London Ed refers us to Understanding Arguments: an Introduction to Informal Logic, Robert Fogelin and Walter Sinnott-Armstrong, and provides this quotation:
Perhaps a bit more surprisingly, our definitions allow 'roses are red and roses are red' to be a substitution instance of 'p & q'. This example makes sense if you compare it to variables in mathematics. Using only positive integers, how many solutions are there to the equation 'x + y = 4'? There are three: 3+1, 1+3, and 2+2. The fact that '2+2' is a solution to 'x + y = 4' shows that '2' can be substituted for both 'x' and 'y' in the same solution. That's just like allowing 'roses are red' to be substituted for both 'p' and 'q', so that 'roses are red and roses are red' is a substitution instance of 'p & q' in propositional logic.
In general, then, we get a substitution instance of a propositional form by uniformly replacing the same variable with the same proposition throughout, but different variables do not have to be replaced with different propositions. The rule is this:
Different variables may be replaced with the same proposition [Ed: Let's call this the London rule], but different propositions may not be replaced with the same variable.
Suppose I am given the task of determining whether the conditional English sentence 'If roses are red, then roses are red' is a tautology, a contradiction, or a contingency. How do I proceed?
Step One is translation, or encoding. Let upper case letters serve as placeholders for propositions. Let '–>' denote the truth-functional connective known in the trade as the material or Philonian conditional. I write 'P –> P.'
Step Two is evaluation. Suppose for reductio that the truth value of 'P –>P' is false. Then, by the definition of the Philonian conditional, we know that the antecedent must be true, and the consequent false. But antecedent and consequent are the same proposition. Therefore, the same proposition is both true and false. This is a contradiction. Therefore, the assumption that conditional is false is itself false. Therefore the conditional is a tautology.
Now that obviously is the right answer since you don't need logic to know that 'If roses are red, then roses are red' is a tautology. (Assuming you know the definition of 'tautology.') But if if Fogelin & Co. are right, and the 'P –>Q' encoding is permitted, then we get the wrong answer, namely, that the English conditional is a contingency.
I am assuming that if 'P–>Q' is a logical form of 'If roses are red, then roses are red,' then 'P –>Q' is a legitimate translation of 'If roses are red, then roses are red.' As Heraclitus said, the way up and the way down are the same. The assumption seems correct.
If I am right, then there must be something wrong with the mathematical analogy. Now there is no doubt that Fogelin and his side kick are right when it comes to mathematics. And I allow that what they say is true about variables in general. Suppose I want to translate into first-order predicate logic with identity the sentence, 'There is exactly one wise man.' I would write, '[(Ex)Wx & (y)(Wy –> x = y)].' Suppose Siddartha is the unique wise man. Then Siddartha is both the value of 'x' and the value of 'y.'
So different variables can have the same value. And they can have the same substituend. In the example, Siddartha is the value and 'Siddartha' is the substituend. But is a placeholder the same as a variable? I don't think so. Here is a little argument:
No variable is a constant Every placeholder is an arbitrary constant Every arbitrary constant is a constant ——- No placeholder is a variable.
A placeholder is neither an abbreviation, nor a variable. It is an arbitrary constant. Thus the logical form of 'Al is fat' is Fa, not Fx. Fa is a proposition, not a propositional function. 'F' is a predicate constant. 'a' is an individual constant. We cannot symbolize 'Al is fat' as Fx. For Fx is not a proposition but a propositional function. If 'a' were not an arbitrary constant, then Fa would not depict the logical form of 'Al is fat,' a form it shares with other atomic sentences.
Here is another argument:
Every variable is either free or bound by a quantifier No placeholder is either free or bound by a quantifier ——- No placeholder is a variable.
Here is a third argument:
Every variable has a domain over which it ranges No placeholder has a domain over which it ranges ——- No placeholder is a variable.
A fourth argument:
There is no quantification over propositions in the propositional calculus ——- There are no propositional variables in the propositional calculus If there are no propositional variables in the propositional calculus, then the placeholders in the propositional calculus cannot be variables ——- The placeholders in the proposition calculus cannot be variables.
Punchline: because placeholders are not variables, the fact that the different variables can have the same value and the same substituend does not show that different placeholders can have the same substituend. 'If roses are red, then roses are red' does not have the logical form 'P –>Q' and the latter form does not have as a substitutution-instance 'If roses are red, then roses are red.'
As I have said many times already, one cannot abstract away from the fact that the same proposition is both antecedent and consequent.
What one could say, perhaps, is that 'P –> P' has the higher order form 'P –> Q.' But this latter form is not a form of the English sentence but a form of the form of the English sentence.
Ed can appeal to authority all he wants, but that is an unphilosophical move, indeed an informal fallacy. He needs to show where I am going wrong.
"The most conspicuous purpose of logic, in its applications to science and everyday discourse, is the justification and criticism of inference." (Emphasis added, Willard Van Orman Quine, Methods of Logic, 2nd revised ed., Holt, Rinehart & Winston, 1959, p. 33.
Perhaps the dispute in the earlier thread could be resolved if we all could agree on the following.
1. The most specific logical form of a deductive argument A is the form relevant for assessing whether the reasoning embodied in A is valid or invalid.
