The Slingshot Argument: An Improved Version Export

@article{citeulike:715947, abstract = {In the paper I exploit Frege's notions of sense and synonymity in order to amend the slingshot argument. The main emendation is to replace the assumption about logical equivalence by an assumption about synonymity. While the replaced assumption begs the question about the reference of sentences, the replacing assumption has much more theoretical support from Frege's general conception of sense and reference and the relation between them. In the paper I use a specific notion of synonymity which I believe is faithful to Frege's discussion of the subject. I notice that if a stronger (and to my mind implausible) notion of synonymity is used, my version of the argument fails. The failure is explained by showing that this stronger notion of synonymity enables the assignment of facts, and not truth values, as the references of sentences.}, author = {Drai, Dalia}, citeulike-article-id = {715947}, citeulike-linkout-0 = {http://www.blackwell-synergy.com/doi/abs/10.1111/1467-9329.00184}, citeulike-linkout-1 = {http://dx.doi.org/10.1111/1467-9329.00184}, comment = {Drai starts with a pretty standard formulation of the slingshot. Standard formulations usually have two assumptions: (I) if you substitute an expression in a sentence by a co-referential expression, the reference of the entire sentence remains the same, and (II) logically equivalent sentences are co-referential. As Drai points out: \begin{quote}The main objection to this argument is that assumption II is unjustified. Logically equivalent sentences have, by definition, the same truth value in every possible world. But only by begging the question about reference can we claim that they have the same reference in every possible world. The only way to justify assumption II, that logically equivalent sentences have the same reference, is by presupposing that sentences refer to their truth values, and this presupposition is not independently plausible. (Drai 2002: 196) \end{quote} So, there are three claims: \ac{Drai1} (II) should be rejected.\bc \ac{Drai2} The only justification of (II) would be that logically equivalent sentences refer to their truth values.\bc \ac{Drai3} There is no independent reason to believe that logically equivalent sentences refer to their truth values.\bc Drai suggests replacing (II) with another principle: \ac{Drai4} Synonymous (non-indexical) expressions have the same reference.\bc whereas the criterion for synonymity is: \$A\$ and \$B\$ have the same sense iff it is not possible for someone who understands two sentences, \$A\$ and \$B\$, to believe one of them without believing the other. Drai suggests that some logically equivalent sentences are probably not synonymous, but not the other way round.\footnote{{\em Clearly Epistemological Synonymity is stricter than logical equivalence.} (Drai 2002: 200)} However, the claim may be extended. Note that \rref{Drai4} is actually neither stronger nor weaker that (II). There also seem to be sentences in this sense {\em doxastically synonymous}\footnote{Drai speaks of epistemological synonymity. However, since beliefs and not knowledge are suggested as criteria, {\em doxastic} seems more proper.} which are not logically equivalent. For instance, take the sentence `someone believes a sentence' and another sentence `every sentence is logically equivalent to itself': there is quite an intuitive meaning of the verb `to believe' in which it seems that one can hardly believe that every sentence is logically equivalent to itself without believing that someone believes a sentence. Also, unless some bizarre cases are considered, we may assume that everyone believes that every sentence is logically equivalent to itself, if only he understands the claim. Thus, it does not seem possible to believe that someone believes a sentence without believing that every sentence is logically equivalent to itself and {\em vice versa}. But it is far from obvious that, say, `every sentence is logically equivalent to itself' is logically equivalent to `someone believes a sentence'. Now that we have a suspicion that this principle is not weaker that the original one, one can ask why we should be better off accepting \rref{Drai4} instead of (II) anyway. The reason that Drai gives is that \rref{Drai4} is supported by the analogy between sentences and names, whereas (II) is not. In fact, it seems plausible that: \ac{Drai5} Synonymous names denote the same.