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BASIC CONCEPTS
- What is Logic? .................................................................................................. OF LOGIC
- Inferences and Arguments ................................................................................
- Deductive Logic versus Inductive Logic ..........................................................
- Statements versus Propositions .........................................................................
- Form versus Content .........................................................................................
- Preliminary Definitions.....................................................................................
- Form and Content in Syllogistic Logic ...........................................................
- Demonstrating Invalidity Using the Method of Counterexamples .................
- Examples of Valid Arguments in Syllogistic Logic .......................................
- Exercises for Chapter 1 ...................................................................................
- Answers to Exercises for Chapter 1 ................................................................
2 Hardegree, Symbolic Logic
1. WHAT IS LOGIC?
Logic may be defined as the science of reasoning. However, this is not to suggest that logic is an empirical (i.e., experimental or observational) science like physics, biology, or psychology. Rather, logic is a non-empirical science like mathematics. Also, in saying that logic is the science of reasoning, we do not mean that it is concerned with the actual mental (or physical) process employed by a thinking entity when it is reasoning. The investigation of the actual reasoning proc- ess falls more appropriately within the province of psychology, neurophysiology, or cybernetics.
Even if these empirical disciplines were considerably more advanced than they presently are, the most they could disclose is the exact process that goes on in a being's head when he or she (or it) is reasoning. They could not, however, tell us whether the being is reasoning correctly or incorrectly.
Distinguishing correct reasoning from incorrect reasoning is the task of logic.
2. INFERENCES AND ARGUMENTS
Reasoning is a special mental activity called inferring , what can also be called making (or performing) inferences. The following is a useful and simple definition of the word ‘infer’.
To infer is to draw conclusions from premises.
In place of word ‘premises’, you can also put: ‘data’, ‘information’, ‘facts’.
Examples of Inferences:
(1) You see smoke and infer that there is a fire.
(2) You count 19 persons in a group that originally had 20, and you infer that someone is missing.
Note carefully the difference between ‘infer’ and ‘imply’, which are sometimes confused. We infer the fire on the basis of the smoke, but we do not imply the fire. On the other hand, the smoke implies the fire, but it does not infer the fire. The word ‘infer’ is not equivalent to the word ‘imply’, nor is it equivalent to ‘insinuate’.
The reasoning process may be thought of as beginning with input (premises, data, etc.) and producing output (conclusions). In each specific case of drawing (inferring) a conclusion C from premises P 1 , P 2 , P 3 , ..., the details of the actual mental process (how the "gears" work) is not the proper concern of logic, but of psychology or neurophysiology. The proper concern of logic is whether the infer- ence of C on the basis of P 1 , P 2 , P3, ... is warranted (correct).
Inferences are made on the basis of various sorts of things – data, facts, infor- mation, states of affairs. In order to simplify the investigation of reasoning, logic
4 Hardegree, Symbolic Logic
Now let us get back to inferences and arguments. Earlier, we discussed two examples of inferences. Let us see how these can be represented as arguments. In the case of the smoke-fire inference, the corresponding argument is given as follows.
(a1) there is smoke (premise) therefore , there is fire (conclusion)
Here the argument consists of two statements, ‘there is smoke’ and ‘there is fire’. The term ‘therefore’ is not strictly speaking part of the argument; it rather serves to designate the conclusion (‘there is fire’), setting it off from the premise (‘there is smoke’). In this argument, there is just one premise.
In the case of the missing-person inference, the corresponding argument is given as follows.
(a2) there were 20 persons originally (premise) there are 19 persons currently (premise) therefore , someone is missing (conclusion)
Here the argument consists of three statements – ‘there were 20 persons originally’, ‘there are 19 persons currently’, and ‘someone is missing’. Once again, ‘therefore’ sets off the conclusion from the premises.
In principle, any collection of statements can be treated as an argument simply by designating which statement in particular is the conclusion. However, not every collection of statements is intended to be an argument. We accordingly need criteria by which to distinguish arguments from other collections of statements.
