2. Arguments, Validity, and Truth

        Let us consider some examples to illustrate the general  discussion above.

                   (1) All men are bipeds.

            A    (2) Edgar is a man.

 (3) (Therefore) Edgar is a biped.

is an argument in which line A(3) is the conclusion that  is indicated to follow from lines A(1) and A(2). If we  know A(1) and A(2), we can deduce that A(3) is true.  Lines A(1) and A(2) are called premises; line A(3) is  called the conclusion. Here the fact that a conclusion is  sometimes, but not always, marked by a word such as  ëthereforeí or ësoí is shown by enclosing the  indicator in parentheses.

        Consider the following three sentences:

                 (1) All women are bipeds.

         B      (2) Helen is a  woman.

                 (3) (Therefore) Rover is a biped.

        Here, too, we have an argument, which bears some resem blance to A in form. But this time we notice something  strange. The purported conclusion, line B(3), does not  follow from the premises at alL Even if B(1) and B(2)  are true, B(3) is not thereby guaranteed to be true. So  B, like A, is an argument; but B, unlike A, is not a  good argument Its conclusion does not follow from its  premises, and we call such arguments invalid.

        I may know that no one in San Francisco is seven feet tall  and that Jean Jones is a five-foot New Yorker. Still, I can  assert that if it were true that all San Franciscans are  seven feet tall and that Jean Jones lives in San Francisco,  then it would be true that Jean Jones is seven feet talL  My argument would look like this:

                 (1) All San Franciscans are seven feet talL

        C       (2) Jean Jones is a San Franciscan.

                 (3) (Therefore) Jean Jones is seven feet talL

Argument C, like argument A, is valid; that is, both  arguments are such that it must be that if the premises are  true, the conclusion is true. The conclusion of a valid  argument is a logical consequence of its premises, and the premises  are said to imply or to entail the conclusion. But C,  unlike A, has false premises. Thus C is a valid  argument, but it is not sound. Argument C illustrates  the fact that a sentence can be the conclusion of a valid  argument and still be false. For, to say that a sentence is  the conclusion of a valid argument is to say only that its  truth is guaranteed ~fthe premises of the argument are  true. But consider D:

                 (1) All men are mortaL

        D      (2) Socrates is a  man.

                 (3) (Therefore) Jean Jones is seven feet tall.

        Here the conclusion clearly does not follow. Argument  D is invalid. Yet D(3) is the same sentence as C(3).  That sentence is both the conclusion of a valid argument  and the conclusion of an invalid argument. It is helpful in  beginning the study of philosophy to speak only of  arguments, not of sentences, as being valid or invalid and  to speak only of sentences as being true or false.

        To illustrate further the difference between truth and  validity, let us consider the following arguments:
 
 

                  (1) All professional tennis players are athletes.           T

         E       (2) Billie Jean King is a professional tennis  player.     T         V

                  (3)  Billie Jean King is an athlete.                             T
 
 

                  (1) All Oakland Raiders are football players.       T

           F     (2) Ken Stabler is a football player.                   T         I

                  (3) Ken Stabler  is an Oakland Raider.               T
 
 

                  (1) All athletes are professional golfers.              F

          G      (2) Arthur  Ashe is an athlete.                          T         V

                   (3) Arthur Asbe is a professional  golfer.            F
 
 

                  (1) All philosophers are Greeks.                         F

          H      (2) Inge  Broverman is a psychologist.               T         I

                  (3) Inge Broverman is a Greek.                         F
 
 

                  (1) All humans are whales.                                 F

           I      (2) All whales are  mammals.                             T         V

                  (3) All humans are mammals.                             T
 
 

                  (1) All whales are humans.                                  F

           J      (2) All whales are  mammals.                              T         I

                  (3) All humans are mammals.                              T
 
 

                  (1) All humans are dogs.                                       F

          K      (2) Lassie is a human.                                           F         V

                  (3) Lassie is a dog.                                                T
 
 

                  (1) All humans are fish.                                          F

          L      (2) Lassie is a human.                                            F         I

                  (3) Lassie is a dog.                                                T
 
 