2. Every deductive argument has exactly one most specific form.
3. Symmetry Thesis: if the most specific form of A is valid, then A is valid; if the most specific form of A is invalid, then A is invalid.
In case 'most specific logical form' needs explanation, consider the difference between the following valid form from the predicate calculus and the following invalid form from the propositional calculus:
Fa Ga ——- (Ex)(Fx & Gx)
p q ——- r.
The former is the most specific logical form of 'Al is fat, Al is gay, ergo, something is both fat and gay.' The latter, if a form of the argument at all, is less specific: it abstracts from the internal subpropositional logical structure of the constituent propositions.
Now three examples in illustration of (1)-(3).
Example One. Call the following argument 'Charley':
Tom is tall ——- Tom is tall.
Although the above display, which is a written expression of the argument and not the argument itself, shows two tokens of the sentence type 'Tom is tall,' the argument consists of exactly one proposition. Anyone who executes the reasoning displayed infers the proposition *Tom is tall* from itself. (I am using asterisks to mention propositions. So '*Tom is tall*' is an abbreviation of 'the proposition expressed by a tokening of the sentence type "Tom is tall".')
It is perfectly clear that the reasoning embodied by Charley is valid and that its form is 'P ergo P.' The reasoning is not from P to some proposition that may or may not be identical to P. Therefore the concrete episode of reasoning does not have the form 'P ergo Q.'
But let us irenically concede that if one wished, for whatever reason, to abstract not only from the content of the argument but also from the plain fact that the argument involves exactly one proposition, one could view the form 'P ergo P' as a special case of 'P ergo Q.' And I will also concede, to keep peace between Phoenix and London, that the argument instantiates the second invalid form, even though I don't believe that this is the case.
Either way, the Symmetry Thesis stands and the Asymmetry Thesis falls. For as G. Rodrigues in the earlier thread pointed out, 'P ergo P' is the most specific form of Charley.
Example Two. Call the following argument 'Kitty Kat.'
If cats like cream, then cats like cream Cats like cream ——- Cats like cream.
Please note that there is no equivocation in this example: 'Cats like cream' has the same sense in all four of its occurrences.
Kitty Kat's most specific form is 'P –> P, P, ergo P.' This form is valid. So Kitty Kat is valid, notwithstanding the fact, if it is a fact, that Kitty Kat also instantiates the formal fallacy, Affirming the Consequent: P –> Q, Q, ergo P. By (1) above, the fact, if it is a fact, that Kitty Kat instantiates Affirming the Consequent is irrelevant to the assessment of the validity/invalidty of the reasoning embodied in Kitty Kat.
Example Three. Call the following example 'Massey':
If God created something , then God created everything. God created everything. ——- God created something.
This argument fits the pattern of the formal fallacy, Affirming the Consequent:
If p then q q ——- p.
But the argument also has a valid form:
Every x is such that Cgx ——- Some x is such that Cgx.
Please note that if an argument is valid, adding a premise can't make it invalid; this principle is what allows us to disregard the first line.
(Example adapted from Gerald J. Massey, "The Fallacy behind Fallacies," Midwest Studies in Philosophy VI (1981), pp. 489-500)
The most specific form of Massey is the predicate logic form above displayed. Since it is valid, Massey is valid.
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.
………….
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.
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.
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:
Some Englishmen are Londoners ——- Some Londoners are Englishmen.
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.
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,
D1. Argument A is valid =df necessarily, if A's premises are all true, then A's conclusion is true.
This necessity is plainly the necessity of the consequence (necessitas consequentiae), not the necessity of the consequent (necessitas consequentiis): in the majority of cases the premises and conclusion are all contingent propositions.
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 of the conditional on the RHS of (D1)?
Enter logical form.
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
All S are M All M are P ——- All S are P
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.
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:
D2. Argument A is valid =df A is an instance of a valid argument form.
D3. Argument form F is valid =df no instance of F has true premises and a false conclusion.
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.
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:
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.
But there are examples that appear to break the symmetry, e.g.:
If God created something , then God created everything. God created everything. ——- God created something.
This argument fits the pattern of the formal fallacy, Affirming the Consequent:
If p then q q ——- p.
But the argument also has a valid form:
Every x is such that Cgx ——- Some x is such that Cgx.
(Example adapted from Gerald J. Massey, "The Fallacy behind Fallacies," Midwest Studies in Philosophy VI (1981), pp. 489-500)
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:
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.
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.
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. 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, is an instance of the invalid PC form, p, ergo q.
To think of a valid syllogism as having the invalid form p, q, ergo r 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
All men are mortal Socrates is a man ——- Socrates is mortal
has the invalid logical form
All Fs are Gs a is an H ——- a is a G
But why stop there? By the same 'reasoning,' the Socrates syllogism has the invalid form:
All Fs are Gs a is an H ——- b is an I.
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
Tom is tall ——- Tom is tall
has the valid form p ergo p and the invalid form p ergo q. 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.
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.
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.
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.
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 can have two or more forms, an argument in concreto, i.e. a concrete episode of reasoning cannot have more than one form. If this form is valid the argument in concreto is valid. If invalid, the argument in concreto is invalid. To illustrate:
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.
Now the concrete episode of reasoning embodies a pattern. In the example, I reason in accordance with this pattern:
(x) (Fx –> ~Gx) Fa ——- ~Ga
Which is also representable as follows:
No Fs are Gs a is an F ——- a is not a G.