There are no hard and fast rules for telling when a collection of statements is intended to be an argument, but there are a few rules of thumb. Often an argument can be identified as such because its conclusion is marked. We have already seen one conclusion-marker – the word ‘therefore’. Besides ‘therefore’, there are other words that are commonly used to mark conclusions of arguments, including ‘consequently’, ‘hence’, ‘thus’, ‘so’, and ‘ergo’. Usually, such words indicate that what follows is the conclusion of an argument.
Other times an argument can be identified as such because its premises are marked. Words that are used for this purpose include: ‘for’, ‘because’, and ‘since’. For example, using the word ‘for’, the smoke-fire argument (a1) earlier can be rephrased as follows.
(a1′) there is fire for there is smoke
Note that in (a1′) the conclusion comes before the premise.
Other times neither the conclusion nor the premises of an argument are marked, so it is harder to tell that the collection of statements is intended to be an argument. A general rule of thumb applies in this case, as well as in previous cases.
Chapter 1: Basic Concepts 5
In an argument, the premises are intended to support (justify) the conclusion.
To state things somewhat differently, when a person (speaking or writing) advances an argument, he(she) expresses a statement he(she) believes to be true (the conclusion), and he(she) cites other statements as a reason for believing that state- ment (the premises).
3. DEDUCTIVE LOGIC VERSUS INDUCTIVE LOGIC
Let us go back to the two arguments from the previous section.
(a1) there is smoke; therefore, there is fire.
(a2) there were 20 people originally; there are 19 persons currently; therefore, someone is missing.
There is an important difference between these two inferences, which corresponds to a division of logic into two branches.
On the one hand, we know that the existence of smoke does not guarantee (ensure) the existence of fire; it only makes the existence of fire likely or probable. Thus, although inferring fire on the basis of smoke is reasonable, it is nevertheless fallible. Insofar as it is possible for there to be smoke without there being fire, we may be wrong in asserting that there is a fire.
The investigation of inferences of this sort is traditionally called inductive logic. Inductive logic investigates the process of drawing probable (likely, plausi- ble) though fallible conclusions from premises. Another way of stating this: induc- tive logic investigates arguments in which the truth of the premises makes likely the truth of the conclusion.
Inductive logic is a very difficult and intricate subject, partly because the practitioners (experts) of this discipline are not in complete agreement concerning what constitutes correct inductive reasoning.
Inductive logic is not the subject of this book. If you want to learn about inductive logic, it is probably best to take a course on probability and statistics. Inductive reasoning is often called statistical (or probabilistic) reasoning, and forms the basis of experimental science.
Inductive reasoning is important to science, but so is deductive reasoning, which is the subject of this book.
Consider argument (a2) above. In this argument, if the premises are in fact true, then the conclusion is certainly also true; or, to state things in the subjunctive mood, if the premises were true, then the conclusion would certainly also be true.
Chapter 1: Basic Concepts 7
There is another difference between statements and propositions. Whereas statements are always part of a particular language (e.g., English), propositions are not peculiar to any particular language in which they might be expressed. Thus, for example, the following are different statements in different languages, yet they all express the same proposition – namely, the whiteness of snow.
snow is white der Schnee ist weiss la neige est blanche
In this case, quite clearly different sentences may be used to express the same proposition. The opposite can also happen: the same sentence may be used in different contexts, or under different circumstances, to express different proposi- tions, to denote different states of affairs. For example, the statement ‘I am hungry’ expresses a different proposition for each person who utters it. When I utter it, the proposition expressed pertains to my stomach; when you utter it, the proposition pertains to your stomach; when the president utters it, the proposition pertains to his(her) stomach.
5. FORM VERSUS CONTENT
Although propositions (or the meanings of statements) are always lurking be- hind the scenes, logic is primarily concerned with statements. The reason is that statements are in some sense easier to point at, easier to work with; for example, we can write a statement on the blackboard and examine it. By contrast, since they are essentially abstract in nature, propositions cannot be brought into the classroom, or anywhere. Propositions are unwieldy and uncooperative. What is worse, no one quite knows exactly what they are!
