{"id":7980,"date":"2014-05-03T05:43:40","date_gmt":"2014-05-03T05:43:40","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2014\/05\/03\/logical-form\/"},"modified":"2014-05-03T05:43:40","modified_gmt":"2014-05-03T05:43:40","slug":"logical-form","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2014\/05\/03\/logical-form\/","title":{"rendered":"Logical Form and the Supposed Asymmetry of Validity and Invalidity: A Defense of Symmetry"},"content":{"rendered":"

For the 'Londonistas,'  Ed and David, partners in logical investigations.  We are unlikely ever to agree, but clarification of differences is an attainable and worthwhile goal, here, and in every arena of controversy.  Have at it, boys.<\/span><\/p>\n

………….<\/span><\/p>\n

1. Suppose someone reasons as follows. 'Some Englishmen are Londoners; therefore, some Londoners are Englishmen.'  To reason is one thing, to reason correctly another.  So one can ask: Is this specimen of reasoning correct or incorrect?  This is the sort of question with which logic deals.  Logic is the study of inference and argument from a normative point of view.   It seeks to articulate the criteria of correct and incorrect reasoning.  It is analogous to ethics which seeks to articulate the criteria of correct and incorrect action.<\/span><\/p>\n

2. We all take for granted that some reasoning is correct and some incorrect, and we are all more or less naturally good at reasoning correctly.  Almost everyone grasps immediately that if Tom is an Englishman and some Englishmen are Londoners, it does not follow that Tom is a Londoner. What distinguishes the logician is his reflective stance.  He reflects upon reasoning in general and tries to extract and systematize the principles of correct reasoning.  'Extract' is an apt metaphor.  The logician  develops a theory from his pre-theoretical understanding of argumentative correctness.  As every teacher of logic comes to learn, one must already be logical to profit from the study of logic just as one must already be ethical to profit from the study of ethics.  It is a matter of making explicit and raising to the full light of awareness what must already be implicitly present if the e-duc-ation, the drawing out into the explicit is to occur.  This is why courses in logic and ethics are useless for many and positively harmful for some.  But they do make some of us more logical and more ethical.<\/span><\/p>\n

3.  Correctness in deductive logic is called validity, and incorrectness invalidity.   Since one can argue correctly from false premises and incorrectly from true premises, we distinguish validity from truth.  Consider the following argument:<\/span><\/p>\n

Some Englishmen are Londoners<\/span>
——-<\/span>
Some Londoners are Englishmen.<\/span><\/p>\n

We say of neither the premise nor the conclusion that it is either valid or invalid: we say that it is either true or false.  And we do not say of the argument that it is true or false, but that it is either valid or invalid. We also speak of inferences as either valid or invalid. <\/span><\/p>\n

4.  What makes a valid argument valid?  It can't be that it has true premises and a true conclusion.  For there are invalid arguments that satisfy this condition.  Some say that what makes a valid argument valid is the impossibility of the premises' being true and the conclusion false.  Theirs is a modal explanation of validity.  Equivalently,<\/span><\/p>\n

\n

D1. Argument A is valid =df necessarily, if A's premises are all true, then A's conclusion is true.<\/span><\/p>\n<\/blockquote>\n

This necessity is plainly the necessity of the consequence (necessitas consequentiae<\/em>), not the necessity of the consequent (necessitas consequentiis<\/em>):  in the majority of cases the premises and conclusion are all contingent propositions.<\/span><\/p>\n

The modal explanation of validity in (D1) is fine as far as it goes, but it leads to the question: what is the ground of the necessity?  If validity is explained by the RHS of (D1), what explains the necessity?  What explains the necessitas consequentiae<\/em> of the conditional on the RHS of (D1)?<\/span><\/p>\n

Enter logical form.<\/span><\/p>\n

The validity of a given valid argument evidently resides in something distinct from the given argument.  What is this distinct something?  It is the logical form of the argument, the argument form.  The form F of an argument A is distinct from A because F is a universal (a repeatable) while A is a particular (an unrepeatable).  Thus the form<\/span><\/p>\n

All S are M<\/span>
All M are P<\/span>
——-<\/span>
All S are P<\/span><\/p>\n

is a one-in-many, a repeatable.  It is repeated in every argument of that form.  It is the form of  indefinitely many syllogisms, although it is not itself a syllogism, any more than 'All S are M' is a proposition.  A proposition is either true or false, but 'All S are M' is neither true nor false.  To appreciate this, bear in mind that 'S' and 'M' are not abbreviations but placeholders.  If the letters above were abbreviations, then the array above would be an (abbreviated) argument, not  an argument form.  An argument form is not an argument but a form of indefinitely many arguments. <\/span><\/p>\n