The pattern or logical form of my concrete episode of reasoning is assuredly not: p, q, ergo r. This is consistent with saying that the argument in abstracto instantiates the invalid form p, q, ergo r in addition to the valid form above.
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. 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.
One must distinguish among the following:
The temporally extended event of Jones' reasoning. This is a particular mental process.
The content of this reasoning process, the argument in abstracto as sequence of propositions.
The concrete episode of reasoning (i.e. the argument in concreto) which involves both the reasoning and its content.
The verbal expression in written or spoken sentences of the argument.
The form or forms of the argument in abstracto.
The verbal expression of a form or forms in a form diagram(s).
The form of the argument in concreto.
My point, again, is that we can uphold the Symmetry Thesis if we make a distinction between arguments in the concrete and arguments in the abstract. But this is a distinction we need in any case. The Symmetry Thesis holds for arguments in the concrete. But these are the arguments that matter because these are the ones people actually give.
Applying this to the Massey example above, we can say that while the abstract argument expressed by the following display has two forms, one invalid, the other valid:
If God created something , then God created everything. God created everything. ——- God created something
there is no one concrete argument, no one concrete episode of reasoning, that the display expresses. One who reasons in a way that is attentive to the internal subpropositional structure of the constituent propositions reasons correctly. But one who ignores this internal structure reasons incorrectly.
In this way we can uphold the Symmetry Thesis and avoid the absurdities to which the Asymmetry Thesis leads.
I read and excerpted the chapter. I am not mistaken. Also, what he says seems correct to me.
He claims that logic is not formal, insofar as it is concerned with the 'laws of thought'. He says "Thought is a psychical phenomenon, and psychical phenomena have no extension. What is meant by the form of an object that has no extension?" I can't fault this.
I take it that the argument is this:
1. Only spatially extended objects have forms. 2. Neither acts of thinking, nor such objects of thought as propositions, are spatially extended. Therefore 3. If logic studies either acts of thinking or objects of thought, then logic is not a formal study, a study of forms.
If this is the argument, I am not impressed. Premise (1) is false. L.'s notion of form is unduly restrictive. There are forms other than shapes. Consider a chord and an arpeggio consisting of the same notes. The 'matter' is the same, the 'form' is different. In a chord the notes sound at the same time; in an arpeggio at different times. The arrangement of the notes is different. Arrangement and structure are forms. Examples are easily multiplied.
Nor, he says, is it the object of logic to investigate how we are thinking or how we ought to think. "The first task belongs to psychology, the second to a practical art of a similar kind to mnemonics". And then he says "Logic has no more to do with thinking than mathematics has". Isn't that correct?
We can agree that logic is not a branch of psychology: it is not an empirical study and its laws are not empirical generalizations. LNC, for example, is not an empirical generalization. But a case can be made for logic's being normative. It does not describe how we do think, but it does prescribe how we ought to think if we are to arrive at truth. If so, then logic does have a practical side and issues hypothetical imperatives, e.g., "If you want truth, avoid contradictions!"
In a similar vein he notes the formalism of Aristotelian logic. The whole Aristotelian theory of the syllogism is built up on the four expressions 'every' (A), 'no' (E), 'some' (I) and 'not every' (O). "It is obvious that such a theory has nothing more in common with our thinking than, for instance, the theory of the relations of greater and less in the field of numbers". Brilliant.
Why do you call it "brilliant"? Husserl and Frege said similar things. It's old hat, isn't it? Psychologism died with the 19th century at least in the mainstream. Given propositions p, q, logic is concerned with such questions as: Does p entail q? Are they consistent? Are they inconsistent? We could say that logic studies certain relations between and among propositions, which are the possible contents of judgings, but are not themselves judgings or entertainings or supposings or anything else that is mental or psychological.
Again, on the need for logic and science to focus on the expression of thought rather than 'thought', he says "Modern formal logic strives to attain the greatest possible exactness. This aim can be reached only by means of a precise language built up of stable, visually perceptible signs. Such a language is indispensable for any science. Our own thoughts not formed in words are for ourselves almost inapprehensible and the thoughts of other people, when not bearing an external shape [my emphasis] could be accessible only to a clairvoyant. Every scientific truth in order to be perceived and verified, must be put into an external form [my emphasis] intelligible to everybody."
I can't fault any of this. What do you think?
Sorry, but I am not impressed. It is fundamentally wrongheaded. First of all this is a howling non sequitur:
1. Logic does not study mental processes; Therefore 2. Logic studies visually perceptive signs.
Surely it is a False Alternative to suppose that logic must either study mental processes or else physical squiggles and such. There is an easy way between the horns: logic studies propositions, which are neither mental nor physical.
In my last post I can gave two powerful arguments why a perceptible string of marks is not identical to the proposition those marks are used to express.
L. speaks of an external form intelligible to everybody. But what is intelligible (understandable) is not the physical marks, but the proposition they express. We both can see this string:
Yash yetmis ish bitmish
but only I know what it means. (Assuming you don't know any Turkish.) Therefore, the meaning (the proposition), is not identical to the physical string.