\bc On the other hand, Drai claims that: \ac{Drai6} Assumption (II) cannot be justified as an extension of a rule applying to names.\bc Here's the reason given in the paper: \begin{quote} This is because the rule in the old domain must be: logically equivalent expressions have the same reference. But the notion of logical equivalence applies only to sentences and not to sub-sentential expressions such as proper names. That is, it does not apply to expressions in the old domain\ldots it is meaningless when applied to sub-sentential expressions. (Drai 2002: 198) \end{quote} Now that one of the assumptions has been modified by introducing the supposed improvement, the slingshot requires a modification which accommodates this change. Assume (I) and \rref{Drai4}. The argument is: \ac{Drai7} Sentences \$\phi\$ and \$\psi\$ are true.\bc \ac{Drai8} \$\phi\$ is doxastically synonymous to `the truth-value of \$\phi\$ is 1'.\bc \ac{Drai9} \$\psi\$ is doxastically synonymous to `the truth-value of \$\psi\$ is 1'.\bc Thus (where `D' stands for `denotation'): \ac{Drai10} \$D(\phi)=D(\$the truth value of \$\phi\$ is 1)\bc \ac{Drai11} \$D(\psi)=D(\$the truth value of \$\psi\$ is 1)\bc \ac{Drai12} the truth-value of \$\phi\$ = the truth-value of \$\psi\$\bc \ac{Drai13} \$D(\phi)=D(\$the truth value of \$\psi\$ is 1)\bc \ac{Drai14} \$D(\phi)=D(\psi)\$\bc However, there seem to be some problems with Drai's position. In what follows I will discuss them, arguing that \rref{Drai2}, \rref{Drai6}, \rref{Drai8} and \rref{Drai9} are at least debatable. Consider \rref{Drai6}. The claim that (II) cannot be supported by an analogy between names and sentences is based on the claim that predicating `logical equivalence' of names is meaningless. Although `solving' philosophical problems by saying that some claims are meaningless played important part in the history of philosophy, it seems that this kind of linguistic imperialism is a little bit suspicious. Can't we really come up with a compelling interpretation of logical equivalence of singular terms? Let us take a look at the following terms: \ac{ex1} \$(\iota x)(P(x)\jt Q(x))\$\bc \ac{ex2} \$(\iota x)(P(x) \et \n Q(x))\$\bc \ac{ex3}\$(\iota x)(\n P(x) \jt Q(x))\$\bc \ac{ex4} \$(\iota x)(\n (P(x) \jt Q(x)))\$\bc When asked, what the pairs: \rref{ex1} and \rref{ex3}, \rref{ex2} and \rref{ex4} have in common, a plausible answer seems to be that they are, well, in some sense equivalent. One can try to suggest that they are doxastically synonymous; but it seems that such an answer will not be very compelling in a more general case. Take \$(\iota x)(\phi(x))\$, \$(\iota x)(\psi(x))\$ such that \$\ko{x}(\phi(x)\equiv \psi(x))\$ is logically valid, and nevertheless for some \$x\$ the expression \$\phi(x)\$ is epistemologically synonymous to \$\psi(x)\$ (just assume that \$\phi\$ and \$\psi\$ are complicated enough). This suggests that the notion of logical equivalence of terms may be useful and it seems to differ from the notion of doxastic synonymity, and that there may be some point in introducing this notion. Thus, a tentative definition of logical equivalence of singular terms may be: \ac{raf1} Terms \$\alpha\$ and \$\beta\$ are logically equivalent iff \[\ko{x}[x=\alpha \equiv x=\beta]\] is logically valid.\bc So, in some sense, one can employ logical equivalence meaningfully to terms. Moreover, if two terms are logically equivalent, they surely have the same denotation (that is, if they both denote, they both denote the same object). This suggests that \rref{Drai6} and \rref{Drai2} are not as compelling as Drai claims. Of course, this does not constitute any decisive refutation of Drai's position: it just suggests that the situation is not so obvious and that there may be some reasons to reject this position. It is also doubtful that for any sentence \$\phi\$, this sentence is doxastically synonymous with `the truth-value of \$\phi\$ is 1'. If we translate it using Drai's criteria of doxastic synonymity we get: \ac{Drai16} For any \$\phi\$ it is impossible for a competent speaker to believe that \$\phi\$ and not to believe that the truth-value of \$\phi\$ is 1, and it is impossible to believe that the truth-value of \$\phi\$ is 1 and not to believe that \$\phi\$.