Numa entrevista sobre seu livro, Words Without Meaning, Christopher Gauker afirma que a seguinte inferência é um contra-exemplo da lei de instanciação universal em inglês:
(1) Everyone is present.
(2) Therefore Wladmir Putin is present.
(1) é afirmado por um membro de um comitê que está prestes a iniciar uma reunião com todos os seus membros. Wladmir Putin não faz parte desse comitê. Gauker afirma que, nesse contexto, (1) é verdadeira e (2) é falsa. A lei de instanciação pode ser expressa assim:
(x)(Fx) |- Fa
Mas eu diria que, nas circunstâncias em que (1) foi proferida, ela não está expressando algo cuja forma lógica seja (x)(present(x)), mas algo cuja forma é (x)(member of the committee(x) -> presente(x)). O predicado "member of the committee(x)" está implicito, naquele contexto, na expressão "Everyone". Gauker trata a palavra "Everyone" como se fosse uma variação notacional do quantificador "(x)"; como se fosse sempre correto traduzir "Everyone" por "(x)", independentemente do contexto de uso de "Everyone". Sendo assim, a forma do argumento (1)-(2) é a seguinte:
(1') (x)(member of the committee(x) -> present(x))
(2') Therefore, present(Wladmir Putin)
Que, obviamente, não é um caso de instanciação universal.
Formalização é tradução.
Clique no título da postagem para ler as objeções do Prof. Gauker e minhas réplicas.
Dear Prof. Machado: Thank you for bringing my objection to Universal Instantiation to the attention of your audience. I'm afraid I can't accept your defense. (I hope I have not misunderstood the Portugese, which I don't speak.) Certainly, "Everyone on the committee is present" is not a translation in the usual sense of "Everyone is present." No doubt the utterance of "Everyone on the committee" expresses in context the proposition that everyone on the committee is present. But that fact is not a defense of Universal Instantiation; it's just a way of explaining the counterexample. Perhaps I should emphasize that Universal Instantiation is a formal rule of inference; it pertains to sentences, not propositions. In any case, what I am objecting to is only a formal rule of inference pertaining to sentences.
Dear Professor Gauker: Thank you for your kind reply. I think you understood perfectly what I have said, although I still tend to think I am right. But before I answer your reply, let me ask you a question: are you saying that Universal Instantiation is a grammatical rule which does not take into account possible differences between grammatical form and logical form of sentences?
Prof. Machado: I do not take for granted that there are formal rules of logic for natural languages at all. Rather, there may be semantic relations of logical consequence only, and that is something different. But if one supposes (which I do not) that there are formal rules of inference for natural language, they will be defined over the syntactic structures of the sentences of natural language. If we spoke a language with a syntax like the syntax of the artificial languages of formal logic, UI would not be valid in it, because even in such a language we would consider "For all x, x is present" to be true even in contexts lacking Vladimir Putin. Granted, a theoretical representation of the syntactic structure of a sentence of language may bring to the surface lexical material that does not appear in what is spoken or written. But I think it's pretty clear that "on the committee" does not belong to that additional structure that might be revealed through syntactic analysis in the sentence "Everyone is present." Do you seriously contend that "Everyone on the committee is present" is a translation of "Everyone is present"?
Prof. Gauker: Once more, thank you for your comment. UI is a logical rule, not a grammatical rule. There is no purely syntactical grammatical rule equivalent to UI in English. That is why no competent English speaker would infer "Wladmir Putin is here" from "Everyone is here" in the context you described (except as a joke). A competent English speaker knows that "Everyone is here" is not always used to talk about every person in the world. I think we agree on that much.
But if one wants to know whether UI is or is not valid in English, then one has to pay attention to what one says by uttering sentences, not only to their purely syntactical grammatical structure. The logical form of a sentece is the logical form of what is said by uttering it. And what one says by uttering a sentence may vary from context to context. If I were in the final judgment and said "Everyone is here" I would be saying something different from what the president of the committee says in your example. If the president said "Everyone in the committee is here" she would be saying the same thing. Note that the concept of a person is implicit in the use of "everyone" in your example. Why not the concept of a being member of the commettee? Summing up: I don't think that "Everyone in the commettee is here" is the correct translation of "Everyone is here" regardless the context in which the last sentence is used. In the context you decribed, I think it is.
