John is at the movies. The pattern of both arguments is "Either p or q, not q, therefore p," where the lowercase letters p and q stand for simple sentences, such as "The cat has eaten the mouse" and "John is at home. In sentential logic, complete simple sentences such as "John is at home" are taken as unbroken units and are not further analyzed into their component parts. The logical structure is a matter of how these simple sentences are combined with certain logical words, such as "and," "or," and "not," into compound sentences such as "Either John is not at home or the doorbell is broken and the phone is out of order.
The following, for instance, all have the form "either p or not q," where p and q, again, represent complete sentences. Either Andrew starts studying or he won't graduate. Either it snows or we won't go skiing. Either my car starts or I won't get to class on time.
Either you finish your homework or you can't go out. Either Henry campaigns or he won't get elected. Each of the specific sentences is considered an instance of the sentence form. Similarly, the specific arguments 3 and 4 are both instances of the argument form "Either p or q, not q, therefore p.
However, the idea is the same: the form of a sentence is the way in which certain specified logical words are combined with the other elements of the sentence. It should be clear that the following two arguments in predicate logic have the same form, although the form here contains the predicate logic words "all" and "some.
All U. Some U. Some males are generals. All snakes are reptiles. Some snakes are poisonous. Some reptiles are poisonous. Some B's are C's," where A, B, and C represent class terms, common nouns that serve as subjects and predicates of the sentences.
In order to see how form determines validity, we must be very clear about the distinction between aform and its instances. Theform, again, is the general pattern, or structure, which abstracts from all specific subject matter, whereas the instances are the particular meaningful examples that exhibit that form. In sentential logic, an instance is obtained from a form by substituting meaningful sentences consistently for the p's and q's, whereas in predicate logic instances are obtained by substituting class terms for the capital letters A, B, and C.
Having made the distinction between form and instance, we can now be a little more precise in our definition of validity. We will need to distinguish between the validity of an argument instance and the validity of a form, and the former definition will depend on the latter. An argument particular instance will be said to be valid if and only if it is an instance of, or has, a valid form. An argument form will be valid if and only if there are no instances of that form in which all the premises are true and the conclusion is false.
A form will be invalid, therefore, just in case there is an instance of that form with all true premises and a false conclusion. Given these definitions, we are now in a position to explain how it is possible to demonstrate that an argument form is invalid. Consider the following two arguments from sentential logic, which have the same form as argument 2.
If coal is black, then snow is white. Snow is white. Coal is black. If the author of this book is a monkey, then the author of this book is a mammal. The author of this book is a mammal. The author of this book is a monkey. Again, both arguments have the same form; in this case, "If p then q, q, therefore p," where p and q stand for the simple sentences, such as "Snow is white. You will have to take the author's word for this. Such an invalidating instance of a form, an instance with all true premises but a false conclusion, will be called a counterexample to that form, and any form that has a counterexample will be invalid.
Argument 9 below, from predicate logic, is also invalid. We can see that it is invalid by extracting its form and then finding class terms for A, B, and C that give us true premises with a false conclusion, as in argument A seemingly minor change in form can mean the difference between validity and invalidity. Note also that it is not obvious at first glance whether an argument is valid or invalid; argument 9 initially looks quite plausible. All cats are fur-bearing mammals. Some fur-bearing mammals are black.
Some cats are black. Some males have been on the moon. An argument form will be valid, as noted, if and only if it has no counterexample, that is, no instance with true premises and a false conclusion. We will not be able to show that argument forms are valid until we have the means of systematically examining all possible instances; you will learn how to do this in Unit 5, and a more detailed explanation of validity will have to be deferred until we reach that unit.
Truth and Validity The single most important moral of the above story is that the validity of a particular argument the instance depends on its form; again, a particular argument will be valid if and only if its form is valid.
Since form has nothing to do with subject matter, it follows that what an argument says, its content, is irrelevant to its validity. Silly-sounding arguments may be valid, and weighty-sounding arguments may be invalid. Furthermore, the truth or falsity of the specific premises and conclusion is also irrelevant to the validity of an argument, with one exception: a valid argument may never have true premises and a false conclusion.
