A Primer on Logic: Interlude

x
Bookmark

A Primer on Logic

An Interlude on the Inadequacies of Aristotelian Logic

Jason's Pictures May 2009 - February 2010 283

Whether or not the treatment of the cognitions belonging to the concern of reason travels the secure course of a science is something which can soon be judged by its success.

[…]

That from the earliest times logic has traveled this secure course can be seen from the fact that since the time of Aristotle it has not had to go a single step backwards, unless we count the abolition of a few dispensable subtleties or the more distinct determination of its presentation, which improvements belong more to the elegance than to the security of that science. What is further remarkable about logic is that until now it has also been unable to take a single step forward, and therefore seems to all appearance to be finished and complete.

— p. B viii / p. 106 of Immanuel Kant’s Critique of Pure Reason, edited by Paul Guyer and Allen W. Wood, Cambridge University Press, 1998; originally published in 1787 in the preface to the second edition of the Critique.

In this Interlude I’ll expose some inadequacies of Aristotelian logic, so you can see why it was necessary to go beyond it. Let’s return to the first argument we encountered, all the way back at the beginning of Part 1:

All humans are mortal.

Socrates is human.

Socrates is mortal.

Given what we’ve learned in Part 3, we have to acknowledge that this argument looks very much like a syllogism. For one thing, it has two premises and a conclusion. For another, it has three terms, namely ‘human’ (minus its plural ending in the first premise), ‘mortal’ and ‘Socrates’. But what kind of term is ‘Socrates’? According to the system of classification of Part 3, ‘Socrates’ is a singular term. It applies to the historical Socrates, the Gadfly of Athens and the teacher of Plato, and moreover it applies to him as an individual. If the name ‘Socrates’ applies to anyone else it can only apply to them because it is ambiguous, not because it marks the people that it applies to as being members of a kind. As a singular term, ‘Socrates’ cannot occur in a genuine syllogism. Thus, if we wish to reason about Socrates in Aristotelian logic, we must find a corresponding general term which we can use in place of the name ‘Socrates’. And if we are really to reason about Socrates, and not just individuals who happen to belong to the same kinds that Socrates does, we need to find a general term which nevertheless applies to Socrates and him alone.

One such term is ‘individual who is the same person as Socrates’. While this term applies only to Socrates, grammatically it is a general term, for in a sense it does mark the individuals that it applies to as being members of a kind, namely the kind of individuals who are the same person as Socrates. It just turns out that, since any individual who is the same person as Socrates just is Socrates, this kind is one of which Socrates is the sole member! And so while the term ‘individual who is the same person as Socrates’ does in effect specify a kind, it is neither a name of Socrates nor a singular term.

With this term ready to hand, we may now recast our argument as a proper syllogism:

All humans are mortal.

All individuals who are the same person as Socrates are human.

All individuals who are the same person as Socrates are mortal.

This is of the form AAA and is therefore valid. One can reason about individuals using Aristotelian logic, but it is only possible to do so in a rather artificial way.

By contrast, a true limitation of Aristotelian logic is that it cannot handle arguments or argument schemas whose validity depends on the truth-functional composition of their constituent statements or schemas. Take Modus Ponens as an example, which, to recall, is represented by the following schema:

p ⊃ q

p____

Is there any way that Aristotelian logic can handle this argument form? The answer is “No”. There are two reasons why it cannot. First, Modus Ponens is valid no matter what statements one substitutes for ‘p’ and ‘q’. These statements need not be of any of the forms A, E, I, or O, so the second premise and the conclusion of some arguments that satisfy Modus Ponens might not be allowed to occur in a proper syllogism. Second, even if we ignore this problem and use only statements of one of the four admissible forms, there is nothing in Aristotelian logic which corresponds to ‘⊃’, nor is there anything which corresponds to the English ‘if…then’ construction. Even if the statements which one substitutes for ‘p’ and ‘q’ are admissible, the conditional ‘p ⊃ q’ most certainly is not.

Consider also the following argument schemas, which can be shown to be valid using the methods of Part 2:

p & q

p____

p ∨ q

Besides the fact that statements of the forms “p & q” and “p ∨ q” cannot occur in a syllogism, these schemas are also disqualified because they have only one premise. Thus it is apparent that Aristotelian logic cannot so much as express arguments or argument schemas which are valid as a matter of propositional logic.

There are other valid arguments or argument schemas which can be shown to be valid neither by propositional logic nor Aristotelian logic. Here are three examples:

Everyone is happy.

Jones is happy.

Someone is happy.

Everyone knows someone.

Everyone knows someone who knows someone.

These depend for their validity on the meaning of the quantity terms ‘everyone’ and ‘someone’. Logicians and linguists classify such terms as quantifiers, of which more will be said in Part 4. In the above three arguments, the statements formed with these quantifiers resemble the syllogistic forms A and I, but since the arguments have only one premise, they are not syllogistic in form. We might be able to reconstruct them as syllogisms, but even if we could that wouldn’t change the fact that Aristotelian logic can’t really handle them. Consider this argument as an illustration:

A____

Since “A” and “B” are atomic statements, this argument cannot be shown to be valid in propositional logic. In spite of that, the argument could still be valid, say if ‘A’ represented “It is necessary that 12 is greater than 9”, and ‘B’ represented “12 is greater than 9”. Knowing that this argument is valid—if something is necessarily so, it is so—we could try to reconstruct it as an argument which can be shown to be valid in propositional logic, namely:

A ⊃ B

(You can probably recognize that this is just an instance of Modus Ponens with the order of the premises reversed.) Granted, since “A” entails “B” the material implication “A ⊃ B” must be true. But to reconstruct the argument as we have just done misses the point. One can take any two statements p and q, whether p entails q or not, form the material conditional “p ⊃ q”, take p and “p ⊃ q” as premises and q as the conclusion, and—presto!—end up with a valid argument. While the original argument was valid because of the meaning of the phrase ‘it is necessary that’, the reconstruction is valid because of the meaning of the truth function ‘⊃’. The point is this: If you can only show a given valid argument to be valid within a certain logical system by “reconstructing” it in such a way that the resulting argument is valid in that system for a different reason than the original argument was, you haven’t really shown the original argument to be valid within that system. The same is true of any attempt to show the above three quantificational arguments to be valid using Aristotelian logic, no matter what extra premise(s) you might add. (Neither can propositional logic explain their validity, for propositional logic does not make use of quantifiers.)