\bc But what does one have to believe in when one believes that the value of \$\phi \$ is \$1\$ to make the slingshot viable? Notice that steps like \rref{Drai12} and \rref{Drai13} indicate that the argument uses (I) and replaces `the value of \$\phi\$' with `the value of \$\psi\$'. Thus, it seems that the interpretation of the identity in that claim has to be {\em objectual}. This means that in order to believe that the value of \$\phi\$ is \$1\$ one has to believe that (i) there is a unique object which is the value of \$\phi\$, (ii) there is a unique object which is 1, and (iii) these objects are identical. That being the case, it seems that \rref{Drai16} is false. We can indicate a possible situation where one believes \$\phi\$ without believing (i)-(iii). Suppose I am a radical nominalist. Whether I am right does not matter at all: what matters here is what one can believe. Suppose also that I believe that \$\phi\$. However, I have pretty strong feelings against the claim that there is a unique object which is the truth-value of \$\phi\$, or that there is a unique object that is called `number 1'. So I do not believe that the truth-value of \$\phi\$ is 1 in any sense which would make the application of (I) valid. Thus, Drai's claim that: \begin{quote} Anyone who understands the notions of truth value and True, knows that to say [that the truth-value of \$\phi\$ = 1, R.U.] is no more and no less than to say [that \$\phi\$]. (Drai 2002: 202) \end{quote} is misleading: my commitment when I believe that \$\phi\$ is different from the commitment I make when I believe that the truth-value of \$\phi\$ =1. Of course, there is a reading of `the truth-value of \$\phi\$ is 1' which allows also a nominalist to believe it. It is the sense in which `the truth value of \$\phi\$' and `1' are not interpreted objectually as in fact referring to objects, and the whole sentence is treated just as a fancy and ontologically misleading way of saying that \$\phi\$ is true. Indeed, in that case to say that \$\phi\$ may be no more and no less than to say that the truth-value of \$\phi\$ is 1. But in this reading the slingshot does not work, because there is no real singular term one can substitute for.\footnote{This kind of reading is based on a suggestion by Tadeusz Kotarbi\' nski, a representative of Lvow-Warsaw school, who was a radical nominalist. A very basic claim of his {\em semantic nominalism} was that noun phrases that on the face of it are taken to name something, but do not name concrete individuals only do not name anything in fact, and their role, insofar as their use is meaningful, is to abbreviate or reword some expressions that do not contain such `abstract' noun phrases. A very basic example would be that in `whiteness is a colour.' neither `whiteness' nor `colour' names anything and the whole sentence works as an abbreviation for `everything that is white is coloured'. Abstract names of that sort, because they `pretend' to be names but they do not name anything were called by Kotarbi\' nski {\em onomatoids}.} Since there is nothing contradictory or impossible in the fact that there exists a nominalist who has such beliefs (it does not matter how debatable his beliefs may be), it seems that \rref{Drai8} and \rref{Drai9} are false. Note that the above argument has nothing to do with the assumption about the existence of truth-values. Even if we do assume that there are truth values and even if we believe that the truth-value of \$\phi\$ =1, and we take all singular terms in this sentence to refer to their respective unique objects, it has no bearing upon the fact that there may exist a nominalist who concedes \$\phi\$ and does not concede that the truth value of \$\phi =1\$. This fact alone, no matter what our position about truth-values is, falsifies \rref{Drai8} and \rref{Drai9}.}, doi = {10.1111/1467-9329.00184}, journal = {Ratio}, keywords = {analytic, logic, paradox, slingshot}, number = {2}, pages = {194--204}, posted-at = {2006-06-29 19:02:58}, priority = {0}, title = {The Slingshot Argument: An Improved Version}, url = {http://dx.doi.org/10.1111/1467-9329.00184}, volume = {15}, year = {2002} }

TY - JOUR ID - citeulike:715947 L3 - citeulike-article-id:715947 N2 - In the paper I exploit Frege's notions of sense and synonymity in order to amend the slingshot argument. The main emendation is to replace the assumption about logical equivalence by an assumption about synonymity. While the replaced assumption begs the question about the reference of sentences, the replacing assumption has much more theoretical support from Frege's general conception of sense and reference and the relation between them. In the paper I use a specific notion of synonymity which I believe is faithful to Frege's discussion of the subject. I notice that if a stronger (and to my mind implausible) notion of