In ordinary second-order logics, is the first-order fragment of the logic complete? That is, are all propositions that are true in all models and expressible strictly in terms of first-order quantification also provable?
Also, intuitively, when one plays a role-playing video game such as Final Fantasy, Breath of Fire, or Dragon Warrior, does one pretend to be the character(s) one plays?
Friday, July 11, 2008
Wednesday, July 9, 2008
Comments on The Philosophy of Philosophy
In Chapter 2 of The Philosophy of Philosophy, “Taking Philosophical Questions at Face Value”, Timothy Williamson argues that a certain philosophical or “proto-philosophical” question is not, explicitly or implicitly, about language. This, what he calls the original question, is: “Was Mars always either dry or not dry?” He shows how a number of ways of answering the original question, through the consideration of intuitionistic, three-valued, and fuzzy logics, still don’t make it a linguistic question, since the answers are not about (i.e. don’t refer to) linguistic items. The answers are “Mars was always either dry or not dry”, “Mars was not always either dry or not dry”, and “It is indefinite whether Mars was always either dry or not dry.” Since none of these answers is about language, the question is not a question about language.
Let’s focus on yes-no questions for now. Say that A is a straightforward answer to a yes-no question Q, stated in language L, iff A is stated in L and expresses what “yes” would express or expresses what “no” would express. Clearly, not every non-straightforward answer to a question is about language. If Susan asks “You ate lunch at Vinny’s last night?”, and Tim responds “Actually, I went to Aunt Suzie’s”, Tim does not give a straightforward answer, but neither does he give a linguistic answer. Nor is every linguistic answer non-straightforward. If Tim responded “True” – as in “What you just said is true, stated indicatively” – I think the answer is both straightforward, because equivalent to “yes”, and linguistic, because about a sentence. Still, most linguistic answers are non-straightforward. If Tim responded “Depends on what you mean by ‘lunch’”, because, say, he ate a borderline meal of salad and an omelet at 11:45, that would be a typical non-straightforward, linguistic answer.
Note also that linguistic questions – questions about language – admit of non-straightforward and perhaps also straightforward non-linguistic answers.
The point is that questions that are about language admit of non-linguistic answers, and questions that aren’t about language admit of linguistic answers.
One way for a question to be implicitly, but not explicitly, about language, relative to a kind of answer K, is for all of the members of K to be explicitly about language. Williamson has shown that the original question is not in this way implicitly about language, relative to its philosophical answers, since the philosophical answers are not explicitly about language. But might the question be implicitly about language because the philosophical answers are implicitly about language? I kinda think so, for the following reasons.
(1) and (3) are obvious. Although I’m not sure Quine would agree with me on (2), I think Williamson would. (4) is probably the most controversial, but I take it that Williamson should agree with me on that as well, judging by what he has to say about Vann McGee in his paper “Understanding and Inference.” But (6) straightforwardly follows from (1)-(4).
Now, once we get to (6), it’s not obvious that every answer in L1 to a question in some other language L2 is thereby a linguistic answer. After all, if a bilingual speaker asks me how the weather is in English, and I answer “Hace fresco”, I have not thereby given a linguistic answer. But, I want to say, that is because it was merely a manner of speaking for me to answer in Spanish. The philosopher who answers in a three-valued language, or a fuzzy language, or an intuitionistic language, or a classical language thinks she has to answer in that language, because that is the right language in which to answer the question, or the only (kind of) language in which to state her theory of vagueness, then the answer is not merely a manner of speaking. The step of translation from the logical language to natural English is a necessary step for the philosopher to give the sort of answer she wants to give. I want to say that it is in virtue of this necessity that the original question is linguistic, at least relative to these sorts of answers grounded in logical metareflection.
