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Logic: A Complete Introduction: Teach Yourself

Siu-Fan Lee

First annotation on .

33 quotes

philosophy

Chapter 1

  • Example (): It has been raining for a month now. So, it is likely to rain again tomorrow. Example (): If God exists, then there would not be any suffering. There is suffering. Therefore, God does not exist. Example (): Mr Smith must be either a monster or a strict teacher because only a monster or a strict teacher would beat children, and he beats them.Jun 3 2024 1:58AM
  • Premise-indicators: because, since, for, as, follows from, as shown by, inasmuch as, as indicated by, the reason is that, for the reason that, may be inferred/derived/deduced from, in view of the fact that…Jun 3 2024 1:36AM
  • Conclusion-indicators: therefore, so, hence, thus, in consequence, consequently, accordingly, as a result, for this reason, it proves that, it follows that, we may infer, which allow us to infer that, which shows/means/entails/implies that, which points to the conclusion that…Jun 3 2024 1:37AM
  • Sometimes an argument may have a missing premise, or premises, or even a missing conclusion. These arguments are called elliptical arguments. In such cases, one has to fill in the missing parts in order to reveal the full reasoning behind them. Doing so often involves contextual interpretation and as such controversy may arise as to whether the interpretation is correctJun 3 2024 1:49AM
  • Below is an example. Example (): Why are you still here? All students should report to the examination hall minutes before the examination begins. It represents the following argument: (Premise ) All students should report to the examination hall minutes before the examination begins. (Premise 2, missing) You are a student going to sit in an examination now. (Premise 3, missing) You are here. (Premise 4, missing) Here is not the examination hall. (Conclusion, implied) You should not be here.Jun 3 2024 1:49AM
  • Example () is also an example of a rhetorical question. Sometimes a question is asked or an exclamation is made, yet the point of the utterance is not really to enquire or to express emotion. Rather, an implicit proposition is presented, though in the form of a different mood of speech. The question ‘why are you still here?’ in () does not just ask for an answer but really implies that the speaker assumes the audience should not be here.Jun 3 2024 1:50AM
  • There are at least two main types of inferences. For example, compare the arguments in (6), (7) and (8). () operates very differently from () and () in that even if the premise of () is true, the conclusion is only probably true, rather than definitely true. In other words, the conclusion is likely to be true though it could be false even when the premises are true. We call this type of argument induction. However, for () and (8), if the premises are true, we know already whether the conclusion must be true or not. In other words, even though the premises may not actually be true, we can already decide whether the conclusion would follow or not. We can say that the truth of the premises completely determines the truth of the conclusion given the validity of the argument. This form of argument is called deduction.Jun 3 2024 1:57AM
  • Induction is commonly used in scientific inferences to generalize claims based on particular observations and experiments. Deduction is more often found in mathematics and philosophy because it promises knowledge with certainty from premises which do not need to already be known as true or false.Jun 3 2024 1:58AM
  • The main task of logic is not just to identify an argument but also to evaluate it. Two criteria are used: (i) the truth of individual statements (premises and conclusion), and (ii) the validity of the whole argument.Jun 3 2024 1:59AM
  • Logic aims at truth. Indeed, all reasoning aims at it. Truth is whatever is the case. Intelligent beings like humans want to know what is out there, what the case is and what it is not, what connections there are between things, what has happened and what could have happened. All science aims at establishing such knowledge. Logic, in particular, can be regarded as a special kind of science, in that it is interested in not only what things are, but also what they could be. Hence, although we may not know how things actually are, we do want to know what could follow if things were so and so.Jun 3 2024 2:05AM
  • Example (): (1) It is morally wrong to inflict unnecessary pain on other beings. () Some animals can feel pain. () Therefore, it is morally wrong to inflict unnecessary pain on some animals. We may not already know the truth of the premises. In example (13), the truth of () is subject to moral debate; that of () is subject to scientific investigation. However, we do know that if () and () are true, then () will follow. We can tell this result from the relation between (1), (2) and (3). Such a relation between premises and conclusions is called validityJun 3 2024 2:06AM