This would, by definition, be a counterexample and thus would make the argument form, and so the argument, invalid. All other combinations are possible. Some examples of valid and invalid arguments, with various combinations, are given below. Arguments may, of course, have more or fewer than two premises; the ones below all have two for the sake of uniform comparison.
Note that some are from sentential logic and some are from predicate logic. All cats are green. All green things are immortal. All cats are immortal.
All cats are reptiles. All reptiles have fur. All cats have fur. Either Texas or Oregon is on the West Coast. Oregon is not on the West Coast. Texas is on the West Coast. If dogs don't bark, then cats meow. Cats do not meow. Dogs bark. Some Republicans are wealthy. Some Democrats are not wealthy.
Some Democrats are not Republicans. No Democrats are wealthy people. No wealthy people have visited Mars. If that tree with acorns is a maple, then it's deciduous. That tree with acorns is deciduous. That tree with acorns is a maple.
Either dogs bark or cats bark. Cats bark. No Democrats have visited Mars. T T It may seem a bit strange to talk about valid arguments with false premises and false conclusions, but in fact we very often want to see what follows-what can be deduced from-false premises.
A scientist may try to figure out, for instance, what would happen if there were a meltdown in a nuclear reactor, although we certainly hope that won't happen. More optimistically, an economist may try to deduce the economic consequences of a breakthrough in solar energy technology, although unfortunately none has yet occurred.
Sometimes we simply do not know whether the premises are true or not, as when we are testing some unknown chemical compound or a new engineering design. The use of logic allows us to figure out ahead of time what would happen if certain hypotheses or premises were true, and this is obviously essential in all kinds of situations. Think how expensive it would be if, instead of sitting down at a computer and simply deducing the flight characteristics of certain designs, an aeronautical engineer had to actually build an example of each new design he or she thought up and test it out!
Of course, some valid arguments do have true premises, and a special term is used for these cases. Such arguments, with true premises and therefore also with true conclusions , are said to be sound. A sound argument is "one up" on a valid argument, since in addition to being valid it also has true premises and a true conclusion. In general, however, logicians are not concerned with the truth of the premises of particular arguments; their job is to determine validity.
This, again, is not a matter of the truth or falsity of premises and conclusion, but of the connection 14 Unit 1 Introduction to Logic between them.
Logic is concerned solely with whether the conclusionfollowsfrom the premises, and this, as we have seen, is a matter of the form, rather than of the truth, falsity, or content of an argument. The Nature of Symbolic Logic Detailed systems of logic have existed at least since the time of Aristotle to B. The advantages of using symbols in logic are the same as in mathematics; symbols are easier to manipulate, they provide an economical shorthand, and they allow us to see at a glance the overall structure of a sentence.
By using symbols, we are able to deal with much more complicated arguments and thus take logic much further than we otherwise could. In fact, since the development of symbolic logic there has been an explosion of knowledge in this area and fascinating developments that would not have been possible without it. In some of the extra-credit sections you will be introduced to some of these more advanced topics. Beginning students, especially those who have had trouble with mathematics, sometimes worry about whether they will be able to cope with a course in symbolic logic.
They may be convinced that they are "just no good with symbols" or that they simply can't understand formal material. These fears are unfounded. For one thing, the material in this book presupposes no previous acquaintance with symbol systems; it builds from the ground up, so no student is at a disadvantage. Furthermore, the kinds of symbols we will be using are quite simple and are really just abbreviations for common terms in English, such as "and," "or," and "not. Some students may feel that logic is unnecessary or is just an unwelcome constraint on the free flow of their imaginations.
But without logic, without the ability to reason correctly, we would not survive for long; whether we recognize it or not, logic is fundamental to our very existence.
Furthermore, logic itself, like mathematics, can be a highly creative enterprise and as you will see when you come to proofs can require the exercise of considerable ingenuity. Thus, it can actually enhance your creative powers, rather than serve as just an intellectual fetter. In any case, the use of symbols should facilitate rather than impede your understanding of logic. In this book we use two sets of symbols, corresponding to the kinds of logic we will be investigating.