Now validity is a property of argument forms primarily, and secondarily of arguments having valid forms. What makes a valid argument valid is the validity of its form:<\/span><\/p>\n

\n

D2. Argument A is valid =df A is an instance of a valid argument form.<\/span><\/p>\n

D3. Argument form F is valid =df no  instance of F has true premises and a false conclusion.<\/span><\/p>\n<\/blockquote>\n

Validity is truth-preserving: a valid argument form will never take you from true premises to a false conclusion.  (Exercise for the reader: show that invalidity is not falsehood preserving.)  In sum, an argument is valid in virtue of having a valid form, and a form is valid if no argument of that form has true premises and a false concusion. The logical form of a valid argument is what makes it impossible for the premises to be true and the conclusion false.<\/span><\/p>\n

5.  If a valid argument is one with a valid form, one will be tempted to to say that an invalid argument is one with an invalid form.  Call this the Symmetry Thesis:<\/span><\/p>\n

\n

ST. If an argument  is an instance of a valid form, then it is valid, and if it is an instance of an invalid form, then it is invalid.<\/span><\/p>\n<\/blockquote>\n

But there are examples that appear to break the symmetry, e.g.:<\/span><\/p>\n

If God created something , then God created everything.<\/span>
God created everything.<\/span>
——-<\/span>
God created something.<\/span><\/p>\n

This argument fits the pattern of the formal fallacy, Affirming the Consequent:<\/span><\/p>\n

If p then q<\/span>
q<\/span>
——-<\/span>
p.<\/span><\/p>\n

But the argument also has a valid form:<\/span><\/p>\n

Every x is such that Cgx<\/span>
——-<\/span>
Some x is such that Cgx. <\/span><\/p>\n

(Example adapted from Gerald J. Massey, "The Fallacy behind Fallacies," Midwest Studies in Philosophy<\/em> VI (1981), pp. 489-500)<\/span><\/p>\n

So which is it? Is the argument valid or invalid?   It can't be both and it can't be neither.  One option is to abandon the Symmetry Thesis and maintain that having a valid form is sufficient for an argument to be valid, but that having an invalid form is not sufficient for it to be invalid. One would then be adopting the following Asymmetry Thesis:<\/span><\/p>\n

\n

AT.  Having a valid form suffices for an argument to be valid, but having an invalid form does not suffice for an argument to be invalid.<\/span><\/p>\n<\/blockquote>\n

Another option is to hold to the Symmetry Thesis and maintain that the Massey argument is really two arguments, not one.  But before exploring this option, let us consider the unintuitive consequences of holding that one and the same argument can have two different forms, one valid, the other invalid. <\/span><\/p>\n

6. Consider any valid syllogism.  A syllogism, by definition, consists of exactly three different propositions: a major premise, a minor premise, and a conclusion.  So every valid syllogism has the invalid form: p, q, ergo r<\/em>.  Generalizing, we can say that any argument whose validity hinges upon the internal subpropositional logical structure of its constituent propositions will instantiate an invalid form from the propositional calculus (PC).  For example, any argument of the valid form, Some S are P; ergo, Some P are S<\/em>, is an instance of the invalid PC form, p, ergo q<\/em>. <\/span><\/p>\n

To think of a valid syllogism as having the invalid form p, q, ergo r<\/em> is to abstract away from the internal subpropositional logical structure that the syllogism's validity pivots on.  But if this abstraction is permitted, one may permit oneself to abstract away from the requirement that the same terms in an argument be replaced by the same placeholders.  One might then maintain that<\/span><\/p>\n

All men are mortal<\/span>
Socrates is a man<\/span>
——-<\/span>
Socrates is mortal<\/span><\/p>\n

has the invalid logical form<\/span><\/p>\n

All Fs are Gs<\/span>
a is an H<\/span>
——-<\/span>
a is a G<\/span><\/p>\n

But why stop there?  By the same 'reasoning,' the Socrates syllogism has the invalid form:<\/span><\/p>\n

All Fs are Gs<\/span>
a is an H<\/span>
——-<\/span>
b is an I.<\/span><\/p>\n

But if one abstracts away from the requirement that the same term or sentence be replaced by the same placeholder, then we get the result that the obviously valid<\/span><\/p>\n