There is another important reason for concentrating on statements rather than propositions. Logic analyzes and classifies arguments according to their form , as opposed to their content (this distinction will be explained later). Whereas the form of a statement is fairly easily understood, the form of a proposition is not so easily understood. Whereas it is easy to say what a statement consists of, it is not so easy to say what a proposition consists of.
A statement consists of words arranged in a particular order. Thus, the form of a statement may be analyzed in terms of the arrangement of its constituent words. To be more precise, a statement consists of terms , which include simple terms and compound terms. A simple term is just a single word together with a specific grammatical role (being a noun, or being a verb, etc.). A compound term is a string of words that act as a grammatical unit within statements. Examples of compound terms include noun phrases, such as ‘the president of the U.S.’, and predicate phrases, such as ‘is a Democrat’.
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For the purposes of logic, terms divide into two important categories – descriptive terms and logical terms. One must carefully note, however, that this distinction is not absolute. Rather, the distinction between descriptive and logical terms depends upon the level (depth) of logical analysis we are pursuing.
Let us pursue an analogy for a moment. Recall first of all that the core mean- ing of the word ‘analyze’ is to break down a complex whole into its constituent parts. In physics, matter can be broken down (analyzed) at different levels; it can be analyzed into molecules, into atoms, into elementary particles (electrons, protons, etc.); still deeper levels of analysis are available (e.g., quarks). The basic idea in breaking down matter is that in order to go deeper and deeper one needs ever increasing amounts of energy, and one needs ever increasing sophistication.
The same may be said about logic and the analysis of language. There are many levels at which we can analyze language, and the deeper levels require more logical sophistication than the shallower levels (they also require more energy on the part of the logician!)
In the present text, we consider three different levels of logical analysis. Each of these levels is given a name – Syllogistic Logic, Sentential Logic, and Predicate Logic. Whereas syllogistic logic and sentential logic represent relatively superficial (shallow) levels of logical analysis, predicate logic represents a relatively deep level of analysis. Deeper levels of analysis are available.
Each level of analysis – syllogistic logic, sentential logic, and predicate logic
- has associated with it a special class of logical terms. In the case of syllogistic logic, the logical terms include only the following: ‘all’, ‘some’, ‘no’, ‘not’, and ‘is/are’. In the case of sentential logic, the logical terms include only sentential connectives (e.g., ‘and’, ‘or’, ‘if...then’, ‘only if’). In the case of predicate logic, the logical terms include the logical terms of both syllogistic logic and sentential logic.
As noted earlier, logic analyzes and classifies arguments according to their form. The (logical) form of an argument is a function of the forms of the individual statements that constitute the argument. The logical form of a statement, in turn, is a function of the arrangement of its terms, where the logical terms are regarded as more important than the descriptive terms. Whereas the logical terms have to do with the form of a statement, the descriptive terms have to do with its content.
Note, however, that since the distinction between logical terms and descriptive terms is relative to the particular level of analysis we are pursuing, the notion of logical form is likewise relative in this way. In particular, for each of the different logics listed above, there is a corresponding notion of logical form.
The distinction between form and content is difficult to understand in the ab- stract. It is best to consider some actual examples. In a later section, we examine this distinction in the context of syllogistic logic.
As soon as we can get a clear idea about form and content, then we can discuss how to classify arguments into those that are deductively correct and those that are not deductively correct.
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The classification of a given argument is based on the answers to these two questions. In particular, we have the following definitions.
An argument is factually correct if and only if all of its premises are true.
An argument is valid if and only if its conclusion follows from its premises.
An argument is sound if and only if it is both factually correct and valid.
Basically, a factually correct argument has good content , and a valid argument has good form , and a sound argument has both good content and good form.
Note that a factually correct argument may have a false conclusion; the defini- tion only refers to the premises.
Whether an argument is valid is sometimes difficult to decide. Sometimes it is hard to know whether or not the conclusion follows from the premises. Part of the problem has to do with knowing what ‘follows from’ means. In studying logic we are attempting to understand the meaning of ‘follows from’; more importantly perhaps, we are attempting to learn how to distinguish between valid and invalid arguments.