I think it is reasonable to say that there is a sense in which a question is implicitly about language, relative to a kind of answer K, iff every member of K is in another language because it must, for the speaker’s most cherished purposes, be in another language. So it is, apparently, with the original question and its philosophical-type answers – or at least the original question and the philosophical-type answers that Williamson has on offer. I guess that if the deconstructionist wants to say (in English) that Mars was always both dry and not dry, because binary distinctions are always unstable and every inscription of both “Mars has always been dry” and “Mars has always not been dry” is internally contradictory, then we have a philosophical English-language answer to our English-language question. But that’s not the kind of philosophy we were talking about, right? Weren’t we looking for the right philosophy of analytic philosophy?
Let’s focus on yes-no questions for now. Say that A is a straightforward answer to a yes-no question Q, stated in language L, iff A is stated in L and expresses what “yes” would express or expresses what “no” would express. Clearly, not every non-straightforward answer to a question is about language. If Susan asks “You ate lunch at Vinny’s last night?”, and Tim responds “Actually, I went to Aunt Suzie’s”, Tim does not give a straightforward answer, but neither does he give a linguistic answer. Nor is every linguistic answer non-straightforward. If Tim responded “True” – as in “What you just said is true, stated indicatively” – I think the answer is both straightforward, because equivalent to “yes”, and linguistic, because about a sentence. Still, most linguistic answers are non-straightforward. If Tim responded “Depends on what you mean by ‘lunch’”, because, say, he ate a borderline meal of salad and an omelet at 11:45, that would be a typical non-straightforward, linguistic answer.
Note also that linguistic questions – questions about language – admit of non-straightforward and perhaps also straightforward non-linguistic answers.
Tim: “… but then I had to jet to the supermarket.”Tim’s first answer might be straightforward. His second answer is non-straightforward. Neither answer is about language.
Susan: “What does ‘jet’ mean?”
Tim: “Somebody jets somewhere whenever they try to get there very quickly.”
Or
Tim: “I had to get there very quickly.”
The point is that questions that are about language admit of non-linguistic answers, and questions that aren’t about language admit of linguistic answers.
One way for a question to be implicitly, but not explicitly, about language, relative to a kind of answer K, is for all of the members of K to be explicitly about language. Williamson has shown that the original question is not in this way implicitly about language, relative to its philosophical answers, since the philosophical answers are not explicitly about language. But might the question be implicitly about language because the philosophical answers are implicitly about language? I kinda think so, for the following reasons.
1) The original question is stated in English.
2) Languages are partially constituted by their logics. Two things with different logics cannot be the same language.
3) The language of each answer has some formal logic – three-valued, fuzzy, intuitionistic, classical, etc.
4) English has no formal logic – neither three-valued, nor fuzzy, nor intuitionistic, nor classical, etc.
5) Therefore, English does not have the same logic as the language of any of the answers.
6) Therefore, the language of each of the answers is not the same as the language of the original question.
(1) and (3) are obvious. Although I’m not sure Quine would agree with me on (2), I think Williamson would. (4) is probably the most controversial, but I take it that Williamson should agree with me on that as well, judging by what he has to say about Vann McGee in his paper “Understanding and Inference.” But (6) straightforwardly follows from (1)-(4).
Now, once we get to (6), it’s not obvious that every answer in L1 to a question in some other language L2 is thereby a linguistic answer. After all, if a bilingual speaker asks me how the weather is in English, and I answer “Hace fresco”, I have not thereby given a linguistic answer. But, I want to say, that is because it was merely a manner of speaking for me to answer in Spanish. The philosopher who answers in a three-valued language, or a fuzzy language, or an intuitionistic language, or a classical language thinks she has to answer in that language, because that is the right language in which to answer the question, or the only (kind of) language in which to state her theory of vagueness, then the answer is not merely a manner of speaking. The step of translation from the logical language to natural English is a necessary step for the philosopher to give the sort of answer she wants to give. I want to say that it is in virtue of this necessity that the original question is linguistic, at least relative to these sorts of answers grounded in logical metareflection.