For the first half of the book, which covers sententiallogic, we use an extremely easy symbolic system consisting only of 1 single letters to stand for simple sentences, such as "Jane is blond," 2 special symbols for the five logical terms "and," "or," "not," "if-then," and "if and only Unit 1 Introduction to Logic 15 if," and 3 grouping symbols such as parentheses.
We will be concerned only with how simple sentences are compounded into more complex sentences by using the five logical terms, and we will not attempt an internal analysis of the simple sentences themselves.
In the second half of the book, which covers predicate logic, we go a bit deeper into sentence structure and analyze the simple sentences into their component parts, such as subject and predicate.
For this we need a few additional symbols, but the symbolism will remain surprisingly simple. Even students who have "hang-ups" about symbols need not worry about symbolic logic; the symbols are quickly grasped, and they are simply a way of making logic much easier than it would otherwise be.
There are some limitations to the use of symbols. Since we have only five symbols to represent ways of compounding sentences, and English is an extremely subtle and complex language, something is bound to be lost in translation.
We are going to have to squeeze all the richness of English into just a few symbols, and this will mean, unfortunately, violating the full sense of the sentence on many occasions. What logic lacks in subtlety, however, it more than makes up for in clarity, and you will probably appreciate the ease of manipulating the logical system.
Just don't look for poetry, or you will be disappointed. The Scope of Symbolic logic a. Levels of Logical Structure. As noted earlier, there are various kinds of logic, depending on how deeply we analyze the structure of the English sentences. At the simplest level, sentential or propositional logic, we analyze only how complete sentences are compounded with others by means of logical words such as "and," "or," and "not," which are called sentential operators. We do not attempt an internal analysis of simple sentences in terms of their grammatical parts such as subjects and predicates.
In this book, sentential logic is discussed in Units 2 through 9. In one-variable predicate logic, presented in Units 10 through 16, our logical analysis goes deeper, into the internal structure of the simple sentences, and instead of single letters to represent complete sentences, we have various symbols to represent the various parts of a sentence.
We have individual terms to represent single individual objects, predicate letters to represent predicates roughly, what is said about an individual object , and two new logical words, the very important quantifiers "all" and "some. Examples of such predicates are "is 16 Unit 1 Introduction to Logic taller than" and "lives next to," which are two-place predicates used to state a relation between two things, and "between," which is a three-place predicate used to state a relation between three things.
In relational predicate logic, the third level we discuss, we simply add these relational predicates to the already existing machinery of one-variable predicate logic. Thus, relational predicate logic is simply an extension of one-variable predicate logic. Relational predicate logic is the topic of Units 17 and Finally, the fourth and last level we discuss is relational predicate logic with identity, and here we just add one more element: the very important logical relation of identity.
Relational logic with identity is covered in Units 19 and Four Kinds of Logical Investigation. For each of the four levels of logic described above, there are four kinds of inquiry that must be undertaken. We must first ask what is the formal structure of the logical language; such an investigation might be called the logical grammar of the system.
Here we must make very clear exactly which symbols are to be used and how they fit together properly into meaningful formulas. The grammar of sentential logic is discussed in Unit 2, and the grammars of one-variable predicate logic, relational predicate logic, and predicate logic with identity are explored, somewhat diffusely, in Units 10 through 12, 17, and 19, respectively. Once we know the structure of the symbolic system, we must see how it is reflected in ordinary English sentences and arguments.
We might call this the application of the logical system, and it includes being able to put the English sentences into symbolic form, as well as being able to read off from the symbolic form the appropriate English sentence. For this you will always be provided with a "dictionary" that links simple symbolic elements with their English counterparts.
In Unit 4, we undertake a careful analysis of the application of sentential logic to English, and in Units 10 through 14 and 17 through 20, we cover various aspects of the application of predicate logic to English. The third kind of inquiry we must undertake for any branch of logic is its semantics.
