Welcome, readers, to the 76th fortnightly Philosophers’ Carnival!
Enigman asks what philosophical reasons mathematicians have for assuming the axiom of infinity in his post Philosophy of Mathematics. It’s not clear what sorts of reasons he’s looking for; fundamental questions about mathematical truth and the role of axioms seem to be lurking just below the surface here. The comments thread hasn’t grown prohibitively long yet, so hop on over and pitch in your $.02.
Alexander Pruss criticizes several ways of construing the supernaturalness of magic in Magic, science, and the supernatural. I’m not convinced by a lot of what he says, but the discussion is very clear and open-minded. Peruse the other entries while you’re there, if you haven’t visited before – it’s a nice blog.
Avery Archer works on a theory of rational agency in Why Questions and Rational Agents (more about the latter than the former). I like this post, even though I don’t like a lot (of the little) I have read elsewhere on rationality. It’s not clear to me that the appearances of the good (allegedly) involved in desire are reflections of a perspective held by some subsystem of an agent which is involved in producing the agent’s desires, just because I’m not sure that subsystems of agents are the sorts of things that can have perspectives. This might be a quibble. When a person’s reasoned course of action conflicts with her desires, there obviously does seem to be some sub-agential system bearing some interesting relationship to the course of action desired but not taken, or a mental representation of that course of action. It might be useful to spell out what is not quite “perspectival” about that relationship, though. I have more to say about this, but you don’t need to read it.
Over at Possibly Philosophy, Andrew Bacon weighs in on Counterexamples to Modus Ponens. I’m not sure I understand why he thinks that a syntactic characterization of modus ponens won’t work, and I don’t understand accessibility (between possible worlds) well enough to follow the rest of the argument. The McGee counterexample is super-interesting, though, and deserves attention from those of you out there with more logical competence than your humble host.
Thom Brooks of The Brooks Blog lets us in on his Five Secrets to Publishing Success, published on InsideHigherEd.com. Helpful to those looking for, well, publishing success.
Richard Chappell offers a brief but convincing discussion of Fair Shares and Others’ Responsibilities. He argues that, in the interest of fairness, we should pick up the slack for others’ moral failings. I think I agree, although I do not live up to the conclusion in my own life. Also, it’s not clear to me how well this sits with Richard’s views on the “demandingness objection” and the permissibility of living a basically decent life expressed here.
Bryan Norwood presents some objections to epistemological internalism, with an alternative, in Internalist Justification vs. Virtuoso Expertise. There is a lot I don’t understand here – the distinction between subjective and objective blame, the relationships between foundationalism and this distinction, and the relation between internalism and K = JTB. Still, I think there are some good ideas about epistemic blameworthiness brewing here.
Chris Hallq discusses Gettier and the purpose of analyzing “knowledge” in The case against Gettier. Some of the literature on what knowledge is for – the relation between knowledge and assertion, or knowledge and the attribution of other factive mental states – could help here. Still, the basic point, that philosophers interested in a concept need to keep the distinctive intended uses of the concept, is worth reiterating.
Lastly, Gualtiero Piccinini disambiguates “connectionism” for us, and spells out some of the morals of the disambiguation in The Ambiguity of "Connectionism". I was taught that connectionism is the view that the brain does most everything using Parallel Distributed Processing, but these other senses of “connectionism” are useful to distinguish as well.
That wraps up this edition of the Philosophers’ Carnival. If you’re still jonesing for more philosophy after all that, I invite you to check out some of the posts here on Think It Over. And, as always, keep your eye out for the next edition upcoming at Kenny Pearce’s blog.
Showing posts with label logic. Show all posts
Showing posts with label logic. Show all posts
Monday, August 25, 2008
Monday, July 21, 2008
Existence-Likeness
I would like to take a stab at defining “existence-like” as characterized in Eklund (2008).
An expression is existence-like if it is translated by “exists” in English.