                  (1) All senators are Democrats.                                F

         M      (2) Gerald Ford  is a senator.                                   F         V

                  (3) Gerald Ford is a Democrat.                                 F
 
 

                (1) All Federal judges are Republicans.                        F

         N     (2)  Barbara Jordan is a Republican.                            F         I

                (3) Barbara Jordan is a Federal judge                           F
 
 

                (1) All Republican senators are U.S. citizens.                 T

        O     (2)  All Democratic senators are U.S. citizens.                T         I

                (3) All  Republican senators are Democratic senators.      F
 
 

        grounds and the third as a  conclusion which is purported to follow from the  premises. It is convenient to speak of the truth value  of a sentence in referring to its truth, if the sentence is  true, or to its falsity, if the sentence is false. For many of  these arguments, their validity or invalidity is intuitively  obvious. Since a proof of the validity or invalidity of  these arguments is beyond the scope of this book, we will  capitalize on the readerís intuitions in using these  arguments to illustrate some important points about  relations between an argumentís validity or invalidity  and the truth values of its premises and conclusion.

        We note, for example, that E is a valid argument with  true premises and a true conclusion, while F is an  invalid argument although, as in the case of E, each of  its premises and its conclusion are true. We can  schematize this situation, as we have done to the right of  the above arguments, indicating the truth value of  premises and conclusion (using ëTí for trueí and  ëFí for ëfalseí) and the validity or invalidity of  each argument (using ëVí for ëvalidí and ëIí for  ëinvalidí).

        Argument G is a valid argument with one false premise,  one true premise, and a false conclusion. However H, while  it is like G in having one false premise, one true premise, and  a false conclusion, is unlike G in a most important respect:  H 2.  Arguments, Validity, and Truth (11 is an invalid argument. The pair of arguments G and H  (as well as each of the pairs E and F, I, and J, K  and L, M and N) illustrates that two arguments  may be exactly alike in respect to the truth value of their  premises and their respective conclusions while differing  in an all-important respect: their validity or invalidity.  This fact is not surprising if we recall that an argument is  valid just in case it is not possible for its premises to  be true while its conclusion is false.

        How does F fare in light of this informal account of  validity? Although its conclusion is in fact true, it is  possible for the premises of F to be true and its  conclusion false. Stabler could be traded to another  football team, or he might play out his option and sign  with a rival club. In fact, even though the conclusion of  F is true when this page is being written, that sentence  may have a different truth value when you are reading  this book. Similarly, many other sentences in our  example arguments may have different truth values as  you read this book than they had when we indicated their  truth values. The truth of the premises of F does not  exclude these possibilities; the truth of the premises of F  does not guarantee the truth of its conclusion. It is beyond  the scope of this book to provide the reader with skills  needed to demonstrate that the truth of the premises of  E does guarantee the truth of its conclusion.  Nonetheless, the possibility that Billie Jean King is not an  athlete is excluded by the truth of the premises of E.

        Arguments E, G, I, K, and M are all valid. Yet  only E, I, and K have true conclusions. So we see  that the conclusion of a valid argument may be a false  sentence. A valid argument guarantees the preservation  of truth in that its conclusion is true if all its premises are  true. This requirement on a valid argument says nothing  about the truth value of the conclusion of a valid  argument with one or more false premises. The  conclusion of such an argument may be true (consider I  and K); the conclusion of such an argument may be  false (consider G and M). While we cannot know that the conclusion  of a valid argument is true on the basis of knowing that  the argument is valid, we do know that the conclusion of  a valid argument would be true if all its premises were  true. Consider G; if it were true that all athletes are  professional golfers and that Arthur Ashe is an athlete,  then it would be true that Arthur Ashe is a professional  golfer.

        If the premises of G were true, it would be a sound  argument, a valid argument all of whose premises are  true. The truth value of the sentences that constitute the  premises and conclusion of an argument may differ from  one time to another.   Thus the soundness of an argument  may differ at different times. But the validity of an  argument, the relation between its premises and its  conclusion which we have expressed informally as  guaranteeing the preservation of truth, cannot change. Of  course, our ability to recognize or to demonstrate its  validity may change. At the time this book is written, E  is a sound argument. But if, for instance, Billie Jean King  were no longer a professional tennis player, E would no  longer be a sound argument, although its conclusion  might remain true. But it would remain a valid argument,  no less so if its conclusion were false.