  • An argument usually consists of several statements, i.e. a conclusion and some premises. A    Can a statement (a premise or a conclusion) be valid or invalid? B    Likewise, can an argument be true or false? No, they can’t. A statement is true or false; yet only an argument can be valid or invalid. It is because validity is a relation between statements, rather than a property of an individual statement by itself. Validity indicates whether the conclusion follows if the premises are true. Although we sometimes hear of people making ‘valid’ statements in ordinary language, this is in fact a mistake. Similarly, only a statement can be true or false. An argument contains statements which are true or false, but the argument itself is not true or false. If an argument is valid, then its conclusion must be true when the premises are true. However, we say the argument is valid, not that it is true.Jun 3 2024 2:07AM
  • An argument is valid if and only if the conclusion follows the premises. That means, the conclusion cannot be false if all the premises are true. AnyJun 3 2024 2:08AM
  • Example (): (1) If it rains, then the ground is wet. () It rains. () Therefore, the ground is wet. The argument is valid because the conclusion follows from the premises. However, the premises do not need to be true. () does not have to be true; it is just a contingent fact of our actual world. () is false, too, if it is uttered on a sunny day. However, this does not alter the fact that if () and () are both true, then () must be true.Jun 3 2024 2:09AM
  • Contrast this with another argument: Example (): (1) If it rains, then the ground is wet. () It does not rain. () Therefore, the ground is not wet. This is an invalid argument because even if the premises () and () are true, (3) can still be false. For example, someone may splash water on the ground, or break a water bottle accidentally and so the ground becomes wet, such that even when it does not rain, the ground may still be wet. The truth of the premises thus does not guarantee the truth of the conclusion; hence the argument is invalid.Jun 3 2024 2:10AM
  • Example (): (1) Confucius was a great teacher. () There are seven days in a week. () Therefore, 2 + = 7 In (16), the premises and the conclusion are all true. The conclusion is even necessarily true because mathematical truths do not depend on contingent matters in the empirical world. However, the premises are not relevant to the conclusion. So, although the argument can be called valid, it is only vacuously valid. And it does not represent good reasoning.Jun 3 2024 2:13AM
  • Example (): (1) All spiders are insects. () All insects have eight legs. () Therefore, all spiders have eight legs. In (17), both () and () are false. Indeed, spiders are not insects and insects have six legs instead of eight. However, the argument is valid. It is because if the premises were true, then the conclusion would follow and be trueJun 3 2024 2:17AM
  • Example (): (1) All insects are spiders. () All insects have eight legs. () Therefore, all spiders have eight legs. The premises of () are false; however, the argument is invalid, too. It is because even if () and () were true, it still does not guarantee that () is true. Premise () states that all insects are spiders. If this is true, the set of spiders is supposedly larger than that of insects. This follows that even if all insects have eight legs, the set of insects does not coincide with all spiders, so there still can be spiders that are not insects and so do not have eight legsJun 3 2024 2:18AM
  • Example (): (1) All mothers are women. () All mothers have children. () Therefore, all women have children. The construction of () in contrast with that of () illustrates a good strategy in arguing called logical analogy. It is a strategy in which the reasoner provides two arguments with the same argument form, one obviously valid or invalid while the other is not so obvious. By pointing out the validity of one argument, it shows how the other, although more obscure in outlook, shares the same logical properties.Jun 3 2024 2:19AM
  • A valid argument may have: •  true premises and true conclusion •  false premises and true conclusion •  false premises and false conclusion A valid argument just can never have: •  true premises and false conclusionJun 3 2024 2:19AM
  • Example (): (1) No spiders are insects. () All insects have six legs. () Therefore, no spiders have six legs. This is an invalid argument because the premises do not state that only insects have six legs. Perhaps some other types of animals have six legs, too, and spiders are one of them. So, although the premises and the conclusion of this argument are indeed true, the premises do not guarantee the truth of the conclusion and the argument is still invalidJun 3 2024 2:23AM