There is also an equivocation on 'thought' to beware of, as between thinking and object of thought. As you well know, in his seminal essay Der Gedanke Frege was not referring to anything psychological.
I will grant L. this much, however. Until one has expressed a thought, it is not fully clear what that thought is. But I insist that the thought — the proposition — must not be confused with its expression.
The real problem here is that you wrongly think that one is multiplying entities beyond necessity if one makes the sorts of elementary distinctions that I am making.
We can't say that an argument is invalid because it instantiates an invalid form. The argument Socrates is a man; all men are mortal; ergo Socrates is mortal instantiates the invalid form a is F; all Hs are G; ergo a is G, but modulo equivocation, it is truth-preserving. Instantiation of form is just pattern-matching, and the argument does match the pattern of the invalid form.
I reject this of course.The sample argument is an example of correct reasoning. But anyone who argues in accordance with the schema argues incorrectly. Why? Because the schema is not truth-preserving. Therefore the sample argument does not instantiate the invalid form.
I don't think Brightly understands 'truth-preserving.' This is a predicate of argument forms, primarily, and the same goes for 'valid' and 'invalid.' Here are some definitions:
D1. An argument form is truth-preserving =df no argument of that form has true premises and a false conclusion.
D2. An argument form F is valid =df F is truth-preserving.
D3. A particular argument A is valid =df A instantiates a valid form. (This allows for the few cases in which an argument has two forms, one valid and one invalid.)
D4. A particular argument A is invalid =df there is no valid form that it instantiates.
Now what is it for an argument to instantiate an argument form? To answer this question we need to know what an argument is. Since deductive arguments alone are under consideration, I define:
D5. A deductive argument is a sequence of propositions together with the claim that one of them, the conclusion, follows from the others, the premises, taken together.
If the claim holds, the argument is valid; if not, invalid.
Now the main point for present purposes is that an argument is composed of propositions. A proposition is not a complex physical object such as a string of marks on paper. Thus what you literally SEE when you see this:
7 + 5 = 12
is not a proposition, but a spatiotemporal particular, a physical item subject to change: it can be deleted. But the proposition it expresses cannot be deleted by deleting what you just literally SAW. That suffices to show that the proposition expressed by what you saw is not identical to what you saw. Whatever propositions are (and there are different theories), they are not physical items.
What's more, you did not SEE (with your eyes) the proposition, or that it is true, but you UNDERSTOOD the proposition and that it is true. (A proposition and its being true are not the same even if the proposition is true.) So this is a second reason why a proposition is not identical to its physical expression.
Now what holds for propositions also holds for arguments: you cannot delete an argument by deleting physical marks, and you cannot understand an argument merely by seeing a sequence of strings of physical marks.
An argument is not a pattern of physical marks. So there is no question of matching this physical pattern with some other physical pattern. Instantiation of logical form is not just pattern-matching.
If a sentence contains a sign like 'bank' susceptible of two or more readings, then no one definite proposition is expressed by the sentence. Until that ambiguity is resolved one does not have a definite proposition, and without definite propositions no definite argument. But once one has a definite argument then one can assess its validity. If it instantiates a valid form, then it is valid; if it instantiates an invalid form, then it is invalid.
It is as simple as that. But one has to avoid the nominalist mistake of thinking that arguments are just collections of physical items.
It is a well-known and puzzling fact that proper names are ambiguous. According to the US telephone directory, Frodo Baggins is a real person (who lives in Ohio). But according to LOTR, Frodo Baggins is a hobbit. Not a problem. The name ‘Frodo Baggins’ as used in LOTR, clearly has a different meaning from when used to talk about the person in Ohio. So the argument below is invalid:
Frodo Baggins is a hobbit Frodo Baggins is not a hobbit Some hobbit is not a hobbit.
This is because both premisses could be true, but the conclusion could not be true. So your claim that the validity of arguments using fictional names has ‘nothing to do with any semantic property’ is incorrect.
Well, ex contradictione quodlibet. Since anything follows from a contradiction, the conclusion of the above syllogism follows from the premises. So the above argument is valid in that it instantiates a valid argument-form, namely:
p ~p — q
Obviously, there is no argument of the above form that has true premises and a false conclusion. So every argument of that form is valid or truth-preserving.
You invoke a Moorean fact. But we have to be very clear as to the identity of this fact.
It is a Moorean fact that proper names, taken in abstraction from the circumstances of their thoughtful use, are not, well, proper. They are common, or ambiguous as you say. It is no surprise that some dude in Ohio rejoices under the name 'Frodo Baggins.'
But so taken, a name has no semantic properties: it doesn't mean anything. It is just a physical phenomenon, whether marks on paper or a sequence of sounds, etc. Pronounce the sounds corresponding to 'bill,' 'john, 'dick.' Is 'dick' a name or a common noun, and for what? How many dicks in this room? How many detectives? How many penises? How many disagreeable males, 'pricks'? How many men named 'Dick'? Consider the multiple ambiguity of 'There are more dicks than johns in the room but the same number of bills.'
A name that has meaning (whether or not it refers to anything) is always a name used by a mind (not a voice synthesizing machine) in definite circumstances. For example, if the context is a discussion of LOTR, then my use and yours of 'Frodo' has meaning: it means a character in that work, despite the fact that in reality there is no individual named. And as long as we stay in that context, the name has the same meaning.