Dear Prof. Machado: Your claim seems to be that the correct procedure for evaluating an argument for validity is to first use some kind of psychological reasoning to figure out what the speaker had in mind in speaking the premises and conclusion, write that out in the form of eternal sentences (sentences whose truth value does not vary from context to context) and then evaluate the resulting argument in the usual way. I don't want to settle philosophical questions by taking a vote, but I think it is at least noteworthy that one will not find such an account in any of the standard logic textbooks or in the contemporary journal literature in semantics. But moreover, it is not a workable procedure because we do not think in eternal sentences either. Even our thoughts have their semantic values only relative to a context of thought. (This is something that I have argued for in some detail in a paper in Mind from 1997 and in the fourth chapter of Words without Meaning.)
Dear Prof. Gauker: I am not sure I can recognize what I said in your rephrasing. What I am saying is way much simpler than the sophisticated theory you sketched. As much as we have to pay attention to what is said to decide whether the sentence,
(1) The whale is a mammal.
has this logical form
Ma
or this
(x)(Wx -> Mx)
in order to decide whether the following inference is or is not valid
(1) The whale is a mammal. (2) Therefore, something is a mammal.
we have to pay attention to what is said by the president of the committee in order to decide what is the logical form of the sentence “Everyone is here” in order to evaluate the validity of her inference. The sentence “The whale is a mammal” does not have the same logical form as “The capital of Bahia is a big city”, although both sentences have the same grammatical form: “The S is a P”. “The whale is a mammal” has the same logical form as “All snakes are reptiles”, although their grammatical form differ from each other. Even if one disagrees on what I have just said about these examples, the disagreement would be based on and should be settled by an analysis of what is said by means of the utterance of these sentences, not by regarding only the grammatical form of the sentences.
In order to evaluate the validity of an inference, one does not have to rewrite it in eternal sentences. I don’t need such a thing in order to know that “This spot is completely black; therefore it is not completely white” is valid. But if one wants to know whether it is or is not an instance of a formal rule of inference which one expresses in a formal language, then one has to take into account the logical form of the inference and to do that one has to take into account what is said by means of the utterance of the sentences of the inference, for one has to translate them into that formal language. What I have in mind here is something like the point raised by V. McGee in “A counterexample to modus ponens” (Journal of Philosophy, 82, 1982) and specially Frank Jackson’s answer in “Conditionals” (The Backwell Guide to the Philosophy of Language, chap. 11). You wrote a book on conditionals. So you must know this discussion much better than me. “The best reply, in my view, to this argument points out that we sometimes need to do a certain amount of massaging of the surface linguistic structure in order to display logical form.” (p. 216)
Prof. Machado: OK, I can concede that I was wrong to interpret you as proposing to use psychology to decide the form of an argument. I reply instead as follows:
We don't need to worry about the syntax of Portugese or English. I claim that UI would be invalid even in a language with the syntax of the artificial languages of logic. So, AxFx/Fa is an instance of UI but invalid. The premise may be true in a context in which every object in the contextually-determined domain of discourse is in the extension of F, while the conclusion is false in that same context because the denotation of a is not in the extension of F.
You do not deny that the premise may be true while the conclusion is false in such a context. You claim that the argument is not an instance of UI. It is not, you say, because in order to decide whether a pair of sentences is an instance UI, we must find certain other sentences and decide whether they constitute an instance of UI. To which I reply: If AxFx/Fa is not an instance of UI even though it has the proper form, then UI is not a form of argument after all and in that case I no longer know what it is and do not know how to tell whether that other pair of sentences is an instance of UI or not.
Prof. Gauker: First I would like to say that it has been a pleasure to discuss with you. You have made me think harder and harder on the subject.
If in a certain context my domain of discourse is not the whole universe, but something smaller, then my expression in a logical notation should make that clear. As far as I know, one can do it either by means of a special kind o variable -- which is done in mathematics by means of “n” and “m”, for instance -- or by means of conditionalization. If my domain contains only numbers, for instance, then I can express a universal quantification on that domain by saying
(x)(Nx -> Fx)
for instance, where “Nx” is the predicate “x is a number”. In either case I would have a valid instance of UI.
(n)((Em)(m is the successor of n)) (Em)(m is the successor of 0)
(x)(Nx -> Ey(Ny & y is the successor of x)) Ey(Ny & y is the successor of 0)
UI says that we can infer an instance of a universal quantification. But the instance is determined by the domain of the quantifier, so that if the domain of (x) is D, then an instance of (x)(Fx) is only those sentences resulting from replacing the free variable in Fx by a name of something in D. If one says that (x)(Fx) and then says that Fa, where a is a name of something that is not in D, then one is not following UI. -- What I said about the difference between logical form and grammatical form is true also for the sentences of artificial formal languages.