synonymity is used, my version of the argument fails. The failure is explained by showing that this stronger notion of synonymity enables the assignment of facts, and not truth values, as the references of sentences. IS - 2 JF - Ratio EP - 204 TI - The Slingshot Argument: An Improved Version VL - 15 SP - 194 KW - analytic KW - logic KW - paradox KW - slingshot N1 - Drai starts with a pretty standard formulation of the slingshot. Standard formulations usually have two assumptions: (I) if you substitute an expression in a sentence by a co-referential expression, the reference of the entire sentence remains the same, and (II) logically equivalent sentences are co-referential. As Drai points out: \begin{quote}The main objection to this argument is that assumption II is unjustified. Logically equivalent sentences have, by definition, the same truth value in every possible world. But only by begging the question about reference can we claim that they have the same reference in every possible world. The only way to justify assumption II, that logically equivalent sentences have the same reference, is by presupposing that sentences refer to their truth values, and this presupposition is not independently plausible. (Drai 2002: 196) \end{quote} So, there are three claims: \ac{Drai1} (II) should be rejected.\bc \ac{Drai2} The only justification of (II) would be that logically equivalent sentences refer to their truth values.\bc \ac{Drai3} There is no independent reason to believe that logically equivalent sentences refer to their truth values.\bc Drai suggests replacing (II) with another principle: \ac{Drai4} Synonymous (non-indexical) expressions have the same reference.\bc whereas the criterion for synonymity is: $A$ and $B$ have the same sense iff it is not possible for someone who understands two sentences, $A$ and $B$, to believe one of them without believing the other. Drai suggests that some logically equivalent sentences are probably not synonymous, but not the other way round.\footnote{{\em Clearly Epistemological Synonymity is stricter than logical equivalence.} (Drai 2002: 200)} However, the claim may be extended. Note that \rref{Drai4} is actually neither stronger nor weaker that (II). There also seem to be sentences in this sense {\em doxastically synonymous}\footnote{Drai speaks of epistemological synonymity. However, since beliefs and not knowledge are suggested as criteria, {\em doxastic} seems more proper.} which are not logically equivalent. For instance, take the sentence `someone believes a sentence' and another sentence `every sentence is logically equivalent to itself': there is quite an intuitive meaning of the verb `to believe' in which it seems that one can hardly believe that every sentence is logically equivalent to itself without believing that someone believes a sentence. Also, unless some bizarre cases are considered, we may assume that everyone believes that every sentence is logically equivalent to itself, if only he understands the claim. Thus, it does not seem possible to believe that someone believes a sentence without believing that every sentence is logically equivalent to itself and {\em vice versa}. But it is far from obvious that, say, `every sentence is logically equivalent to itself' is logically equivalent to `someone believes a sentence'. Now that we have a suspicion that this principle is not weaker that the original one, one can ask why we should be better off accepting \rref{Drai4} instead of (II) anyway. The reason that Drai gives is that \rref{Drai4} is supported by the analogy between sentences and names, whereas (II) is not. In fact, it seems plausible that: \ac{Drai5} Synonymous names denote the same.\bc On the other hand, Drai claims that: \ac{Drai6} Assumption (II) cannot be justified as an extension of a rule applying to names.\bc Here's the reason given in the paper: \begin{quote} This is because the rule in the old domain must be: logically equivalent expressions have the same reference. But the notion of logical equivalence applies only to sentences and not to sub-sentential expressions such as proper names. That is, it does not apply to expressions in the old domain\ldots it is meaningless when applied to sub-sentential expressions. (Drai 2002: 198) \end{quote} Now that one of the assumptions has been modified by introducing the supposed improvement, the slingshot requires a modification which accommodates this change. Assume (I) and \rref{Drai4}. The argument is: \ac{Drai7} Sentences $\phi$ and $\psi$ are true.