And the same holds in the context of argument. In your argument above 'Frodo Baggins' has the same meaning in both premises.
You can't have it both ways: you can't maintain that 'Frodo Baggins' is a meaningless string that could mean anything in any occurrence (a fictional character, a real man, his dog, a rock group, a town, etc.) AND that it figures as a term in an argument.
To sum up. Whether a deductive argument is valid or not depends on its logcal form. If there is a valid form it instantiates, then it is valid. The validity of the form is inherited by the argument having that form. But form abstracts from semantic content. So the specific meaning of a name is irrelevant to the evaluation of the validity of an argument in which the name figures. But of course it is always assumed that names are used in the same sense in all of their occurrences in an argument. So only in this very abstract sense is meaning relevant to the assessment of validity.
Cicero was a Roman Tully was a philosopher —– Some Roman was a philosopher.
Quite simply, there is no middle term. The example is an instance of the dreaded quaternio terminorum. But of course we learned at Uncle Willard's knee that Cicero = Tully. Add that fact as a premise and the above argument becomes valid. As a general rule, any invalid argument can be rendered valid by adding one or more premises.
So sameness of reference is not sufficient for sameness of name. 'Cicero' and 'Tully' have the same reference, but they are different names. They are both token- and type-different. Since they are different names, that fact must be accommodated in the form diagram, which looks like this:
Fa Gb — (Ex)(Fx & Gx).
This form is clearly invalid. The most one can squeeze out of these premises using Existential Generalization is '(Ex)Fx & (Ex)Gx.'
It is worth pointing out that the use of the different signs 'a' and 'b' does not entail that a is not identical to b; it leaves open both the possibility that a = b and the possibility that ~(a = b). It is because of the second of these possibilities that the argument-form is invalid.
Commenter Edward Ockham in a comment on the old blog wanted to know why, given that we had to add a premise to make the Cicero argument valid, we don't have to add a premise to make the Alexander argument valid. That argument, from the days when men were men and went around 'seizing' women, proceeds thusly:
Alexander seized Helen Alexander did not seize Helen —– Someone seized and did not seize Helen.
Ockham wants to know why we don't have to add an identity premise to secure the validity of this argument. But what premise would he have us add? It can't be 'Alexander is Alexander' for that is necessarily true and therefore true whether or not both occurrences of 'Alexander' in the original argument are coreferential. Presumably, Ockham wants us to supply '"Alexander" is coreferential in both of its occurrences.' But this goes without saying. There in no need to affirm this in a separate premise since it is implied by the fact that 'Alexander' in both occurrences is a token of the same word-type. We needn't say what is plainly shown. (He said with a sidelong glance in old Ludwig's direction.)
Ockham is bothered by the possibility of equivocation. Well, either there is an equivocation on 'Alexander' or there isn't. If there is an equivocation, then the argument instantiates an invalid form, and Ockham's contention collapses. If there is no equivocation, then the argument instantiates a valid form but it is not the case that both premises are true; so again Ockham's contention collapses. Either way, his contention collapses.
Either we capture the reference [of a name] in the form, and my objection collapses. Or you concede that the form covers only the visible or audible outward form of the word. In which case, my specious Alexander argument really does have the right form, and we have to add on the condition about reference, and my point stands.
I grasp something like the first horn. If 'a' occurs two or more times in a form diagram, then no argument of that form has an equivocation on a term whose place is held by 'a.' This is to say that the form diagram enforces coreferentiality on any terms whose place is held by 'a' in the form schema. Otherwise, the argument would not be of the form in question.
Ockham wants to have it both ways at once. He wants his argument A to be of valid form F without F enforcing coreferentiality on the occurrences of 'a' in A. This is just impossible. If there is an equivocation on 'a,' then A does not instantiate F. But if A does instantiate F, then there cannot be any equivocation of 'a.' Why? Because the form does not permit it. The form enforces coreferentiality.
Now look back at the Cicero argument. It is invalid because its form (depicted above) is invalid and the argument has no valid form. But I don't say that the invalid form enforces lack of coreferentiality on the singular terms whose place is held in the diagram by 'a' and 'b.' I say instead that the invalid form permits coreferentiality of these terms. Thus there is an asymmetry between the Alexander and Cicero cases.
I demanded an argument valid in point of logical form all of whose premises are purely factual but whose conclusion is categorically (as opposed to hypothetically or conditionally) normative. Recall that a factual proposition is one which, whether true or false, purports to record a fact, and that a purely factual proposition is a factual proposition containing no admixture of normativity.
My demand is easily, if trivially, satisfied.
Ex contradictione quodlibet. From a contradiction anything, any proposition, follows. This is rigorously provable within the precincts of the PC (the propositional calculus). As follows:
1. p & ~p 2. p (from 1 by Simplification) 3. p v q (from 2 by Addition) 4. ~p & p (from 1 by Commutation) 5. ~p (from 4 by Simplification) 6. q (from 3, 5 by Disjunctive Syllogism)
Now plug in 'Obama is a liar' for p and 'One ought to be kind to all sentient beings' for q. The result is:
Obama is a liar Obama is not a liar Ergo One ought to be kind to all sentient beings.
My demands have been satisifed. The above is an argument valid in point of logical form whose premises are all purely factual and whose conclusion is categorically normative.