You have described a context in which the sentence (x)(Fx) is true and the sentence Fa is false. But I don’t think you have described a context in which Fa is an instance of (x)(Fx).
Dear Prof. Machado: You write: "If one says that (x)(Fx) and then says that Fa, where a is a name of something that is not in D, then one is not following UI." I am sorry, but I cannot agree. UI is a formal rule. UI is not formulated in semantic terms. It says nothing of denotations. Yes, for any given universal quantification, if it is true in a context, then we may drop the outer quantifier and substitute a name of any object in the domain and the resulting sentence will be true. But that is not the rule UI and not what I intended to criticize.
Dear Prof. Gauker: imagine a context C in which (x)(Gx) is equivalent to (x)(Fx -> Gx), for the domain of (x) in C is all Fs. Now suppose that a is the name of something such that ~Fa. According to what you said, if in that context one infers Ga from (x)(Gx), then one would be following UI. If that was true, then UI would be blind for the logical form of the inference. I would say that UI would not be, then, a logical rule of inference. But UI is introduced in logic as a logical rule of inference.
If the domain of (x) in C is all Fs, then in a logical notation one should make that clear, just to avoid thinking that something is an instance of a quantification when it is not. So to express what is said in C in logical notation by means of (x)(Gx) would be failing to fulfill this requirement of a good expression of what is said in logical notation. If we are going to leave the domain presupposed in the use of logical notation, as we do in ordinary language, whereas we can make it explicit, then I can see no advantage in the use logical notation. If it does not itch, one does not scratch. But if it does itch, one should scratch.
Prof. Gauker: Commenting the fact that we can translate English sentences into logical symbolism by means of differente translation schemes, C. Allen and M. Hand say:
"The choice of whether to represent English phrases with one-place or many-place predicates is dependent on the degree of structure that must be included in order for an argument to be analyzed adequately. In general, more detail is better than less detail, since arguments may be labeled invalid erroneously if insufficient detail is represented." (Logic Primer, Harvard: MIT, 2001, p. 69)
One of the details, of course, is the relation between a quantification and its instances.
Dear Prof. Machado: Thank you for your good-natured perseverance. I'm afraid we've hit an impasse. I think it is undeniable that AxFx/Fa is an instance of UI. The quotation from Allen and Hand is about the adicity of predicates, which is not at issue here.
Dear Prof. Gauker: Thank you for your patience in answering my objections. I agree that we've hit an impasse. But let me just explain why I have quoted Allen and Hand. I agree that they are talking about something else. But what I was trying to illustrate by means of that quotation was the attitude towards formalization they express in the final sentence. I just didn't want to quote it completely out of context. Such an attitude is what has been animating my answers to your replies, like the one just before the quotation. I will read carefully your paper on UI in order to give a more substantial account of my disagreement, or else to end up agreeing with you!
Dear Prof. Machado: Thank you for bringing my objection to Universal Instantiation to the attention of your audience. I'm afraid I can't accept your defense. (I hope I have not misunderstood the Portugese, which I don't speak.) Certainly, "Everyone on the committee is present" is not a translation in the usual sense of "Everyone is present." No doubt the utterance of "Everyone on the committee" expresses in context the proposition that everyone on the committee is present. But that fact is not a defense of Universal Instantiation; it's just a way of explaining the counterexample. Perhaps I should emphasize that Universal Instantiation is a formal rule of inference; it pertains to sentences, not propositions. In any case, what I am objecting to is only a formal rule of inference pertaining to sentences.
ResponderExcluirDear Professor Gauker: Thank you for your kind reply. I think you understood perfectly what I have said, although I still tend to think I am right. But before I answer your reply, let me ask you a question: are you saying that Universal Instantiation is a grammatical rule which does not take into account possible differences between grammatical form and logical form of sentences?
ResponderExcluirProf. Machado: I do not take for granted that there are formal rules of logic for natural languages at all. Rather, there may be semantic relations of logical consequence only, and that is something different. But if one supposes (which I do not) that there are formal rules of inference for natural language, they will be defined over the syntactic structures of the sentences of natural language. If we spoke a language with a syntax like the syntax of the artificial languages of formal logic, UI would not be valid in it, because even in such a language we would consider "For all x, x is present" to be true even in contexts lacking Vladimir Putin. Granted, a theoretical representation of the syntactic structure of a sentence of language may bring to the surface lexical material that does not appear in what is spoken or written. But I think it's pretty clear that "on the committee" does not belong to that additional structure that might be revealed through syntactic analysis in the sentence "Everyone is present." Do you seriously contend that "Everyone on the committee is present" is a translation of "Everyone is present"?