  • ‘Can there be valid arguments with false premises and a false conclusion?’ Yes, below is one example. The premise () is false; however, if it is true, then the conclusion () is true. So the conclusion does follow the premise. Example (): (1) Two is greater than three. () Therefore, two squared is greater than three squared.Jun 3 2024 2:25AM
  • A better type of argument is indeed where the argument is valid and the premises are all true. We call such arguments sound arguments. They are better because validity guarantees that if the premises are true, then the conclusion must be true, and now all premises are true indeed so it follows that the conclusion of a sound argument is true, too. That is just exactly what we want: knowledge about truth with certainty.Jun 3 2024 2:30AM
  • The following are examples of a sound argument: Example (): (2) If 289 × 312 = 90,168, then 289 is a factor of 90,168. () Therefore, 289 is a factor of 90,168. Example (): (1) Every man dies. () Socrates is a man. () Therefore, Socrates dies.Jun 3 2024 2:31AM
  • However, not all sound arguments have premises that are evidentially true. We thus further distinguish sound arguments into two types: cogent and not cogent. The argument in () is sound but not cogent whereas () is sound and cogent. () is not cogent because people may not have actually multiplied 289 by 312 and found out that it equals to 90,168. So, (1) is not evidently true. Moreover, the truth may not be easily accessible to some people (e.g. those who are innumerate). In contrast, all premises in () are obvious and evident. So () is much easier to accept.Jun 3 2024 2:32AM
  • •  Logic is the study of good and bad reasoning. Reasoning is presented in arguments, which take the form of a conclusion supported by premises. •  A statement or a proposition is either true or false. •  An argument is evaluated in the following order: validity, soundness and cogency.Jun 3 2024 2:34AM
  • •  A valid argument is one which, if all premises are true, then the conclusion cannot be false. •  A sound argument is a valid argument in which all premises are true. •  A cogent argument is a strong argument whose premises are evidentially true.Jun 3 2024 2:34AM
  • Take for example the game Sudoku, which seeks to fill a × 9 grid with numbers 1–9 such that each column, row and × 3 square contains all digits with no repetition. It is a logical game but not a mathematical one because the whole game can be performed without using numerals. The game is about manipulating symbols according to certain rules and patterns, and any set of symbols can fulfil the same taskJun 3 2024 2:37AM
  • Bertrand Russell (1872–1970) pointed out in 1901 a paradox (later coined Russell’s paradox) of this project, and defeated it. This paradox concerns the idea of an empty set and its power set. An empty set is a set that has no element within it; a power set is a set made of sets. If we construct a power set containing an empty set, intuitively the empty set will become an element of itself. So the set of an empty set is not empty. Yet an empty set, by definition, should have no element. It thus seems that we do get something out of nothing.Jun 3 2024 2:39AM
  • The difficulty in assessing an argument is often found not in running the validity test, but rather in clarifying the argument, translating it into proper forms and justifying the logic systems we should use. The first two of these difficult tasks concern language and meaning. The last concerns understanding the nature of each systemJun 3 2024 2:40AM

Chapter 2

  • ‘When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean – neither more nor less.’ ‘The question is,’ said Alice, ‘whether you can make words mean so many different things.’ ‘The question is,’ said Humpty Dumpty, ‘which is to be master – that’s all.’ […] ‘When I make a word do a lot of work like that,’ said Humpty Dumpty, ‘I always pay it extra.’ Lewis Carroll (1871), Through the Looking Glass, ChapterJun 3 2024 2:41AM
  • Example (): If the Tories win, then there will be more budget cuts. The Tories will not win. Therefore, there will not be more budget cuts. Example (): If you bake a cake for too long, then you ruin it. You have not baked the cake for too long. Therefore, you have not ruined it.Jun 3 2024 2:50AM
  • Let A stand for the sentence ‘the Tories win’, B stand for ‘there will be more budget cuts’. The first argument becomes: (’) If A, then B. Not-A. Therefore, not-B. Similarly, for the second argument, let A stands for ‘you bake a cake for too long’ and B stands for ‘you will ruin it’. (’) If A, then B. Not-A. Therefore, not-B.Jun 3 2024 2:50AM

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