Although logic can teach us something about validity and invalidity, it can teach us very little about factual correctness. The question of the truth or falsity of individual statements is primarily the subject matter of the sciences, broadly con- strued.
As a rough-and-ready definition of validity, the following is offered.
An argument is valid if and only if it is impossible for the conclusion to be false while the premises are all true.
An alternative definition might be helpful in understanding validity.
To say that an argument is valid is to say that if the premises were true, then the conclusion would necessarily also be true.
Chapter 1: Basic Concepts 11
These will become clearer as you read further, and as you study particular examples.
7. FORM AND CONTENT IN SYLLOGISTIC LOGIC
In order to understand more fully the notion of logical form, we will briefly examine syllogistic logic, which was invented by Aristotle (384-322 B.C.).
The arguments studied in syllogistic logic are called syllogisms (more pre- cisely, categorical syllogisms). Syllogisms have a couple of distinguishing characteristics, which make them peculiar as arguments. First of all, every syllogism has exactly two premises, whereas in general an argument can have any number of premises. Secondly, the statements that constitute a syllogism (two premises, one conclusion) come in very few models, so to speak; more precisely, all such statements have forms similar to the following statements.
(1) all Lutherans are Protestants all dogs are collies (2) some Lutherans are Republicans some dogs are cats (3) no Lutherans are Methodists no dogs are pets (4) some Lutherans are not Democrats some dogs are not mammals
In these examples, the words written in bold-face letters are descriptive terms, and the remaining words are logical terms, relative to syllogistic logic.
In syllogistic logic, the descriptive terms all refer to classes , for example, the class of cats, or the class of mammals. On the other hand, in syllogistic logic, the logical terms are all used to express relations among classes. For example, the statements on line (1) state that a certain class (Lutherans/dogs) is entirely contained in another class (Protestants/collies).
Note the following about the four pairs of statements above. In each case, the pair contains both a true statement (on the left) and a false statement (on the right). Also, in each case, the statements are about different things. Thus, we can say that the two statements differ in content. Note, however, that in each pair above, the two statements have the same form. Thus, although ‘all Lutherans are Protestants’ differs in content from ‘all dogs are collies’, these two statements have the same form.
The sentences (1)-(4) are what we call concrete sentences ; they are all actual sentences of a particular actual language (English). Concrete sentences are to be distinguished from sentence forms. Basically, a sentence form may be obtained from a concrete sentence by replacing all the descriptive terms by letters, which serve as place holders. For example, sentences (1)-(4) yield the following sentence forms.
(f1) all X are Y (f2) some X are Y (f3) no X are Y (f4) some X are not Y
Chapter 1: Basic Concepts 13
8. DEMONSTRATING INVALIDITY USING THE METHOD
OF COUNTEREXAMPLES
Earlier we discussed some of the basic ideas of logic, including the notions of validity and invalidity. In the present section, we attempt to get a better idea about these notions.
We begin by making precise definitions concerning statement forms and argu- ment forms.
A substitution instance of an argument/statement form is a concrete argument/statement that is obtained from that form by substituting appropriate descriptive terms for the letters, in such a way that each occur- rence of the same letter is replaced by the same term.
A uniform substitution instance of an argument/ statement form is a substitution instance with the additional property that distinct letters are replaced by distinct (non-equivalent) descriptive terms.
In order to understand these definitions let us look at a very simple argument form (since it has just one premise it is not a syllogistic argument form):
(F) all X are Y / some Y are Z
Now consider the following concrete arguments.
(1) all cats are dogs / some cats are cows
(2) all cats are dogs / some dogs are cats
(3) all cats are dogs / some dogs are cows
These examples are not chosen because of their intrinsic interest, but merely to illustrate the concepts of substitution instance and uniform substitution instance.
First of all, (1) is not a substitution instance of (F), and so it is not a uniform substitution instance either (why is this?). In order for (1) to be a substitution in- stance to (F), it is required that each occurrence of the same letter is replaced by the same term. This is not the case in (1): in the premise, Y is replaced by ‘dogs’, but in the conclusion, Y is replaced by ‘cats’. It is accordingly not a substitution in- stance.