I think it is reasonable to say that there is a sense in which a question is implicitly about language, relative to a kind of answer K, iff every member of K is in another language because it must, for the speaker’s most cherished purposes, be in another language. So it is, apparently, with the original question and its philosophical-type answers – or at least the original question and the philosophical-type answers that Williamson has on offer. I guess that if the deconstructionist wants to say (in English) that Mars was always both dry and not dry, because binary distinctions are always unstable and every inscription of both “Mars has always been dry” and “Mars has always not been dry” is internally contradictory, then we have a philosophical English-language answer to our English-language question. But that’s not the kind of philosophy we were talking about, right? Weren’t we looking for the right philosophy of analytic philosophy?
Monday, June 30, 2008
What Grammatical Structures Say and the Linguistic Theory of Logical Truth
In the last chapter of Philosophy of Logic, Quine discusses the “linguistic theory of logical truth.” This is the theory that “[a] sentence is logically true if true by virtue purely of its grammatical structure. […] It is language that makes logical truths true – purely language, and nothing to do with the nature of the world.” (2nd ed., 95) Quine offers a few different reasons not to buy into this doctrine, most of them familiar from “Two Dogmas” and “Truth by Convention”. The freshest argument, I think, is the one he offers in the paragraph immediately following the previous quote. Here it is, in full:
Logical truths are about the world, or are true because they “reflect” features of the world, because their grammatical structures are about the world or reflect features of the world. In what sense could a grammatical structure possibly be about the world, say anything about the world, or “reflect” features of the world? First, we should note that grammatical structures are not about the world in the same way that sentences, names, or predicates are. Grammatical structures as such aren’t true or false like (truth-apt) sentences. By all appearances, grammatical structures don’t refer to anything in the world; Tarski’s definition of truth gets along just fine without assigning semantic values to grammatical structures. Nor do they have (Fregean) senses on any theory that I know of. Nor, intuitively, are they meaningful. If a person were to speak or write down a grammatical structure – say, by speaking or writing a sequence of particles and schematic variables for grammatical categories – I can’t see why anyone would want to say that she, or her utterance or inscription, meant anything.
We might want to say that grammatical structures say something about the world in a different sense – viz., in the sense that sentences with the same non-logical constants but different grammatical structures have different truth-conditions. “(Ax)(Cat(x))” says something different from “~(Ax)(Cat(x))” because of the difference in grammatical structure between the two. We might say that a negation symbol says that the negated sentence is false, a universal quantifier over a variable says that the sentence in the scope of the quantifier is true for all values of the bound variable, and so on.* In this way, by specifying what all of the particles or logical constants say, we can state more or less precisely what an entire grammatical structure, paired with a particular sentence instantiating it, says about the world. But two observations are in order. First, it is not clear how we should construe what the grammatical structures of atomic sentences say.** Second, and more importantly, the worldliness of a grammatical structure, in this sense, is dependent on the worldliness of the non-logical constants in the sentence instantiating it. For instance, the grammatical structure of “~(Cat(Dora))” says something about the world because “(Cat(Dora))” says something about the world – it is either true or false depending on the actual features of the thing called “Dora”. The grammatical structures of uninterpreted schemata say nothing about the world, because the talk of truth, falsity, and values of variables in our sketchy specification of what the particles say presupposes an interpretation of the lexical items in the sentence. When we admit, with Quine, that a logical truth “admittedly depends upon none of those features of the world that are reflected in lexical distinctions”, then the grammatical structure of a logical truth cannot derive its worldliness from the worldliness of the “lexical distinctions” marked by the non-logical constants in a sentence instantiating the structure. Briefly, since the worldliness of grammatical structures derives from the worldliness of the terms in the sentences instantiating them, and since the putative worldliness of logical truths does not depend on the worldliness of these terms, the grammatical structures of logical truths have nothing from which to derive their putative worldliness.
There might be some way of being about the world or reflecting features of the world that I haven’t grasped yet. Perhaps we should understand Quine as saying that we might as well postulate a sui generis mode of being about the world specific to grammatical structures. I can only say in response to this that we might as well not, both for the sake of not multiplying features of the world beyond necessity and for the sake of keeping “about the world” intelligible. Lastly, someone might say that the grammatical structure of logical truths such as “it is raining or it isn’t raining” is about the world because it reflects the fact about the world that things, in general, are or aren’t the case. But this begs the question against the proponent of the linguistic theory of logical truth. What is at issue is whether this fact is about the world.