For starters, this handles the cases of disputes about mereology. If L has some formal mereology, with formal criteria for when things have a fusion, and s is the sentence in L stating that there is no fusion that is a table of a certain number of particles "arranged table-wise", then I think we would translate s in English with something such as "There is no table composed of x, y, z particles." The quantifier in L is translated in English with a "there is" equivalent to "exists".
The definition also handles a number of cases related to deviant logics. Consider a schema with branching quantification such as:
For all w there is an x,
(1) such that F(w, x, y, z).*
for all y there is a z,
This might seem to present a problem, since different theorists will translate (1) different ways. Proponents of branching quantification – and especially proponents of branching quantification as a resource for schematizing the logical form of actual English sentences – will translate sentences of this form more-or-less homophonically (perhaps adding some punctuation or inflection marks). Proponents of classical FOL, such as Quine, will probably translate it with a sentence of the form:
(2) For all x there is an f, and for all y there is a g, such that F(x, f(x), y, g(y).
So a potentially existence-like expression, such as the branched existential quantifier in (1), will be translated multiple ways in English, depending on the syntactic and ontological commitments of the translator. But this does not seem actually to be a problem for our definition of “existence-like”, since both the classical logician and the proponent of branching quantification translate the “there is” of (1) with a “there is”. Where they differ is on the range of values of the existentially quantified variables.
Another problem is the theorist who thinks that “there is” and “exists” are not synonymous or equivalent, or that existential quantifications are not existence claims. This view can manifest itself in a number of ways. The most obvious case would be one in which a theorist uses a language in which the “existential” quantifiers are interpreted substitutionally and are intended to help regiment some fragment of English discourse containing “there is”, but no fragment containing “exists”. Given her preferred regimentation, and the existence of the name “Pegasus” in English, she might consider the following true:
(3) There is at least one winged horse.
The classical logician, who prefers to regiment “there is” discourse with objectual quantification alone, has a few options here. She can translate the substitutionally regimented counterpart of (3), and other sentences of similar logical form, just as (3), but with “there is” interpreted objectually. She can treat the translatum as either true or false depending on her preferred theory and interpretation of fictions and empty names in English. She can also employ semantic ascent, translating the substitutionally regimented counterpart of (3) as:
(4) “There is at least one winged horse” is true.
Again, she can evaluate (4) based on her preferred theory of truth. Perhaps she thinks “true” is implicitly relativized to something more precise than English; perhaps she thinks (3) isn’t truth-apt.
Now, in the case of (3), if the translation is interpreted objectually, then I think it is clear that the existential quantifier in the substitutionally regimented counterpart of (3) satisfies our definition of “existence-like”. Even if (3) is viewed as true, but elliptical (as required by certain theories of fiction or empty names), I can’t see how even the fully explicit interpretation could lack the expression “there is”. And if there is a “there is” in the fully explicit version of (3), I don’t see how we could deny that that “there is” is the translation of the existential quantifier in the substitutionally regimented counterpart of (3), as required by our definition of “existence-like”.
If we take the route of semantic ascent, things get a little weirder. After all “there is” does occur in (4), but it is mentioned, not used. Still, my intuition is that “there is” is the translation in (4) of the substitutionalist’s existential quantifier. One might want to say that “‘there is’”, and not “there is” is the translation, but “‘there is’” does not occur in (4), strictly speaking. (There is no close-quotation mark after the “there is” in (4).) Alternatively, assume that, if a word in a sentence is not treated by a translation as elliptical, or as an auxiliary term in a construction-yielding particle phrase, then part of the translation of the sentence is a translation of the word. Then, since the substitutionalist’s existential quantifier is not being treated as elliptical or auxiliary in the translation (4), there must be a translation in (4) of that quantifier. But that translation obviously has to be “there is”.
(Note that we encounter a similar sort of situation when the classical logician translates second-order quantification. If she doesn’t want to translate second-order quantification objectually into set theory, then she will most likely treat it metalinguistically. For instance, she might translate “There is a P and an x such that P(x)” as “There is a “P” such that ‘There is an x such that P(x)’ is true”. The second-order quantification over P becomes metalinguistic quantification over “P”.)