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Next, (2) is a substitution instance of (F), but it is not a uniform substitution instance. There is only one letter that appears twice (or more) in (F) – namely, Y. In each occurrence, it is replaced by the same term – namely, ‘dogs’. Therefore, (2) is a substitution instance of (F). On the other hand, (2) is not a uniform substitution instance since distinct letters – namely, X and Z – are replaced by the same descriptive term – namely, ‘cats’.
Finally, (3) is a uniform substitution instance and hence a substitution in- stance, of (F). Y is the only letter that is repeated; in each occurrence, it is replaced by the same term – namely, ‘dogs’. So (3) is a substitution instance of (F). To see whether it is a uniform substitution instance, we check to see that the same descrip- tive term is not used to replace different letters. The only descriptive term that is repeated is ‘dogs’, and in each case, it replaces Y. Thus, (3) is a uniform substitu- tion instance.
The following is an argument form followed by three concrete arguments, one of which is not a substitution instance, one of which is a non-uniform substitution instance, and one of which is a uniform substitution instance, in that order.
(F) no X are Y no Y are Z / no X are Z
(1) no cats are dogs no cats are cows / no dogs are cows
(2) no cats are dogs no dogs are cats / no cats are cats
(3) no cats are dogs no dogs are cows / no cats are cows
Check to make sure you agree with this classification.
Having defined (uniform) substitution instance, we now define the notion of having the same form.
Two arguments/statements have the same form if and only if they are both uniform substitution instances of the same argument/statement form.
For example, the following arguments have the same form, because they can both be obtained from the argument form that follows as uniform substitution in- stances.
(a1) all Lutherans are Republicans some Lutherans are Democrats / some Republicans are Democrats
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FUNDAMENTAL PRINCIPLE OF LOGIC
(REWRITTEN)
If an argument is valid, then every argument with the same form is also valid.
If an argument is invalid, then every argument with the same form is also invalid.
There is one more principle that we need to add before describing the method of counterexamples. Since the principle almost doesn't need to be stated, we call it the Trivial Principle, which is stated in two forms.
THE TRIVIAL PRINCIPLE
No argument with all true premises but a false conclu- sion is valid.
If an argument has all true premises but has a false conclusion, then it is invalid.
The Trivial Principle follows from the definition of validity given earlier: an argument is valid if and only if it is impossible for the conclusion to be false while the premises are all true. Now, if the premises are all true, and the conclusion is in fact false, then it is possible for the conclusion to be false while the premises are all true. Therefore, if the premises are all true, and the conclusion is in fact false, then the argument is not valid that is, it is invalid.
Now let's put all these ideas together. Consider the following concrete argu- ment, and the corresponding argument form to its right.
(A) all cats are mammals (F) all X are Y some mammals are dogs some Y are Z / some cats are dogs / some X are Z
First notice that whereas the premises of (A) are both true, the conclusion is false. Therefore, in virtue of the Trivial Principle, argument (A) is invalid. But if (A) is invalid, then in virtue of the Fundamental Principle (rewritten), every argument with the same form as (A) is also invalid.
In other words, every argument with form (F) is invalid. For example, the following arguments are invalid.
(a2) all cats are mammals some mammals are pets / some cats are pets
(a3) all Lutherans are Protestants some Protestants are Democrats / some Lutherans are Democrats
Chapter 1: Basic Concepts 17
Notice that the premises are both true and the conclusion is true, in both arguments (a2) and (a3). Nevertheless, both these arguments are invalid.
To say that (a2) (or (a3)) is invalid is to say that the truth of the premises does not guarantee the truth of the conclusion – the premises do not support the conclu- sion. For example, it is possible for the conclusion to be false even while the prem- ises are both true. Can't we imagine a world in which all cats are mammals, some mammals are pets, but no cats are pets. Such a world could in fact be easily brought about by a dastardly dictator, who passed an edict prohibiting cats to be kept as pets. In this world, all cats are mammals (that hasn't changed!), some mammals are pets (e.g., dogs), yet no cats are pets (in virtue of the edict proclaimed by the dictator).