        An invalid argument does not guarantee the preservation  of truth. The above discussion of F indicates that an  argument can fail to guarantee that if its premises are true  so also is its conclusion, even if its premises and  conclusion are in fact true. But neither does an invalid  argument guarantee that falsity in the premises will be  preserved in the conclusion. In other words, an invalid  argument may have one or more false premises and a true  conclusion (consider J and L). Since a valid argument does not guarantee the preservation of falsity  (consider 1 and K), we can say that no argument  guarantees that if one or more of its premises is false then  so is its conclusion.

        Of course an invalid argument with one or more false  premises may have a false conclusion, as do arguments  H and N. And an invalid argument, all of whose  premises are true, may have a true conclusion (consider  F) or a false conclusion (consider O)~ In short we  may say that no invalid argument guarantees either that  its conclusion is true or that it is false. From the pair of arguments I and J, as well as K  and 4 we see that a sentence that is the conclusion of an  invalid argument may be the conclusion of a valid  argument also. If a sentence is the conclusion of an  invalid argument, that argument does not guarantee its  truth; it remains an open question whether there is a valid  or a sound argument of which it is the conclusion.

        Argument 0 has true premises and a false conclusion;  so 0 is an obviously invalid argument. Among the  invalid arguments above, 0 is the only one whose  invalidity is obvious merely from the assertion of section  1 that an argument is valid just in case it is not possible  for all its premises to be true while its conclusion is false.  Argument 0 alone presents us with an actual instance  of that possibility which is excluded for a valid argument.  It is for this reason that no argument is paired with 0 ëas each other invalid argument is grouped with a  valid argument. For there can be no valid argument  whose premises are true and whose conclusion is false.

        Of course not all invalid arguments display their  invalidity in the truth values of their premises and  conclusion; not all invalid arguments have premises that  are in fact true and a conclusion that is in fact false.  Argument F, for instance, has a true conclusion. To  argue that it is invalid, we attempted to describe situations in which its premises would be true  but its conclusion would be false. We did not have to  argue that any such situation actually is the case. For an  argument is invalid if it is possible that its premises are  true while its conclusion is false. Argument H, for  example, has a false premise; so its invalidity is not  obvious from the truth values of its premises and  conclusion alone. Again we attempt to describe a  situation in which its premises would be true and argue  that H is invalid by showing that in such a situation its  conclusion might be false. Suppose it were true that all  philosophers are Greeks and that Inge Broverman is a  psychologist; it would still be possible that she is not  Greek. Roughly speaking, even if the premises of H  were true, they would offer no support, and certainly no  guarantee, of the truth of the conclusion of H.

         In arguing that F and H are invalid, we have relied  on the informal account that an argument is valid if but  only if it is not possible for its premises all to be true  and its conclusion to be false. This technique of arguing  that an argument is invalid is not singly adequate.  Consider the following argument:

            (1) All squares are polygons.    T

    P      (2) All  rectangles are polygons    T I

            (3) All squares are rectangles    T

        An  attempt to describe a situation in which the conclusion of  P would be false while its premises are true will faiL  But it will fail not because the truth of the premises of P  guarantees the truth of its conclusion. The truth of the  conclusion of P is guaranteed by information which is  no part of the argument. While this information from  geometry supports the truth of the conclusion of P, it in  no way strengthens the support which argument P  provides for its conclusion. Indeed the conclusion of P could be the conclusion of a sound  argument, but P is not that argument; for P is not a valid argument and so cannot be a sound argument. To  argue that P is invalid, we might produce another  argument which has the same logical form as P  and has true premises and a false conclusion.

