BIV. β: I do not know that I have hands. The Sceptic’s Argument is valid in virtue of its Modus Ponens form. Thus, to explain the validity of the Sceptic’s Argument, we do not need to resort to its atomic propositional form. 41 / 50 Modus Tollens Another well-known valid (propositional) argument form is Modus Tollens. Modus Tollens: If α then β, not-β. So, not-α. Atheism, No Mammal and No Determinism all have Modus Tollens form. In A defence of common sense (1925), G.E. Moore rebuts the Sceptic’s Argument: Moore’s Argument (BIV = Brain In a Vat) P1 If I don’t know that I am not a BIV, then I don’t know that I have hands. P2 I do not not know (i.e. I know) that I am not a BIV. C I do know that I have hands. Moore’s argument has Modus Tollens form: α: I do not know that I am not a BIV. β: I do not know that I have hands. This illustrates the saying that: One philosopher’s modus ponens is another philosopher’s modus tollens. 42 / 50 20 Modus Tollens and the UP You are familiar with Modus Tollens reasoning from the previous lecture: Five persons need a heart, kidney, lung, liver and brain transplant respectively. Without the transplant they die. Some other person, call him David, is perfectly healthy. All six persons are strangers to you. By harvesting David’s organs, you can save the life of 5 persons. What should you do? An argument against the Utilitarian Principle: P1 If the UP is true, then we should harvest David’s organs. P2 We should not harvest David’s organ. C So, the UP is not true. This argument against the UP is an instantiation of Modus Tollens: Modus Tollens: If α then β, not-β. So, not-α. 43 / 50 Thee valid argument forms We have seen three examples of valid argument forms: Disjunctive Syllogism: α or β, not-α. So, β. Modus Ponens: If α then β, α. So, β. Modus Tollens: If α then β, not-β. So, not-α. These three forms are all examples of propositional argument forms: One obtains a propositional form of an argument by replacing propositions that occur in the argument with letters, using the same letter for propositions that occur more than once. As we will now illustrate: Arguments can be valid in virtue of having a valid argument form, but this valid form need not be a propositional one. To do so, let us revisit Musician. 44 / 50 21 Validity beyond propositional form Musician is a valid argument All musicians can read music. John is a musician. So, John can read music. The propositional form of Musician: p, q. So, r. Remember, one obtains the propositional form of an argument by replacing propositions that occur in the argument with letters, using the same letter for propositions that occur more than once. So Musician is valid, but does not instantiate a valid propositional form. To see this, here is an invalid argument with the same propositional form: Fikir Amlak is not a valid argument Fikir Amlak is the greatest musician alive. Grass is green. So, 2 + 2 = 5 45 / 50 The predicate form of Musician Musician All musicians can read music. John is a musician. So, John reads music. The valid predicate form of Musician For every object x: if x is M then x is R Object j is M So, object j is R. Musician is valid because it instantiates a valid predicate form: For every object j and properties M , R, that we substitute in this form, the resulting argument is valid. Predicate logic captures reasoning about objects, properties and relations. Predicate logic does not abstract away from e.g. “all” and “some”. 46 / 50 22 Logical forms and validity So, the logical form of an argument is that which remains of it when one abstracts away from the specific content of the premises and the conclusion leaving only logical constants, i.e. those elements that are common to discourse and reasoning about any subject, such as: 1. and, not, or, if then, if and only if. 2. and, not, or, if then, if and only if, all, some. 3. and, not, or, if then, if and only if, all, some, necessarily, possibly. The (in)validity of these different logical forms are studied by, respectively: 1. Propositional logic 2. Predicate logic 3. Modal logic That there are different logics associated with different argument forms illustrates that validity is in general determined by logical form. Let us now illustrate this in general qualification. 47 / 50 Validity beyond logical form Car My car is red. Therefore, my car is coloured. Car is valid: it’s impossible that its premise is true and its conclusion false. But the propositional / predicate / modal form of Car are all invalid. More generally, the validity of Car depends on the relation between ‘coloured’ and ‘red’ , neither of which are logical constants i.e. neither of which are “elements that are common to discourse and reasoning about any subject” So Car is valid but not in virtue of its logical form. 48 / 50 23 Logic and valid forms Since none of the predicates [... ] ‘is red’ and ‘is coloured’ would be counted among the logical vocabulary of standard systems of logic, the necessary preservation of truth between the premisses and the conclusion of these arguments cannot be traced to the behaviour of any logical constant, and the arguments in question are therefore classified as invalid by standard systems of logic. Koslicki (2018). The Structure of Objects. Broadly conceived, logic is that branch of philosophy which systematically investigates the foundational concepts and principles that are at stake when discussing the question: how should we reason? Narrowly conceived: Logic is the study of the (in-)validity of argument forms. 49 / 50 Wrapping up P1 It is wrong to make animals suffer. P2 If it is wrong to make animals suffer, then it is wrong to kill animals. P3 If it is wrong to kill animals, then it is wrong to eat meat. C It is wrong to eat meat. The atomic propositional form of McPherson’s argument: s, if s then k, if k then e. Therefore, e. McPherson’s argument is valid in virtue of its form. This means that the conclusion must be true if all of the premises are true. The validity of McPherson’s argument also means that if you disagree with his conclusion, you should explain which of his premisses is false and why it is false: you should give an argument against (at least) one of his premisses. 50 / 50 24