Tom is tall<\/span>
——-<\/span>
Tom is tall<\/span><\/p>\n

has the valid form p ergo p<\/em> and the invalid form p ergo q<\/em>.  Here we are abstracting away from the fact that a proposition entails itself and ascending to the higher level of abstraction at which  a proposition entails a proposition.  After all, it is surely true that in our example a proposition entails a proposition.<\/span><\/p>\n

I submit, however, that our example's having an invalid form is an intolerable result.  Something has gone wrong.  Surely the last argument has no invalid form.  Surely one cannot lay bare the form of an argument, in an serious sense of 'argument,'  if one abandons the requirement that the same term or sentence be replaced by the same placeholder. To do that is to engage in vicious abstraction.  It is vicious because an argument in any serious sense of the term is not just a sequence of isolated propositions, but a sequence of propositions together with the idea that one of them is supposed to follow from the others.  An argument in any serious sense of the term is a sequence of propositions that has the property of being putatively such that one of them, the conclusion, follows from the others, the premises.  But no sequence of propositions can have this property if the argument's form allows for different terms\/propositions to have different placeholders.<\/span><\/p>\n

7.  So I suggest that we abandon the Asymmetry Thesis and adopt the Symmetry Thesis according to which no valid argument has any invalid forms.  Let me now try to motivate this proposal.<\/span><\/p>\n

An argument form is an abstraction from an argument.  But it is also true that an argument is an abstraction from a concrete episode of reasoning by a definite person at a definite time.  Clearly, the same argument can be enacted by the same person at different times, and by the same or different persons at different times.  I can 'run through' the argument that the null set is unique any number of times, and so can you.  An argument in this sense is not a concrete episode of arguing (reasoning) but a sequence of propositions.  A proposition, of course, is not the same as a sentence used to express  it.<\/span><\/p>\n

Now I grant that an argument taken in abstraction from an episode of reasoning (and as the content of that reasoning) can instantiate two or more argument forms.  But I deny that a concrete episode of reasoning by a definite person at a definite time can instantiate two or more argument forms. So my claim is that while an argument in abstracto<\/em> can have two or more forms, an argument in concreto<\/em>, i.e. a concrete episode of reasoning cannot have more than one form.  If this form is valid the argument in concreto<\/em> is valid.  If invalid, the argument in concreto<\/em> is invalid.  To illustrate:<\/span><\/p>\n

Suppose I know that no Democrat supports capital punishment.  Then I learn that Jones is a Democrat.  Putting together these two pieces of information, I infer that Jones does not support capital punishment. By 'the concrete episode of reasoning,' I mean the reasoning process together with its content.  One first thinks of the first proposition, then the second, then one infers the third, and all of this in the unity of one consciousness.  The content is the argument considered in abstraction from any particular diachronic mental enactment by a particular person at a particular time.  The reasoning process as a datable temporally extended mental process is also an abstraction from the concrete episode of reasoning which must include both, the reasoning and its content.<\/span><\/p>\n

Now the concrete episode of reasoning embodies a pattern.  In the example, I reason in accordance with this pattern:<\/span><\/p>\n

(x) (Fx –> ~Gx)<\/span>
Fa<\/span>
——-<\/span>
~Ga<\/span><\/p>\n

Which is also representable as follows:<\/span><\/p>\n

No Fs are Gs<\/span>
a is an F<\/span>
——-<\/span>
a is not a G.<\/span><\/p>\n

The pattern or logical form of my concrete episode of reasoning is assuredly not: p, q, ergo r<\/em>.  This is consistent with saying that the argument in abstracto<\/em> instantiates the invalid form p, q, ergo r<\/em> in addition to the valid form above.<\/span><\/p>\n

The point I am making is this.  If we take an argument in abstraction from the concrete episode of reasoning in which it is embodied, then we may find that it instantiates more than one form.  There is no denying that every valid syllogism, considered by itself and apart from the mental life of an agent who thinks it through, instantiates the invalid form p, q, ergo r.<\/em>  But no one who reasons syllogistically reasons in accordance with that invalid form.  Syllogistic reasoning, whether correct or incorrect, is reasoning that is sensitive to the internal subpropositional logical structure of the syllogism's constituent propositions.  The invalid form is not a form of the argument in concreto<\/em>.<\/span><\/p>\n

 One must  distinguish among the following:<\/span><\/p>\n