Lastly, another limitation of Aristotelian logic is that has trouble with some arguments that contain relational statements. These are statements which say that something is related to something, possibly itself, in a certain way. The statements “Wyman is taller than Ortcutt”, “The Civil War took place before World War II”, and “The Morning Star is the same thing as the Evening Star” are all relational. In the first the relation is ‘is taller than’, in the second it is ‘took place before’, and in third it is ‘is the same thing as’. For the sake of brevity, from this point onwards I will call the relation ‘is the same thing as’ the identity relation or simply identity, and so instead of saying “The Morning Star is the same thing as the Evening Star” I will say “The Morning Star is identical to the Evening Star”. This relation should not be confused with another relation that goes by the same name, namely the relation of indiscernibility or extreme similarity which holds, for example, between identical twins.

Consider the following argument, all of whose statements are relational:

The Morning Star is identical to the Evening Star.

The Evening Star is identical to Venus.

The Morning Star is identical to Venus.

As you have probably guessed, propositional logic cannot show arguments like this to be valid. It cannot handle them because the atomic statements and schematic letters it deals with have no internal structure, and complex statements and schemas can only be formed by means of truth functions, which means that propositional logic cannot express relations. Relational statements can be symbolized by atomic statement letters, and can also be substituted for schematic letters, but because it has no terms for the relations themselves propositional logic is blind to the relational character of these statements.

In Aristotelian logic one could express the relational statements composing the preceding argument as A statements. If we paraphrase away the names involved, much as we did earlier with the name ‘Socrates’, we can modify the argument to get this syllogism:

All things which are identical to the Morning Star are things which are identical to the Evening Star.

All things which are identical to the Evening Star are things which are identical to Venus.

All things which are identical to the Morning Star are things which are identical to Venus.

Let’s use ‘H’ to represent ‘things which are identical to the Morning Star’, ‘F’ to represent ‘things which are identical to the Evening Star’, and ‘G’ to represent ‘things which are identical to Venus’. The appropriate schematic syllogism is then:

All H are F.

All F are G.

All H are G.

(Note that here the first premise is the minor premise and the second is the major premise.) Here we have a valid syllogism of the form AAA.

All is not well, however. One can construct an argument which has the same logical form as the original argument about the Morning Star, the Evening Star, and Venus, but which is clearly invalid. We simply replace the identity relation with the relation ‘is ten feet distant from’, and replace the terms ‘the Morning Star’, ‘the Evening Star’, and ‘Venus’ with the dummy terms ‘a’, ‘b’, and ‘c’, which can be the names of any objects you please as long as they occupy space. The argument is then:

a is ten feet distant from b.

b is ten feet distant from c.

a is ten feet distant from c.

The problem is this: How can the one argument be valid and the other not when—so it appears—they have the same form? The answer is that the first argument is valid because the identity relation is transitive: If any entity—call it e₁—is identical to e₂, and e₂ is identical to e₃, it follows that e₁ is identical to e₃. The second argument is invalid because the relation ‘is ten feet distant from’ is not transitive: If a, b, and c all lie on a straight line, and a is ten feet distant from b and b is ten feet distant from c, a will be twenty feet distant from c, not ten. And if we were to try to turn this argument into a syllogism, about the best we could do is:

All things which are identical to a are ten feet distant from things which are identical to b.

All things which are identical to b are ten feet distant from things which are identical to c.

All things which are identical to a are ten feet distant from things which are identical to c.

Clearly there is something wrong here. To see what it is, we can use ‘H’ to represent ‘things which are identical to a’, ‘F’ to represent ‘things which are identical to b’, and ‘G’, to represent ‘things which are identical to c’. The appropriate schema is:

All H are ten feet distant from F.

All F are ten feet distant from G.

All H are ten feet distant from G.

This is similar to the syllogistic form AAA, but similarity is not enough. Here the relation ‘are ten feet distant from’ takes the place of the usual syllogistic copula ‘is’ or ‘are’. Thus the statements in the above schema are not really A statements, which means that the schema is not really a schematic syllogism. The modified argument about the Morning Star, the Evening Star, and Venus is syllogistically valid because, given the inherent transitivity of the identity relation and the fact that it can only relate a thing to itself, it is not necessary to assert that the identity relation holds between the terms. If we do assert that, we get an argument of this form:

All H are identical to F.

All F are identical to G.

All H are identical to G.

In this schema the identity relation takes the place of the syllogistic copula. Although it is valid, is not a valid syllogism.

If, then, we are to sift the valid relational arguments from the invalid ones as one would the wheat from the chaff, we must use a logical system that is more powerful than Aristotelian logic.

To sum up this Interlude, Aristotelian logic has difficulties when it comes to arguments involving individuals, truth functions, some quantified statements, and some relational statements. In the next Part we shall explore a new logical system, called predicate logic, which can handle all these problems in a rather simple yet elegant way.