Here, as the title indicates, we explain exactly what our logical words such as "and," "not," and "all" mean. These meanings are given by stating precisely the conditions under which sentences containing them will be true or false; we give the logical meaning of "and," for instance, by saying that a conjunction "p and q" is to be true if and only if each of the components p and q is true. There are a great many other logical concepts that can be explained in semantic terms, that is, in terms of their truth conditions.
We have already seen that validity is defined in terms of the possible truth combinations for premises and conclusion: an argument form is valid if and only if there is no possible instance of the form that has all the premises true with the conclusion false.
Other semantic concepts are consistency, Unit 1 Introduction to Logic 17 equivalence, and contingency. We examine the semantics for sentential logic in considerable detail in Units 3, 5, and 6 and discuss the semantics for predicate logic in Unit Finally, the fourth part of a study of any branch of logic is its proof methods.
Here we set out formal rules for deriving, that is, proving, certain symbolic formulas from others. It is interesting that this procedure is theoretically independent of any semantics-we can learn to do proofs of formulas without even knowing what they mean. The proof methods for sentential logic are developed in Units 7, 8, and 9; predicate logic proof methods are discussed in Units 15, 18, and In Unit 9, we discuss the relationship between proof methods and semantics for the various branches of logic.
The material covered in Units 1 through the grammars, applications, semantics, and proof methods of the four branches of logic-forms the solid core of symbolic logic.
As noted earlier, there are many extensions of and alternatives to this basic logic, which you might study in a more advanced course. An argument is a set of sentences consisting of one or more premises, which contain the evidence, and a conclusion, which should follow from the premises.
A deductive argument is an argument in which the premises are intended to provide absolute support for the conclusion. An inductive argument is an argument in which the premises are intended to provide some degree of support for the conclusion. An argument particular instance is valid if and only if it is an instance of, or has, a valid form. An argument form is valid if and only if there are no instances of that form in which all premises are true and the conclusion is false.
A counterexample to an argument form is an instance of that form a particular example in which all the premises are true and the conclusion is false. A sound argument is a valid deductive argument in which all the premises are true. What is the advantage of thinking logically? What are the two ways in which the study of logic can improve your reasoning ability? What theoretical aspect of logic can you expect to learn in this course?
Give a brief statement of what logic is about. What is a valid deductive argument? What is the difference between deductive and inductive arguments? Give an example of two different arguments with the same form. How is the form of an argument related to its validity or invalidity?
Can a valid argument have false premises with a true conclusion? What other combinations are possible? Give an example of your own of an invalid argument with true premises and a true conclusion. Give an example of your own of a valid argument with false premises and a false conclusion. What are the advantages of using symbols in logic? What is one disadvantage of using symbols in logic?
What are the four branches or levels of logic, and what are the fundamental differences among them? What are the four areas of investigation for a branch of logic? Describe each briefly. For each of the following, 1 determine whether it is an argument. If it is an argument, then 2 identify the premises and conclusion, and 3 indicate whether it is inductive or deductive.
That eight-foot crocodile looks ferocious, with its huge teeth and mean eyes. That crocodile hasn't eaten for two weeks, so it is probably hungry. Since crocodiles are reptiles, and that crocodile is a man-eater, some reptiles are man-eaters. Anyone who goes into the cage of a hungry crocodile is extremely foolish. Although many people think that global warming is caused by humans, it is really just a part of Earth's natural cycle.
Atmospheric carbon dioxide has increased substantially since , and it is known to have a "greenhouse" effect, so it is likely that global warming is the result of burning fossil fuels. It was so hot yesterday that I couldn't work in the garden, but it is supposed to be cooler today. Anyone who either insults his or her boss or can't use email properly deserves to be fired. I can't use email properly, so I deserve to be fired. If I live close to school I'll pay a lot in rent, and if I don't live close to school I'll pay a lot for gas, so I'll pay a lot for either rent or gas.
If I take up smoking, my health insurance rates will go up and I'll also get sick. My car insurance rates will probably go up, since I got a speeding ticket and was thrown in jail for evading an officer. The forest fire was caused by arson, since it could only have been caused by arson or lightning, and it definitely wasn't caused by lightning. Decide whether each of the arguments below is valid or invalid. If you think it is invalid, try to give a counterexample.