ResponderExcluirProf. Gauker: Once more, thank you for your comment. UI is a logical rule, not a grammatical rule. There is no purely syntactical grammatical rule equivalent to UI in English. That is why no competent English speaker would infer "Wladmir Putin is here" from "Everyone is here" in the context you described (except as a joke). A competent English speaker knows that "Everyone is here" is not always used to talk about every person in the world. I think we agree on that much.
ResponderExcluirBut if one wants to know whether UI is or is not valid in English, then one has to pay attention to what one says by uttering sentences, not only to their purely syntactical grammatical structure. The logical form of a sentece is the logical form of what is said by uttering it. And what one says by uttering a sentence may vary from context to context. If I were in the final judgment and said "Everyone is here" I would be saying something different from what the president of the committee says in your example. If the president said "Everyone in the committee is here" she would be saying the same thing. Note that the concept of a person is implicit in the use of "everyone" in your example. Why not the concept of a being member of the commettee? Summing up: I don't think that "Everyone in the commettee is here" is the correct translation of "Everyone is here" regardless the context in which the last sentence is used. In the context you decribed, I think it is.
Dear Prof. Machado: Your claim seems to be that the correct procedure for evaluating an argument for validity is to first use some kind of psychological reasoning to figure out what the speaker had in mind in speaking the premises and conclusion, write that out in the form of eternal sentences (sentences whose truth value does not vary from context to context) and then evaluate the resulting argument in the usual way. I don't want to settle philosophical questions by taking a vote, but I think it is at least noteworthy that one will not find such an account in any of the standard logic textbooks or in the contemporary journal literature in semantics. But moreover, it is not a workable procedure because we do not think in eternal sentences either. Even our thoughts have their semantic values only relative to a context of thought. (This is something that I have argued for in some detail in a paper in Mind from 1997 and in the fourth chapter of Words without Meaning.)
ResponderExcluirDear Prof. Gauker: I am not sure I can recognize what I said in your rephrasing. What I am saying is way much simpler than the sophisticated theory you sketched. As much as we have to pay attention to what is said to decide whether the sentence,
ResponderExcluir(1) The whale is a mammal.
has this logical form
Ma
or this
(x)(Wx -> Mx)
in order to decide whether the following inference is or is not valid
(1) The whale is a mammal.
(2) Therefore, something is a mammal.
we have to pay attention to what is said by the president of the committee in order to decide what is the logical form of the sentence “Everyone is here” in order to evaluate the validity of her inference. The sentence “The whale is a mammal” does not have the same logical form as “The capital of Bahia is a big city”, although both sentences have the same grammatical form: “The S is a P”. “The whale is a mammal” has the same logical form as “All snakes are reptiles”, although their grammatical form differ from each other. Even if one disagrees on what I have just said about these examples, the disagreement would be based on and should be settled by an analysis of what is said by means of the utterance of these sentences, not by regarding only the grammatical form of the sentences.
In order to evaluate the validity of an inference, one does not have to rewrite it in eternal sentences. I don’t need such a thing in order to know that “This spot is completely black; therefore it is not completely white” is valid. But if one wants to know whether it is or is not an instance of a formal rule of inference which one expresses in a formal language, then one has to take into account the logical form of the inference and to do that one has to take into account what is said by means of the utterance of the sentences of the inference, for one has to translate them into that formal language. What I have in mind here is something like the point raised by V. McGee in “A counterexample to modus ponens” (Journal of Philosophy, 82, 1982) and specially Frank Jackson’s answer in “Conditionals” (The Backwell Guide to the Philosophy of Language, chap. 11). You wrote a book on conditionals. So you must know this discussion much better than me. “The best reply, in my view, to this argument points out that we sometimes need to do a certain amount of massaging of the surface linguistic structure in order to display logical form.” (p. 216)
Prof. Machado: OK, I can concede that I was wrong to interpret you as proposing to use psychology to decide the form of an argument. I reply instead as follows:
ResponderExcluirWe don't need to worry about the syntax of Portugese or English. I claim that UI would be invalid even in a language with the syntax of the artificial languages of logic. So, AxFx/Fa is an instance of UI but invalid. The premise may be true in a context in which every object in the contextually-determined domain of discourse is in the extension of F, while the conclusion is false in that same context because the denotation of a is not in the extension of F.