* - It is much easier to fill in the “and so on” for a formalized language than for a natural language. What does a particle like “if” say? It seems we need a worked-out semantics for conditionals to fill out a proposal like this. If you aren’t satisfied by my hand-waving here, then that probably goes to show that it is even more difficult to make Quine’s argument come out sound.
** - This probably isn’t such a big deal, since the only atomic sentence that is a logical truth is “x = x”, and Quine seems to reckon “=” a particle.
Granted, grammatical structure is linguistic; but so is lexicon. The lexicon is used in talking about the world; but so is grammatical structure. A logical truth, staying true as it does under all lexical substitutions, admittedly depends upon none of those features of the world that are reflected in lexical distinctions; but may it not depend on other features of the world, features that our language reflects in grammatical constructions rather than its lexicon? It would be pointless to protest that grammar varies from language to language, for so does lexicon. Perhaps the logical truths owe their truth to certain traits of reality which are reflected in one way by the grammar of our language, in another way by the grammar of another language, and in a third way by the combined grammar and lexicon of a third language. (ibid., my italics)
Logical truths are about the world, or are true because they “reflect” features of the world, because their grammatical structures are about the world or reflect features of the world. In what sense could a grammatical structure possibly be about the world, say anything about the world, or “reflect” features of the world? First, we should note that grammatical structures are not about the world in the same way that sentences, names, or predicates are. Grammatical structures as such aren’t true or false like (truth-apt) sentences. By all appearances, grammatical structures don’t refer to anything in the world; Tarski’s definition of truth gets along just fine without assigning semantic values to grammatical structures. Nor do they have (Fregean) senses on any theory that I know of. Nor, intuitively, are they meaningful. If a person were to speak or write down a grammatical structure – say, by speaking or writing a sequence of particles and schematic variables for grammatical categories – I can’t see why anyone would want to say that she, or her utterance or inscription, meant anything.
We might want to say that grammatical structures say something about the world in a different sense – viz., in the sense that sentences with the same non-logical constants but different grammatical structures have different truth-conditions. “(Ax)(Cat(x))” says something different from “~(Ax)(Cat(x))” because of the difference in grammatical structure between the two. We might say that a negation symbol says that the negated sentence is false, a universal quantifier over a variable says that the sentence in the scope of the quantifier is true for all values of the bound variable, and so on.* In this way, by specifying what all of the particles or logical constants say, we can state more or less precisely what an entire grammatical structure, paired with a particular sentence instantiating it, says about the world. But two observations are in order. First, it is not clear how we should construe what the grammatical structures of atomic sentences say.** Second, and more importantly, the worldliness of a grammatical structure, in this sense, is dependent on the worldliness of the non-logical constants in the sentence instantiating it. For instance, the grammatical structure of “~(Cat(Dora))” says something about the world because “(Cat(Dora))” says something about the world – it is either true or false depending on the actual features of the thing called “Dora”. The grammatical structures of uninterpreted schemata say nothing about the world, because the talk of truth, falsity, and values of variables in our sketchy specification of what the particles say presupposes an interpretation of the lexical items in the sentence. When we admit, with Quine, that a logical truth “admittedly depends upon none of those features of the world that are reflected in lexical distinctions”, then the grammatical structure of a logical truth cannot derive its worldliness from the worldliness of the “lexical distinctions” marked by the non-logical constants in a sentence instantiating the structure. Briefly, since the worldliness of grammatical structures derives from the worldliness of the terms in the sentences instantiating them, and since the putative worldliness of logical truths does not depend on the worldliness of these terms, the grammatical structures of logical truths have nothing from which to derive their putative worldliness.