Recall that our substitutionalist distinguishes between “there is” and “exists” as between substitutional and objectual quantification. What happens when she wants to translate from another substitutional language into English? She will translate existential quantification with “there is”, but not with “exists”. We have seen that, in these cases, the classical logician, who treats “there is” and “exists” as equivalent, can use either of these translations. In this case, then, not only is it unclear what translatum sentence to use (which is not necessarily a problem for our definition of “existence-like”), it is also unclear whether the translatum should contain “exists” as the translation of a given non-English expression. Since the questions about whether and when to use substitutional or objectual quantification are fundamental ontological questions, and we are defining “existence-like” in order to help answer these questions, it would be silly to require that we settle the questions about whether and when to use substitutional or objectual quantification in order to apply our definition correctly. I think we need to modify the definition. The only modification I can think of that works (and is also the simplest) is this:
An expression is existence-like if it is translated either by “exists” or “there is” in English.
This solves our problem, since both the substitutionalist and the classical logician satisfy this definition. This also solves a similar problem, which we haven’t explored, for Meinongian non-English languages.
This solution might create a problem of its own, however. The substitutionalist and the Meinongian have sought to create an interesting distinction between “exists” and “there is”. To some extent, our modified definition erases that distinction. We just observed that we don’t want our definition to trivially settle ontological questions. Has our modified definition done just that?
I think not. The point of a definition of existence-likeness is to enable us to survey what meanings it is theoretically possible to assign to “exists” and “there is” when doing ontology. From a Carnapian point of view, we could say that the point was to find out what terms in what languages have semantic rules that we can use to explicate “exists” and “there is” in English. Our definition indicates that we can use (separately) the Meinongian and the substitutional rules for “there is”, as well as other rules. But surely if these rules determine existence-like uses for expressions, and the rules do not prohibit the use of other rules for distinct expressions (such as the Meinongian or objectual “exists”), then our definition does not prohibit the use of both these sorts of expressions to state an ontology. Simply because we could use the Meinongian rules for “there is” to explicate “exists” does not mean that we could not use distinct rules to explicate “exists” a different way in the same language. So our definition is not problematic for that sort of reason.
* - I'm having trouble writing branching quantifiers on Blogger. The two quantifier strings on the different lines are supposed to be on two different branches.
An expression is existence-like if it is translated by “exists” in English.
For starters, this handles the cases of disputes about mereology. If L has some formal mereology, with formal criteria for when things have a fusion, and s is the sentence in L stating that there is no fusion that is a table of a certain number of particles "arranged table-wise", then I think we would translate s in English with something such as "There is no table composed of x, y, z particles." The quantifier in L is translated in English with a "there is" equivalent to "exists".
The definition also handles a number of cases related to deviant logics. Consider a schema with branching quantification such as:
For all w there is an x,
(1) such that F(w, x, y, z).*
for all y there is a z,
This might seem to present a problem, since different theorists will translate (1) different ways. Proponents of branching quantification – and especially proponents of branching quantification as a resource for schematizing the logical form of actual English sentences – will translate sentences of this form more-or-less homophonically (perhaps adding some punctuation or inflection marks). Proponents of classical FOL, such as Quine, will probably translate it with a sentence of the form:
(2) For all x there is an f, and for all y there is a g, such that F(x, f(x), y, g(y).
So a potentially existence-like expression, such as the branched existential quantifier in (1), will be translated multiple ways in English, depending on the syntactic and ontological commitments of the translator. But this does not seem actually to be a problem for our definition of “existence-like”, since both the classical logician and the proponent of branching quantification translate the “there is” of (1) with a “there is”. Where they differ is on the range of values of the existentially quantified variables.