I thank Tully Borland for pushing the discussion in this fascinating direction.
A
Affirming the Consequent is an invalid argument form. Ergo One ought not (it is obligatory that one not) give arguments having that form.
B
Modus Ponens is valid Ergo One may (it is permissible to) give arguments having that form.
C
Correct deductive reasoning is in every instance truth-preserving. Ergo One ought to reason correctly as far as possible.
An argument form is valid just in case no (actual or possible) argument of that form has true premises and a false conclusion. An argument form is invalid just in case some (actual or possible) argument of that form has true premises and a false conclusion. Deductive reasoning is correct just in case it proceeds in accordance with a valid argument form. 'Just in case' is but a stylistic variant of 'if and only if.'
Now given these explanations of key terms, it seems that validity, invalidity, and correctness are purely factual, and thus purely non-normative, properties of arguments/reasonings. If so, how the devil do we get to the conclusions of the three arguments above?
View One: We don't. A, B, and C are each illicit is-ought slides.
View Two: Each of the above arguments is valid. Each of the key terms in the premises is normatively loaded from the proverbial 'git-go,' in addition to bearing a descriptive load.. Therefore, there is no illict slide. The move is from the normative to the normative. Validity, invalidity, and correctness can be defined only in terms of truth and falsity which are normative notions.
View Three: We have no compelling reason to prefer one of the foregoing views to the other. Each can be argued for and each can be argued against. Thus spoke the Aporetician.
Consider the argument: Bill is a brother —– Bill is a sibling.
Is this little argument valid or invalid? It depends on what we mean by 'valid.' Intuitively, the argument is valid in the following generic sense:
D1. An argument is (generically) valid iff it is impossible that its premise(s) be true and its conclusion false.
(D1) may be glossed by saying that there are no possible circumstances in which the premises are true and the conclusion false. Equivalently, in every possible circumstance in which the premises are true, the conclusion is true. In short, validity is immunity to counterexample.
(D1), though correct as far as it goes, leaves unspecified the source or ground of a valid argument's validity. This is the philosophically interesting question. What makes a valid argument valid? What is the ground of the impossibility of the premises' being true and the conclusion being false? One answer is that the source of validity is narrowly logical or purely syntactic: the validity of a valid argument derives from its subsumability under logical laws or (what comes to the same thing) its instantiation of valid argument-forms.
Now it is obvious that the validity of the above argument does not derive from its logical form. The logical form is
Fa —– Ga
where 'a' is an arbitrary individual constant and 'F' an arbitrary predicate constant. The above argument-form is invalid since it is easy to interpret the place-holders so as to make the premise true and the conclusion false: let 'a' stand for Al, 'F' for fat and 'G' for gay.
We now introduce a second, specific sense of 'valid,' one that alludes to the source of validity:
D2. An argument is syntactically valid iff it is narrowly-logically impossible that there be an argument of that form having true premises and a false conclusion.
According to (D2), a valid argument inherits its validity from the validity of its form, or logical syntax. So on (D2) it is primarily argument-forms that are valid or invalid; arguments are valid or invalid only in virtue of their instantiation of valid or invalid argument-forms. (D2) is thus a specification of the generic (D1).
But there is a second specification of (D1) according to which validity/invalidity has its source in the constituent propositions of the arguments themselves and so depends on their extra-syntactic content:
D3. An argument is extra-syntactically valid iff (i) it is impossible that its premises be true and its conclusion false; and (ii) this impossibility is grounded neither in any contingent matter of fact nor in logic proper, but in some necessary connection between the senses or the referents of the extra-logical terms of the argument.
A specification of (D3) is
D4. An argument is semantically valid iff (i) if it is impossible that its premises be true and its conclusion false; and (ii) this impossibility is grounded in the senses of the extra-logical terms of the argument.
Thus to explain the semantic validity of the opening argument we can say that the sense of 'brother' includes the sense of 'sibling.' There is a necessary connection between the two senses, one that does not rest on any contingent matter of fact and is also not mediated by any law of logic. Note that logic allows (does not rule out) a brother who is not a sibling. Logic would rule out a non-sibling brother only if 'x is F & x is not G' had only false substitution-instances — which is not the case. To put it another way, a brother that is not a sibling is a narrowly-logical possibility. But it is not a broadly-logical possibility due to the necesssary connection of the two senses.
So it looks as if analytic entailments like Bill is a brother, ergo, Bill is a sibling show that subsumability under logical laws is not necessary for (generically) valid inference. Sufficient, but not necessary. Analytic entailments appear to be counterexamples to the thesis that inferences in natural language can be validated only by subsumption under logical laws.
One might wonder what philosophers typically have in mind when they speak of validity. I would say that most philosophers today have in mind (D1) as specified by (D2). Only a minority have in mind (D3) and its specification (D4). I could easily be wrong about that. Is there a sociologist of philosophers in the house?
Consider the Quineans and all who reject the analytic/synthetic distinction. They of course will have no truck with analytic entailments and talk of semantic validity. Carnapians, on the other hand, will uphold the analytic/synthetic distinction but validate all entailments in the standard (derivational) way by importing all analytic truths as meaning postulates into the widened category of L-truths.