There might be some way of being about the world or reflecting features of the world that I haven’t grasped yet. Perhaps we should understand Quine as saying that we might as well postulate a sui generis mode of being about the world specific to grammatical structures. I can only say in response to this that we might as well not, both for the sake of not multiplying features of the world beyond necessity and for the sake of keeping “about the world” intelligible. Lastly, someone might say that the grammatical structure of logical truths such as “it is raining or it isn’t raining” is about the world because it reflects the fact about the world that things, in general, are or aren’t the case. But this begs the question against the proponent of the linguistic theory of logical truth. What is at issue is whether this fact is about the world.
* - It is much easier to fill in the “and so on” for a formalized language than for a natural language. What does a particle like “if” say? It seems we need a worked-out semantics for conditionals to fill out a proposal like this. If you aren’t satisfied by my hand-waving here, then that probably goes to show that it is even more difficult to make Quine’s argument come out sound.
** - This probably isn’t such a big deal, since the only atomic sentence that is a logical truth is “x = x”, and Quine seems to reckon “=” a particle.
Sunday, June 29, 2008
Saturday, June 28, 2008
Quantifiers and the Grammatical Definition of Logical Truth
The grammatical definition of logical truth, discussed here, is probably inadequate for all sorts of interesting languages, including English. The definition, from Quine's Philosophy of Logic, is this:
For Quine - and I think he's right on this count - we treat a class of words as a grammatical category, as opposed to a class of particles yielding new grammatical constructions, just in case the category is big enough. For instance, in a language with lots of intransitive verbs, we treat those as comprising a grammatical category. If an L-structure has an infinite stock of variables, we treat variables (or argument-terms, more generally) as a grammatical category; if it has three variables, we might do well to treat each as a particle.
I take it that English has an infinite - or at least a very large - stock of quantifiers. This is because I think that, in English, the quantifiers translated by "(E_)" and "(A_)" in first-order logic belong to the same grammatical category as expressions such as "There are many", "There are ten", "There are one million", and "There are innumerable". If the literature on quantifiers in natural language says otherwise, please correct me. Also, there are usually an infinite number of quantifier-expressions in languages that support generalized quantification, right? Anyway, if quantifiers all belong to the same grammatical category, and we assume the grammatical definition of logical truth, then I can't think of a single logical truth containing a quantifier. For instance, "If Steve and Janice are cats, then there are some cats" would fail to be a logical truth, since "If Steve and Janice are cats, then there are innumerable cats" - gotten by "substituting for lexicon" - is false.
I imagine the Quinean response to all of this would be to say that, given a prior commitment to standard FOL, we should translate "There are n Fs" as "The class of all Fs has cardinality n." But it seems obvious to me that the average English speaker does not, as a matter of linguistic anthropology, commit herself to the existence of the class of all Fs in uttering "There are n Fs." The nominalist cannot properly respond, "No, there is no such thing as the set of all Fs." And besides, what if we substitute "self-member" for "F"?
"a logical truth is a truth that cannot be turned false by substituting for lexicon. When for its lexical elements we substitute any other strings belonging to the same grammatical categories, the sentence is true." (2nd ed., 58)
For Quine - and I think he's right on this count - we treat a class of words as a grammatical category, as opposed to a class of particles yielding new grammatical constructions, just in case the category is big enough. For instance, in a language with lots of intransitive verbs, we treat those as comprising a grammatical category. If an L-structure has an infinite stock of variables, we treat variables (or argument-terms, more generally) as a grammatical category; if it has three variables, we might do well to treat each as a particle.
I take it that English has an infinite - or at least a very large - stock of quantifiers. This is because I think that, in English, the quantifiers translated by "(E_)" and "(A_)" in first-order logic belong to the same grammatical category as expressions such as "There are many", "There are ten", "There are one million", and "There are innumerable". If the literature on quantifiers in natural language says otherwise, please correct me. Also, there are usually an infinite number of quantifier-expressions in languages that support generalized quantification, right? Anyway, if quantifiers all belong to the same grammatical category, and we assume the grammatical definition of logical truth, then I can't think of a single logical truth containing a quantifier. For instance, "If Steve and Janice are cats, then there are some cats" would fail to be a logical truth, since "If Steve and Janice are cats, then there are innumerable cats" - gotten by "substituting for lexicon" - is false.