Another problem is the theorist who thinks that “there is” and “exists” are not synonymous or equivalent, or that existential quantifications are not existence claims. This view can manifest itself in a number of ways. The most obvious case would be one in which a theorist uses a language in which the “existential” quantifiers are interpreted substitutionally and are intended to help regiment some fragment of English discourse containing “there is”, but no fragment containing “exists”. Given her preferred regimentation, and the existence of the name “Pegasus” in English, she might consider the following true:
(3) There is at least one winged horse.
The classical logician, who prefers to regiment “there is” discourse with objectual quantification alone, has a few options here. She can translate the substitutionally regimented counterpart of (3), and other sentences of similar logical form, just as (3), but with “there is” interpreted objectually. She can treat the translatum as either true or false depending on her preferred theory and interpretation of fictions and empty names in English. She can also employ semantic ascent, translating the substitutionally regimented counterpart of (3) as:
(4) “There is at least one winged horse” is true.
Again, she can evaluate (4) based on her preferred theory of truth. Perhaps she thinks “true” is implicitly relativized to something more precise than English; perhaps she thinks (3) isn’t truth-apt.
Now, in the case of (3), if the translation is interpreted objectually, then I think it is clear that the existential quantifier in the substitutionally regimented counterpart of (3) satisfies our definition of “existence-like”. Even if (3) is viewed as true, but elliptical (as required by certain theories of fiction or empty names), I can’t see how even the fully explicit interpretation could lack the expression “there is”. And if there is a “there is” in the fully explicit version of (3), I don’t see how we could deny that that “there is” is the translation of the existential quantifier in the substitutionally regimented counterpart of (3), as required by our definition of “existence-like”.
If we take the route of semantic ascent, things get a little weirder. After all “there is” does occur in (4), but it is mentioned, not used. Still, my intuition is that “there is” is the translation in (4) of the substitutionalist’s existential quantifier. One might want to say that “‘there is’”, and not “there is” is the translation, but “‘there is’” does not occur in (4), strictly speaking. (There is no close-quotation mark after the “there is” in (4).) Alternatively, assume that, if a word in a sentence is not treated by a translation as elliptical, or as an auxiliary term in a construction-yielding particle phrase, then part of the translation of the sentence is a translation of the word. Then, since the substitutionalist’s existential quantifier is not being treated as elliptical or auxiliary in the translation (4), there must be a translation in (4) of that quantifier. But that translation obviously has to be “there is”.
(Note that we encounter a similar sort of situation when the classical logician translates second-order quantification. If she doesn’t want to translate second-order quantification objectually into set theory, then she will most likely treat it metalinguistically. For instance, she might translate “There is a P and an x such that P(x)” as “There is a “P” such that ‘There is an x such that P(x)’ is true”. The second-order quantification over P becomes metalinguistic quantification over “P”.)
Recall that our substitutionalist distinguishes between “there is” and “exists” as between substitutional and objectual quantification. What happens when she wants to translate from another substitutional language into English? She will translate existential quantification with “there is”, but not with “exists”. We have seen that, in these cases, the classical logician, who treats “there is” and “exists” as equivalent, can use either of these translations. In this case, then, not only is it unclear what translatum sentence to use (which is not necessarily a problem for our definition of “existence-like”), it is also unclear whether the translatum should contain “exists” as the translation of a given non-English expression. Since the questions about whether and when to use substitutional or objectual quantification are fundamental ontological questions, and we are defining “existence-like” in order to help answer these questions, it would be silly to require that we settle the questions about whether and when to use substitutional or objectual quantification in order to apply our definition correctly. I think we need to modify the definition. The only modification I can think of that works (and is also the simplest) is this:
An expression is existence-like if it is translated either by “exists” or “there is” in English.
This solves our problem, since both the substitutionalist and the classical logician satisfy this definition. This also solves a similar problem, which we haven’t explored, for Meinongian non-English languages.
This solution might create a problem of its own, however. The substitutionalist and the Meinongian have sought to create an interesting distinction between “exists” and “there is”. To some extent, our modified definition erases that distinction. We just observed that we don’t want our definition to trivially settle ontological questions. Has our modified definition done just that?