Along broadly Carnapian lines one could argue that the above argument is an enthymeme which when spelled out is
Every brother is a sibling Bill is a brother —– Bill is a sibling.
Since this expanded argument is syntactically valid, the original argument — construed as an enthymeme — is also syntactically valid. When I say that it is syntactically valid I just mean that the conclusion can be derived from the premises using the resources of standard logic, i.e. the Frege-inspired predicate calculus one finds in logic textbooks such as I. Copi's Symbolic Logic. In the aboveexample, one uses two inference rules, Universal Instantiation and Modus Ponens, to derive the conclusion.
If this is right, then the source of the argument's validity is not
in a necessary connection between the senses of the 'brother' and 'sibling' but in logical laws.
Here is a little puzzle I call the Stromboli Puzzle. An earlier post on this topic was defective. So I return to the topic. The puzzle brings out some of the issues surrounding existence. Consider the following argument.
Stromboli exists. Stromboli is an island volcano. Ergo An island volcano exists.
This is a sound argument: the premises are true and the reasoning is correct. It looks to be an instance of Existential Generalization. How can it fail to be valid? But how can it be valid given the equivocation on 'exists'? 'Exists' in the conclusion is a second-level predicate while 'exists' in the initial premise is a first-level predicate. Although Equivocation is standardly classified as an informal fallacy, it induces a formal fallacy. An equivocation on a term in a syllogism induces the dreaded quaternio terminorum, which is a formal fallacy. Thus the above argument appears invalid because it falls afoul of the Four Term Fallacy.
Objection 1. "The argument is valid without the first premise, and as you yourself have pointed out, a valid argument cannot be made invalid by adding a premise. So the argument is valid. What's your problem?"
Reply 1. The argument without the first premise is not valid. For if the singular term in the argument has no existing referent, then the argument is a non sequitur. If 'Stromboli' has no referent at all, or has only a nonexisting Meinongian referent, then Existential Generalization could not be performed, given, as Quine says, that "Existence is what existential quantification expresses."
Objection 2: "The first premise is redundant because we presuppose that the domain of quantification is a domain of existents."
Reply 2: Well, then, if that is what you presuppose, then you can state your presupposition by writing, 'Stromboli exists.' Either the argument without the first premise is an enthymeme or it is invalid. If it is an enthymeme, then we need the first premise to make it valid. If it is invalid, then it is invalid.
Therefore, we are stuck with the problem of explaining how the original argument is valid, which it surely is.
My answer is that the original argument is an enthymeme an unstated premise of which links the first- and second-level uses of 'exist(s)' and thus presupposes the admissibility of the first-level uses. Thus we get:
A first-level concept F exists (is instantiated) iff it is instantiated by an individual that exists in the first-level way. Stromboli is an individual that exists in the first-level way. Stromboli is an island volcano. Ergo The concept island volcano exists (is instantiated). Ergo And island volcano exists.
Now what does this rigmarole show? It shows that Frege and Russell were wrong. It shows that unless we admit as logically kosher first-level uses of 'exist(s)' and cognates, a simple and obviously valid argument like the the one with which we started cannot be made sense of.
'Exists(s)' is an admissible predicate of individuals, and existence belongs to individuals: it cannot be reduced to, or eliminated in favor of, instantiation. This has important consequences for metaphysics.
Nicholas Rescher cites this example from Buridan. The proposition is false, but not self-refuting. If every proposition is affirmative, then of course *Every proposition is affirmative* is affirmative. The self-reference seems innocuous, a case of self-instantiation. But *Every proposition is affirmative* has as a logical consequence *No proposition is negative.* This follows by Obversion, assuming that a proposition is negative if and only if it is not affirmative.
Paradoxically, however, the negative proposition, unlike its obverse, is self-refuting. For if no proposition is negative then *No proposition is negative* is not negative. So if it is, it isn't. Plainly it is. Ergo, it isn't.
Rescher leaves the matter here, and I'm not sure I have anything useful to add.
It is strange, though, that here we have two logically equivalent propositions one of which is self-refuting and the other of which is not. The second is necessarily false. If true, then false; if false, then false; ergo, necessarily false. But then the first must also be necessarily false. After all, they are logically equivalent: each entails the other across all logically possible worlds.
What is curious, though, is that the ground of the logical necessity seems different in the two cases. In the second case, the necessity is grounded in logical self-contradiction. In the first case, there does not appear to be any self-contradiction.
It is impossible that every proposition be affirmative. And it is impossible that no proposition be negative. But whereas the impossibility of the second is the impossibility of self-referential inconsistency, the impossibility of the first is not. (That is the 'of' of apposition.)
Can I make an aporetic polyad out of this? Why not?
1. Logically equivalent logically impossible propositions have the same ground of their logical impossibility.
2. The ground of the logical impossibility of *Every proposition is affirmative* is not in self-reference.
3. The ground of the logical impossibility of *No proposition is negative* is in self-reference.
The limbs of this antilogism are individually plausible but collectively inconsistent.
REFERENCES
Nicholas Rescher, Paradoxes: Their Roots, Range, and Resolution, Open Court, 2001, pp. 21-22.