After you have completed the assignment and only after , check the answer section to see which of these are valid. Rather, these are designed to demonstrate that there is a real need for the systematic study of logic. Our intuitions are often wrong! Either Clinton or Dole was president in Dole was not president in Therefore, Clinton was president in Not both Clinton and Dole were president in Not both Clinton and Gore were president in Clinton was not president in Therefore, Gore was not president in If Tinker is a male cat, then Tinker will not have kittens.
Tinker will not have kittens. Therefore, Tinker is a male cat. Tinker will have kittens. Therefore, Tinker is not a male cat. Tinker is not a male cat. Therefore, Tinker will have kittens. If Tinker is either a cat or a dog, then Tinker is a fur-bearing mammal. Therefore, if Tinker is a cat, then Tinker is a fur-bearing mammal. If I neither diet nor exercise, I will gain weight. Therefore, if I do not diet, I will gain weight.
If I exercise, I will neither gain weight nor lose muscle tone. Therefore, if I exercise, I will not lose muscle tone. If I don't exercise, I will gain weight and lose muscle tone. Therefore, if I don't gain weight, I have exercised.
Some wealthy people are not Democrats. Therefore, some Republicans are not Democrats. No man has experienced childbirth. Therefore, no U. Therefore, no man has experienced childbirth. All great works of art are controversial. Andy Warhol's creations are controversial. Therefore, Andy Warhol's creations are great works of art.
No cats are dogs. No dogs are horses. Therefore, no cats are horses. Not all corporate executives are men. All corporate executives are wealthy people.
Therefore, some wealthy people are not corporate executives. Therefore, some wealthy people are not men. All frogs are reptiles. Therefore, some frogs are poisonous. Some preachers are wealthy people. All wealthy people will get to heaven. Therefore, some preachers will get to heaven. Some wealthy people will get to heaven. No one who gets to heaven has committed a mortal sin. Anyone who commits murder has committed a mortal sin.
Therefore, some murderers are not wealthy. In this unit you will learn the basic structure, or grammar, of one branch of symbolic logic, what is generally called sentential, or propositional, logic. The names derive from the fact that in this part of logic our most basic or elementary unit is the complete sentence, such as "Jane is blond" or "Swans are graceful. It is important to note that although there are such studies as the logic of questions and the logic of commands, we will be confining ourselves to the logic of declarative sentences, those that are definitely either true or false, such as "Salamanders are mammals" which is, of course, false.
Whenever the word "sentence" is used hereafter, it should be understood that we are referring to declarative sentences. More will be said about this in Unit 5, when we begin truth tables. Any expression that is used to build up a compound sentence out of simpler sentences will be called a sentential operator, because it "operates" on sentences to produce more complex forms.
Although there are potentially an infinite number of ways of forming compound sentences in English-that is, an infinite number of sentential operators-we will be using only five. These five operators, for which 21 22 Unit 2 The Structure of Sentential Logic we will have special symbols, are "and," "or," "not," "if-then," and "if and only if. Our discussion includes the difference between simple and compound sentences, the definition of "sentential operator," the five sentential operators we will be using, and how compound sentences are constructed from their components by the use of our five operators.
What you will need to learn is stated somewhat more explicitly in the "Objectives" section below. Simple and Compound Sentences It is absolutely essential in logic to be able to analyze the structure of sentences and arguments, since validity is a matter of form, or structure. Of course, there are various levels of structural analysis, but for the first half of the book, we are going to do only the most elementary sort of analysis. We are going to be concerned only with how complete sentences are compounded with others by words such as "and" and "or" into more complex forms.
We will not analyze sentences into their "inner" elements, such as subjects and predicates this will come later, in predicate logic , but will take them as unbroken wholes, our smallest units. The first step in analyzing sentential structure is to be able to distinguish between simple sentences and compound sentences and to be able to identify the simple components of compound sentences.