        The concept of logical form is a complex one which has  been the subject of considerable work among  philosophers, implicitly at least since Leibniz and  explicitly since Frege. Only a brief discussion of logical  form can be undertaken in this book (see Chapter III,  section 1), but the study of logic will provide the student  with tools enabling increasingly detailed analysis of  form. Argument 0 has the same logical form as P.  This fact, for which the reader will find evidence in  Chapter III, section 1, conjoined with the fact that 0  has true premises and a false conclusion, provides an  excellent argument that P is invalid. For an argument  is valid just in case no argument with the same form has  all true premises and a false conclusion.

        We have said that the goal of logic is to preserve truth  and that, pursuant to this goal, the most basic mark of  quality in an argument is validity. The reader may  wonder why, then, in this section we have discussed  techniques which may be used to provide evidence that  an argument is invalid but not that an argument is valid.  Recalling the informal accounts of validity that have  been given will provide a partial answer. An argument is  valid just in case it is not possible that its premises  are all true while its conclusion is false. To capitalize on this account in an effort to establish that an argument  is valid, we would have to survey all possibilities to  determine that none of them provides a situation in which  all the premises of the argument are true while its  conclusion is false. Such a survey itself is not possible in  any finite period of time! But a single situation in which  the premises are all true and the conclusion false  obviously is sufficient to show that such a situation is  possible and that the argument cannot be valid. An  argument is valid if and only if no argument with the  same form has all true premises and a false conclusion.  Analogously, to capitalize on this account in order to  establish that an argument is valid, we would have to  survey all arguments (actual and possible?) which have  the same form. Such a survey is equally beyond what is  possible for finite humans.

        Because of these and other difficulties in establishing the  validity of an argument considered in isolation rather  than as exemplil~ing the logical form on which its  validity or invalidity depends, the tools of formal logic  are extremely valuable. Within the study of logic, special  symbolism or notation is developed which enables us to  study the forms of arguments and to isolate many of the  formal components of sentences, on which components  the validity of arguments depends. Precise rules can be  stated in terms of the forms of sentences, and the  deductive methods available enable us to evaluate  arguments to the extent that their form can be expressed  in the symbolic notation available to us. In addition to the  informal accounts of validity already mentioned, we can  say that an argument is valid if and only if its conclusion  is a logical consequence of its premises. The methods of  formal logic put us in a position better to understand the  content of this and other informal accounts of validity.  Presentation and discussion of these methods are beyond the scope of  this book. However, we shall attempt, in the remaining  section of this chapter and in Chapter II, to introduce the  reader to some of the most basic formal components of  arguments.

        Many philosophers claim that the existing tools of formal  logic are inadequate to express, and therefore to evaluate,  the forms of many interesting arguments. If so, this claim  provides an incentive for further developments in logic,  but it is no criticism of the value of the existing tools of  logic in evaluating the arguments whose logical forms  these tools enable us to reveal

        Not all arguments are of the three-line form we have  been considering. And not all arguments which are of  that form appear at first glance to be so. For example:

           (1) Caesar is emperor.

    Q     (2) (Therefore) Someone is emperor.

is a simple argument that is sound but has a  different form. And:

            (1) Jones is a man.

     R     (2) (Therefore) Jones is mortal.

is an argument that is valid only on the strength of the  suppressed (or unexpressed) premise that all men are  mortal. Such an argument is clearly valid, and of the  familiar three-line form, if we add the missing  premise. If we do not, we can consider the argument as  incomplete rather than invalid

        And of course not all arguments are either as simple in  their structure or as obvious in their validity or invalidity  as those we have considered. In fact few arguments are,  whether they are found in the writings of a famous  philosopher, in oneís own writings, in editorials, in  political debates, or in advertisements. We have  attempted to choose as examples arguments about which  the reader will have accurate intuitions concerning their  validity or invalidity. Since intuitions are not infallible  guides to validity, especially as arguments increase in  their complexity, we have sought to strengthen these  intuitions and to provide the reader with ways to support  his or her evaluation of arguments. In the discussion of  validity and its relations to the actual and possible truth  value of premises and conclusions, we hope the reader  will find an increased understanding of the concept of  validity which will sharpen intuitions involved in  assessing arguments. And we have attempted to suggest  ways of structuring these intuitions so that they may be  extended and applied constructively to less explicitly  structured and more complex arguments.