G. E. Hughes, John Buridan on Self-Reference, Cambidge UP, 1982, p. 34. Cited by Rescher.
I had the pleasure of meeting London Ed, not in London, but in Prague, in person, a few days ago. Ed, a.k.a. 'Ockham,' and I have been arguing over existence for years. So far he has said nothing to budge me from my position. Perhaps some day he will. The following entry, from the old Powerblogs site, whose archive is no more, was originally posted 25 May 2008. Here it is again slightly redacted.
…………….
I am racking my brains over the question why commenter 'Ockham' cannot appreciate that standard quantificational accounts of existence presuppose rather than account for singular existence. It seems so obvious to me! Since I want to put off as long as possible the evil day when I will have to call him existence-blind, I will do my level best to try to understand what he might mean.
Consider the following renditions of a general and a singular existence statement respectively, where 'E' is the 'existential' or, not to beg any questions, the particular quantifier:
1. Cats exist =df (Ex)(x is a cat)
2. Max (the cat) exists =df (Ex)(x = Max).
Objectually as opposed to substitutionally interpreted, what the right-hand sides of (1) and (2) say in plain English is that something is a cat and that something is (identical to) Max, respectively. Let D be the domain of quantification. Now the right-hand side (RHS) of (1) is true iff at least one member of D is a cat. And the RHS of (2) is true iff exactly one member of D = Max. Now is it not perfectly obvious that the members of D must exist if (1) and (2) are to be true? To me that is obvious since if the members of D were Meinongian nonexistent items, then (1) and (2) would be false. (Bear in mind that there is no logical bar to quantifying over Meinongian objects, whatever metaphysical bar there might be. Meinongians, and there are quite a few of them, do it all the time with gusto.)
Therefore, 'Something is a cat' is a truth-preserving translation of 'Cats exist' only if 'Something is a cat' is elliptical for 'Something that exists is a cat.' And similarly for 'Something is Max.' But here is where 'Ockham' balks. He sees no difference between 'something' and 'something that exists' where I do see a difference.
I am sorely tempted to call anyone who cannot understand this difference 'existence-blind' and cast him into the outer darkness, that place of fletus et stridor dentium, along with qualia-deniers, eliminative materialists, deniers of modal distinctions, and the rest of the terminally benighted. But I will resist this temptation for the moment.
And were I to label 'Ockham' existence-blind he might return the 'compliment' by saying that I am hallucinating, or suffering from double-vision. "You've drunk so much Thomist Kool-Aid that you see a distinction where there isn't one!" But then we get a stand-off in which we sling epithets at each other. Not good for those of us who would like to believe in the power and universality of reason. It should be possible for one of us to convince the other, or failing that, to prove that the issue is rationally undecidable.
The issue that divides us may be put as follows. (Of course, it may be that we have yet to locate the exact bone of contention, and in our dance around each other we have succeeded only in 'dislocating' it.)
BV: Because the items in the domain of quantification exist, there has to be more to existence than can be captured by the so-called 'existential' quantifier. Existence is not a merely logical topic. Pace Quine, it is not the case that "Existence is what existential quantification expresses." Existence is a 'thick' topic: there is room for a metaphysics of existence. One can legitimately ask: What is it for a concrete contingent individual to exist? and one can expect something better than the blatantly circular, 'To exist is to be identical to something.' To beat on this drum one more time, this is a circular explanation because D is a domain all of whose members exist. One moves in a circle of embarrassingly short diameter if one maintains that to exist is to be identical to something that exists. Note that I wrote circular explanation, not circular definition. Note also that I am assuming that there is such a thing as philosophical explanation, which is not obvious, and is denied by some.
O: Pace BV, the items in the domain of quantification admit of no existence/nonexistence contrast. Therefore, 'Something is a cat' is indistinguishable from 'Something that exists is a cat.' There is no difference at all between 'something' and 'something that exists,' and 'something' is all we need. Now 'something' is capturable without remainder using the resources of standard first-order predicate logic with identity. 'Exist(s)' drops out completely. There is no (singular) existence and there are no (singular) existents. There are just items, and one cannot distinguish an item from its existence.
Now if that is what O means, then I understand him, but only on the assumption that for individuals
3. Existence = itemhood.
For if to exist = to be an item, if existence reduces to itemhood, then there cannot be an existence/nonexistence contrast at the level of items. It is a logical truth that every item is an item, and therefore an item that is not an item would be a contradiction: 'x is an item' has no significant denial. Therefore, on the assumption that existence = itemhood, there is no difference between 'Some item is a cat' and 'Some item that exists is a cat.' And if there is no such difference, then existence is fully capturable by the quantifier apparatus.
But now there is a steep price to pay. For now we are quantifying over items and not over existents, and sentences come out true that ought not come out true. 'Dragons exist,' for example, which is false, becomes 'Some item is a dragon' which is true. To block this result, O would have to recur to a first-level understanding of existence as contrasting with nonexistence. He would have to say that every item exists, that there are no nonexisting items. But then he can no longer maintain that 'something' and 'something that exists' are indistinguishable.
In defiance of Ed's teacher, C. J. F. Williams, I deny that the philosophy of existence must give way to the philosophy of someness. (Cf. the latter's What is Existence? Oxford, 1981, p. 215) The metaphysics of existence cannot be supplanted by the logic of 'exist(s).' Existence is not a merely logical topic.
Here is an obituary of Williams written by Richard Swinburne.