I imagine the Quinean response to all of this would be to say that, given a prior commitment to standard FOL, we should translate "There are n Fs" as "The class of all Fs has cardinality n." But it seems obvious to me that the average English speaker does not, as a matter of linguistic anthropology, commit herself to the existence of the class of all Fs in uttering "There are n Fs." The nominalist cannot properly respond, "No, there is no such thing as the set of all Fs." And besides, what if we substitute "self-member" for "F"?
Monday, June 23, 2008
Quine on Grammatical Structure and Logical Truth
In Philosophy of Logic, Quine offers the following definition of "logical truth": "a logical truth is a truth that cannot be turned false by substituting for lexicon. When for its lexical elements we substitute any other strings belonging to the same grammatical categories, the sentence is true." (2nd ed., 58)
Later on, considering whether to strengthen FOL to allow for adverbial modification of predicates, Quine claims that, on the definition of "logical truth" lately quoted:
This is interesting. The grammatical definition of logical truth is both epistemologically interesting and clears up a lot of confusions I have about the relationship between formal logic and natural language. I still don't know whether it adequately captures all of the intuitive cases of logical truth.
Really, my only observation here is that it seems that the grammatical definition does not make (5) a logical truth. This is because adverbs can sometimes alienate the predicates they modify. An adverb A alienates a predicate F in a sentence token S iff removal of A from S would change the truth-value of the clause of which F is a part. Briefly, A alienates F (in a certain context) if something can be F A'ly without being F simpliciter. Consider the following cases of adverbial alienation:
(1) Tim indirectly told John about Sally.
(2) Paul is coming home shortly.
(3) Sue allegedly stole the watch.
(4) Esther nearly won the tennis match.
We can imagine cases in which (1), (2), (3), and (4) are true, but their non-adverbialized counterparts aren't. There doesn't seem to be anything syntactically unusual about these adverbs. By all appearances, (1), (2), (3), and (4) have the same grammatical structure, respectively, as the following:
(1`) Tim excitedly told John about Sally.
(2`) Paul is coming home currently.
(3`) Sue actually stole the watch.
(4`) Esther barely won the tennis match.
But if (1), (2), (3), and (4) can be turned from truth to falsehood by transformation into (1`), (2`), (3`), and (4`) then, by the grammatical definition, none of these are logically true.
Later on, considering whether to strengthen FOL to allow for adverbial modification of predicates, Quine claims that, on the definition of "logical truth" lately quoted:
the sentence
(5) ~(Ex)(x walks rapidly . ~(x walks)),
or 'Whatever walks rapidly walks', would qualify as logically true. (76)
This is interesting. The grammatical definition of logical truth is both epistemologically interesting and clears up a lot of confusions I have about the relationship between formal logic and natural language. I still don't know whether it adequately captures all of the intuitive cases of logical truth.
Really, my only observation here is that it seems that the grammatical definition does not make (5) a logical truth. This is because adverbs can sometimes alienate the predicates they modify. An adverb A alienates a predicate F in a sentence token S iff removal of A from S would change the truth-value of the clause of which F is a part. Briefly, A alienates F (in a certain context) if something can be F A'ly without being F simpliciter. Consider the following cases of adverbial alienation:
(1) Tim indirectly told John about Sally.
(2) Paul is coming home shortly.
(3) Sue allegedly stole the watch.
(4) Esther nearly won the tennis match.
We can imagine cases in which (1), (2), (3), and (4) are true, but their non-adverbialized counterparts aren't. There doesn't seem to be anything syntactically unusual about these adverbs. By all appearances, (1), (2), (3), and (4) have the same grammatical structure, respectively, as the following:
(1`) Tim excitedly told John about Sally.
(2`) Paul is coming home currently.
(3`) Sue actually stole the watch.
(4`) Esther barely won the tennis match.
But if (1), (2), (3), and (4) can be turned from truth to falsehood by transformation into (1`), (2`), (3`), and (4`) then, by the grammatical definition, none of these are logically true.
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