I think not. The point of a definition of existence-likeness is to enable us to survey what meanings it is theoretically possible to assign to “exists” and “there is” when doing ontology. From a Carnapian point of view, we could say that the point was to find out what terms in what languages have semantic rules that we can use to explicate “exists” and “there is” in English. Our definition indicates that we can use (separately) the Meinongian and the substitutional rules for “there is”, as well as other rules. But surely if these rules determine existence-like uses for expressions, and the rules do not prohibit the use of other rules for distinct expressions (such as the Meinongian or objectual “exists”), then our definition does not prohibit the use of both these sorts of expressions to state an ontology. Simply because we could use the Meinongian rules for “there is” to explicate “exists” does not mean that we could not use distinct rules to explicate “exists” a different way in the same language. So our definition is not problematic for that sort of reason.
* - I'm having trouble writing branching quantifiers on Blogger. The two quantifier strings on the different lines are supposed to be on two different branches.
Friday, July 18, 2008
Eklund on Carnap
I’m reading Matti Eklund’s really interesting paper “Carnapian Theses in Metaontology and Metaethics”, which raises some of the issues I've addressed recently.
A couple of things. Eklund claims on p. 6 that the internal/external distinction does not require the analytic/synthetic distinction. I disagree because I think a Carnapian should subscribe to something like either of the following arguments.
Suppose that analytic truths are sentences that are true in a language L solely in virtue of L. Every linguistic framework has semantic rules. Linguistic frameworks are languages or language-fragments. If something is true solely in virtue of semantic rules, it is true solely in virtue of language. For some language L, there is at least one sentence that is true in L solely in virtue of L’s semantic rules. Therefore, there is at least one analytic truth in at least one linguistic framework.
Alternatively, suppose that analytic truths are synonyms of logical truths. Every linguistic framework has semantic rules. The semantic rules of a linguistic framework entail all of the synonymy relations between sentences in that framework. For at least one linguistic framework L, at least one logical truth in L has a synonym in L. Therefore, there is at least one analytic truth (which is not just a logical truth) in at least one linguistic framework.
I am presupposing that Carnap is what Eklund calls a “language pluralist”, but I take that to be obvious. I am also presupposing that Carnap thinks true all of the premises of at least one of these arguments, but I think he does. I remember him advocating the view of analyticity in the second argument (Williamson calls it “Frege-analyticity”) somewhere in his correspondence with Quine.
One more thing. Eklund writes: “One less trivial claim would be that ‘there are
numbers’ has different meanings and truth-values in different languages while meaning what it actually means. But this is less trivial at the expense of being committing to some form of relativism, and language pluralism was supposed to be an alternative to relativism.” (11) This isn’t true, since one of the languages might be much more useful than all of the others. If someone can say that there is an objective fact to the effect that we ought to use a language that assigns a certain meaning and value to “there are numbers”, then it seems odd to call her a relativist.
A couple of things. Eklund claims on p. 6 that the internal/external distinction does not require the analytic/synthetic distinction. I disagree because I think a Carnapian should subscribe to something like either of the following arguments.
Suppose that analytic truths are sentences that are true in a language L solely in virtue of L. Every linguistic framework has semantic rules. Linguistic frameworks are languages or language-fragments. If something is true solely in virtue of semantic rules, it is true solely in virtue of language. For some language L, there is at least one sentence that is true in L solely in virtue of L’s semantic rules. Therefore, there is at least one analytic truth in at least one linguistic framework.
Alternatively, suppose that analytic truths are synonyms of logical truths. Every linguistic framework has semantic rules. The semantic rules of a linguistic framework entail all of the synonymy relations between sentences in that framework. For at least one linguistic framework L, at least one logical truth in L has a synonym in L. Therefore, there is at least one analytic truth (which is not just a logical truth) in at least one linguistic framework.