We begin by defining "compound sentence" and then define a simple sentence as one that is not compound. Unit 2 The Structure of Sentential Logic 23 A declarative sentence will be considered to be compound if it contains another complete declarative sentence as a component. The sentence "The person who ate the cake has a guilty conscience," on the other hand, is not compound because it has no sentential components. The phrase "who ate the cake" cannot be a component since it is not a complete declarative sentence.
What does it mean, then, for one sentence to be a component of another? This notion is closely related to the concept of a sentential operator, which will be defined in the next section. For now, we can say that one sentence occurs as a component of a second if, whenever the first sentence is replaced by another declarative sentence, the result is still a grammatical sentence.
Thus, in the sentence "John believes that Mary loves Bill," we can replace "Mary loves Bill" with any other declarative sentence and still have a grammatical sentence. It is not possible, on the other hand, to replace "who ate the cake" in the sentence "The person who ate the cake has a guilty conscience" with any arbitrary sentence.
A few other examples of proper compound sentences are given next, with their simple components italicized. Some examples of simple sentences are "John is going to New York," "Mary is a good student," "Manx cats are friendly," and "Dolphins are highly intelligent. The subject may be modified in various ways, and the predicate may say something rather intricate, requiring lengthy phrases, but as long as no other complete sentence appears as a component, the sentence is still logically simple.
The following is an example of a rather lengthy but simple sentence: "The odd-looking person standing to the right of the woman with the weird hat with the flowers and cherries on it is the one who infuriated the chairman of the board by complaining in public about the high prices ofthe company's inferior products. The criterion of a simple sentence is decidedly not its length.
It is not always such an easy matter, however, to determine whether a sentence is simple or compound. To clarify the concept of a compound sentence, we need to be a little more precise about, and extend our notion of, what it means for one sentence to contain another as a component. In English, if the predicates of the components of a compound sentence are the same, as in "John went to New York and Mary went to New York" where the predicates "went to New York" are identical , we tend to compress, or condense, the compound by using a compound subject, rather than repeating the predicate for each subject.
Thus, in place of the sentence above, we would probably use the more graceful form "John and Mary went to New York," with the compound subject "John and Mary. Examples of compound predicates are "is lucky or intelligent" and "loves puppies and kittens. An example of such a sentence would be "The Democratic presidential and vice-presidential candidates will either be defeated or will win by a narrow margin.
The exceptions will be noted below. Thus, our previous sentence "John and Mary like fish" is compound because it can be paraphrased into the longer version "John likes fish and Mary likes fish," which explicitly contains Unit 2 The Structure of Sentential Logic 25 the two clauses "John likes fish" and "Mary likes fish. We may now define a compound sentence as one that logically contains another sentence as a component. The sentence "Kennedy and Mondale are Democrats and liberals," for instance, is compound and logically contains the following as components: "Kennedy is a Democrat," "Mondale is a Democrat," "Kennedy is a liberal," and "Mondale is a liberal.
In some cases you may not be able to tell whether the sentence is genuinely compound or is just stating a relationship between two individuals. The ambiguous sentence "John and Mary are married," for instance, might mean simply that they both have spouses, in which case it would be a compound sentence, or it might mean that they are married to each other, in which case it would be stating a relationship between them and would not be a compound sentence.
This demonstrates that the art of paraphrase is not exact. Logic, Symbolic and mathematical. K dc22 Copyright , , , , by Pearson Education, Inc. Pearson Prentice Hall. All rights reserved. Printed in the United States of America. This publication is protected by Copyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise.
For information regarding permission s , write to: Rights and Permissions Department. Pearson Education LTD. Pearson Education Malaysia, Pte. Why Study Logic? Conditional Proof 2. Indirect Proof 3. Discharging Assumptions; Restrictions on C. Using c. Proofs of Theorems 6. Invalidity 7. Rules from Q. If you can't read please download the document. Post on Sep 7. Category: Documents download. Dilemma Dil. Would you like to tell us about a lower price?
Philippe Beauchamp rated it really liked it Apr 15, This comprehensive introduction to symbolic logic is designed for students with no prior background in logic, philosophy, or mathematics. This website uses cookies to improve your experience while you navigate through the website.
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