You do not deny that the premise may be true while the conclusion is false in such a context. You claim that the argument is not an instance of UI. It is not, you say, because in order to decide whether a pair of sentences is an instance UI, we must find certain other sentences and decide whether they constitute an instance of UI. To which I reply: If AxFx/Fa is not an instance of UI even though it has the proper form, then UI is not a form of argument after all and in that case I no longer know what it is and do not know how to tell whether that other pair of sentences is an instance of UI or not.
Prof. Gauker: First I would like to say that it has been a pleasure to discuss with you. You have made me think harder and harder on the subject.
ResponderExcluirIf in a certain context my domain of discourse is not the whole universe, but something smaller, then my expression in a logical notation should make that clear. As far as I know, one can do it either by means of a special kind o variable -- which is done in mathematics by means of “n” and “m”, for instance -- or by means of conditionalization. If my domain contains only numbers, for instance, then I can express a universal quantification on that domain by saying
(x)(Nx -> Fx)
for instance, where “Nx” is the predicate “x is a number”. In either case I would have a valid instance of UI.
(n)((Em)(m is the successor of n))
(Em)(m is the successor of 0)
(x)(Nx -> Ey(Ny & y is the successor of x))
Ey(Ny & y is the successor of 0)
UI says that we can infer an instance of a universal quantification. But the instance is determined by the domain of the quantifier, so that if the domain of (x) is D, then an instance of (x)(Fx) is only those sentences resulting from replacing the free variable in Fx by a name of something in D. If one says that (x)(Fx) and then says that Fa, where a is a name of something that is not in D, then one is not following UI. -- What I said about the difference between logical form and grammatical form is true also for the sentences of artificial formal languages.
You have described a context in which the sentence (x)(Fx) is true and the sentence Fa is false. But I don’t think you have described a context in which Fa is an instance of (x)(Fx).
Dear Prof. Machado: You write: "If one says that (x)(Fx) and then says that Fa, where a is a name of something that is not in D, then one is not following UI." I am sorry, but I cannot agree. UI is a formal rule. UI is not formulated in semantic terms. It says nothing of denotations. Yes, for any given universal quantification, if it is true in a context, then we may drop the outer quantifier and substitute a name of any object in the domain and the resulting sentence will be true. But that is not the rule UI and not what I intended to criticize.
ResponderExcluirDear Prof. Gauker: imagine a context C in which (x)(Gx) is equivalent to (x)(Fx -> Gx), for the domain of (x) in C is all Fs. Now suppose that a is the name of something such that ~Fa. According to what you said, if in that context one infers Ga from (x)(Gx), then one would be following UI. If that was true, then UI would be blind for the logical form of the inference. I would say that UI would not be, then, a logical rule of inference. But UI is introduced in logic as a logical rule of inference.
ResponderExcluirIf the domain of (x) in C is all Fs, then in a logical notation one should make that clear, just to avoid thinking that something is an instance of a quantification when it is not. So to express what is said in C in logical notation by means of (x)(Gx) would be failing to fulfill this requirement of a good expression of what is said in logical notation. If we are going to leave the domain presupposed in the use of logical notation, as we do in ordinary language, whereas we can make it explicit, then I can see no advantage in the use logical notation. If it does not itch, one does not scratch. But if it does itch, one should scratch.
Prof. Gauker: Commenting the fact that we can translate English sentences into logical symbolism by means of differente translation schemes, C. Allen and M. Hand say:
ResponderExcluir"The choice of whether to represent English phrases with one-place or many-place predicates is dependent on the degree of structure that must be included in order for an argument to be analyzed adequately. In general, more detail is better than less detail, since arguments may be labeled invalid erroneously if insufficient detail is represented." (Logic Primer, Harvard: MIT, 2001, p. 69)
One of the details, of course, is the relation between a quantification and its instances.
Dear Prof. Machado: Thank you for your good-natured perseverance. I'm afraid we've hit an impasse. I think it is undeniable that AxFx/Fa is an instance of UI. The quotation from Allen and Hand is about the adicity of predicates, which is not at issue here.
ResponderExcluirDear Prof. Gauker: Thank you for your patience in answering my objections. I agree that we've hit an impasse. But let me just explain why I have quoted Allen and Hand. I agree that they are talking about something else. But what I was trying to illustrate by means of that quotation was the attitude towards formalization they express in the final sentence. I just didn't want to quote it completely out of context. Such an attitude is what has been animating my answers to your replies, like the one just before the quotation. I will read carefully your paper on UI in order to give a more substantial account of my disagreement, or else to end up agreeing with you!
ResponderExcluir