I am presupposing that Carnap is what Eklund calls a “language pluralist”, but I take that to be obvious. I am also presupposing that Carnap thinks true all of the premises of at least one of these arguments, but I think he does. I remember him advocating the view of analyticity in the second argument (Williamson calls it “Frege-analyticity”) somewhere in his correspondence with Quine.
One more thing. Eklund writes: “One less trivial claim would be that ‘there are
numbers’ has different meanings and truth-values in different languages while meaning what it actually means. But this is less trivial at the expense of being committing to some form of relativism, and language pluralism was supposed to be an alternative to relativism.” (11) This isn’t true, since one of the languages might be much more useful than all of the others. If someone can say that there is an objective fact to the effect that we ought to use a language that assigns a certain meaning and value to “there are numbers”, then it seems odd to call her a relativist.
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.
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.
Sunday, March 9, 2008
An Unnecessary Criterion of Adequacy for a Definition of "Confirmation?"
One of Carl Hempel's criteria of adequacy for a general definition of "confirmation" in his "Studies in the Logic of Confirmation" (1945) is that it must apply to sentences of any logical form. According to Hempel, a general definition of "confirmation" cannot state only what sorts of sentences confirm, say, universally quantified sentences with a single variable.
Maybe I'm missing something, but I think this is not a good criterion. As far as I can tell, as long as we define “confirmation” for, say, universally quantified sentences of single variable (with negation), then, if we accept Hempel’s “equivalence condition”*, we have a general definition of “confirmation.”
Can’t we eliminate any existential quantifier in terms of a universal quantifier and negation by means of the logical equivalence (Ex)(Fx) <=> ~(Ax)~(Fx)?
Isn’t it true that, for any n-adic predicate R, for any place pi in the predicate, for any ordered set of objects {x1, x2, …, xi-1, xi+1, …, xn} occupying the other places in the predicate, we can define a monadic predicate R`, such that R`xi <=> Rx1,x2,…xi-1,xi,xi+1,…xn?
And, at least in languages without empty names, isn’t it true that Fa <=> (Ex)(x = a & Fa)?
So, if we allow ourselves to define somewhat contrived predicates and help ourselves to languages non-free logics, isn’t every sentence logically equivalent to some sentence in universal form with one variable? This is not, of course, to say that we could actually get by in science or daily life without sentences of more complex logical forms; this is a claim only about what a definition of “confirmation” needs to be truly general. As usual, I reserve the possibility that I’m totally mistaken about the logic here.
______________________________
* - The equivalence condition states that "If an observation report confirms a hypothesis H, then it also confirms every hypothesis which is logically equivalent with H."
Maybe I'm missing something, but I think this is not a good criterion. As far as I can tell, as long as we define “confirmation” for, say, universally quantified sentences of single variable (with negation), then, if we accept Hempel’s “equivalence condition”*, we have a general definition of “confirmation.”
Can’t we eliminate any existential quantifier in terms of a universal quantifier and negation by means of the logical equivalence (Ex)(Fx) <=> ~(Ax)~(Fx)?
Isn’t it true that, for any n-adic predicate R, for any place pi in the predicate, for any ordered set of objects {x1, x2, …, xi-1, xi+1, …, xn} occupying the other places in the predicate, we can define a monadic predicate R`, such that R`xi <=> Rx1,x2,…xi-1,xi,xi+1,…xn?
And, at least in languages without empty names, isn’t it true that Fa <=> (Ex)(x = a & Fa)?
So, if we allow ourselves to define somewhat contrived predicates and help ourselves to languages non-free logics, isn’t every sentence logically equivalent to some sentence in universal form with one variable? This is not, of course, to say that we could actually get by in science or daily life without sentences of more complex logical forms; this is a claim only about what a definition of “confirmation” needs to be truly general. As usual, I reserve the possibility that I’m totally mistaken about the logic here.
______________________________
* - The equivalence condition states that "If an observation report confirms a hypothesis H, then it also confirms every hypothesis which is logically equivalent with H."
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