Thus, in argument (a2), it is possible for the conclusion to be false while the premises are both true, which is to say that (a2) is invalid.
In demonstrating that a particular argument is invalid, it may be difficult to imagine a world in which the premises are true but the conclusion is false. An easier method, which does not require one to imagine unusual worlds, is the method of counterexamples, which is based on the following definition and principle, each stated in two forms.
A. A counterexample to an argument form is any substitution instance (not necessarily uniform) of that form having true premises but a false con- clusion.
B. A counterexample to a concrete argument A is any concrete argument that
(1) has the same form as A (2) has all true premises (3) has a false conclusion
PRINCIPLE OF COUNTEREXAMPLES
A. An argument (form) is invalid if it admits a coun- terexample.
B. An argument (form) is valid only if it does not admit any counterexamples.
The Principle of Counterexamples follows our earlier principles and the definition of the term ‘counterexample’. One might reason as follows:
Chapter 1: Basic Concepts 19
PREMISES CONCLUSION VALID OR INVALID? all true true can't tell; need more info
all true false definitely invalid
not all true true can't tell; need more info
not all true false can't tell; need more info
9. EXAMPLES OF VALID ARGUMENTS IN SYLLOGISTIC
LOGIC
In the previous section, we examined a few examples of invalid arguments in syllogistic logic. In each case of an invalid argument we found a counterexample, which is an argument with the same form, having all true premises but a false con- clusion.
In the present section, we examine a few examples of valid syllogistic argu- ments (also called valid syllogisms ). At present we have no method to demonstrate that these arguments are in fact valid; this will come in later sections of this chapter.
Note carefully: if we cannot find a counterexample to an argument, it does not mean that no counterexample exists; it might simply mean that we have not looked hard enough. Failure to find a counterexample is not proof that an argument is valid.
Analogously, if I claimed “all swans are white”, you could refute me simply by finding a swan that isn't white; this swan would be a counterexample to my claim. On the other hand, if you could not find a non-white swan, I could not thereby say that my claim was proved, only that it was not disproved yet.
Thus, although we are going to examine some examples of valid syllogisms, we do not presently have a technique to prove this. For the moment, these merely serve as examples.
The following are all valid syllogistic argument forms.
(f1) all X are Y all Y are Z / all X are Z
(f2) all X are Y some X are Z / some Y are Z
(f3) all X are Z no Y are Z / no X are Y
20 Hardegree, Symbolic Logic
(f4) no X are Y some Y are Z / some Z are not X
To say that (f1)-(f4) are valid argument forms is to say that every argument obtained from them by substitution is a valid argument.
Let us examine the first argument form (f1), since it is by far the simplest to comprehend. Since (f1) is valid, every substitution instance is valid. For example the following arguments are all valid.
(1a) all cats are mammals T all mammals are vertebrates T / all cats are vertebrates T
(1b) all cats are reptiles F all reptiles are vertebrates T / all cats are vertebrates T
(1c) all cats are animals T all animals are mammals F / all cats are mammals T
(1d) all cats are reptiles F all reptiles are mammals F / all cats are mammals T
(1e) all cats are mammals T all mammals are reptiles F / all cats are reptiles F
(1f) all cats are reptiles F all reptiles are cold-blooded T / all cats are cold-blooded F
(1g) all cats are dogs F all dogs are reptiles F / all cats are reptiles F
(1h) all Martians are reptiles? all reptiles are vertebrates T / all Martians are vertebrates?
In the above examples, a number of possibilities are exemplified. It is possible for a valid argument to have all true premises and a true conclusion – (1a); it is possible for a valid argument to have some false premises and a true conclusion – (1b)-(1c); it is possible for a valid argument to have all false premises and a true conclusion – (1d); it is possible for a valid argument to have all false premises and a false conclusion – (1g).
On the other hand, it is not possible for a valid argument to have all true premises and a false conclusion – no example of this.