\bc \ac{Drai8} $\phi$ is doxastically synonymous to `the truth-value of $\phi$ is 1'.\bc \ac{Drai9} $\psi$ is doxastically synonymous to `the truth-value of $\psi$ is 1'.\bc Thus (where `D' stands for `denotation'): \ac{Drai10} $D(\phi)=D($the truth value of $\phi$ is 1)\bc \ac{Drai11} $D(\psi)=D($the truth value of $\psi$ is 1)\bc \ac{Drai12} the truth-value of $\phi$ = the truth-value of $\psi$\bc \ac{Drai13} $D(\phi)=D($the truth value of $\psi$ is 1)\bc \ac{Drai14} $D(\phi)=D(\psi)$\bc However, there seem to be some problems with Drai's position. In what follows I will discuss them, arguing that \rref{Drai2}, \rref{Drai6}, \rref{Drai8} and \rref{Drai9} are at least debatable. Consider \rref{Drai6}. The claim that (II) cannot be supported by an analogy between names and sentences is based on the claim that predicating `logical equivalence' of names is meaningless. Although `solving' philosophical problems by saying that some claims are meaningless played important part in the history of philosophy, it seems that this kind of linguistic imperialism is a little bit suspicious. Can't we really come up with a compelling interpretation of logical equivalence of singular terms? Let us take a look at the following terms: \ac{ex1} $(\iota x)(P(x)\jt Q(x))$\bc \ac{ex2} $(\iota x)(P(x) \et \n Q(x))$\bc \ac{ex3}$(\iota x)(\n P(x) \jt Q(x))$\bc \ac{ex4} $(\iota x)(\n (P(x) \jt Q(x)))$\bc When asked, what the pairs: \rref{ex1} and \rref{ex3}, \rref{ex2} and \rref{ex4} have in common, a plausible answer seems to be that they are, well, in some sense equivalent. One can try to suggest that they are doxastically synonymous; but it seems that such an answer will not be very compelling in a more general case. Take $(\iota x)(\phi(x))$, $(\iota x)(\psi(x))$ such that $\ko{x}(\phi(x)\equiv \psi(x))$ is logically valid, and nevertheless for some $x$ the expression $\phi(x)$ is epistemologically synonymous to $\psi(x)$ (just assume that $\phi$ and $\psi$ are complicated enough). This suggests that the notion of logical equivalence of terms may be useful and it seems to differ from the notion of doxastic synonymity, and that there may be some point in introducing this notion. Thus, a tentative definition of logical equivalence of singular terms may be: \ac{raf1} Terms $\alpha$ and $\beta$ are logically equivalent iff \[\ko{x}[x=\alpha \equiv x=\beta]\] is logically valid.\bc So, in some sense, one can employ logical equivalence meaningfully to terms. Moreover, if two terms are logically equivalent, they surely have the same denotation (that is, if they both denote, they both denote the same object). This suggests that \rref{Drai6} and \rref{Drai2} are not as compelling as Drai claims. Of course, this does not constitute any decisive refutation of Drai's position: it just suggests that the situation is not so obvious and that there may be some reasons to reject this position. It is also doubtful that for any sentence $\phi$, this sentence is doxastically synonymous with `the truth-value of $\phi$ is 1'. If we translate it using Drai's criteria of doxastic synonymity we get: \ac{Drai16} For any $\phi$ it is impossible for a competent speaker to believe that $\phi$ and not to believe that the truth-value of $\phi$ is 1, and it is impossible to believe that the truth-value of $\phi$ is 1 and not to believe that $\phi$.\bc But what does one have to believe in when one believes that the value of $\phi $ is $1$ to make the slingshot viable? Notice that steps like \rref{Drai12} and \rref{Drai13} indicate that the argument uses (I) and replaces `the value of $\phi$' with `the value of $\psi$'. Thus, it seems that the interpretation of the identity in that claim has to be {\em objectual}. This means that in order to believe that the value of $\phi$ is $1$ one has to believe that (i) there is a unique object which is the value of $\phi$, (ii) there is a unique object which is 1, and (iii) these objects are identical. That being the case, it seems that \rref{Drai16} is false. We can indicate a possible situation where one believes $\phi$ without believing (i)-(iii). Suppose I am a radical nominalist. Whether I am right does not matter at all: what matters here is what one can believe. Suppose also that I believe that $\phi$. However, I have pretty strong feelings against the claim that there is a unique object which is the truth-value of $\phi$, or that there is a unique object that is called `number 1'. So I do not believe that the truth-value of $\phi$ is 1 in any sense which would make the application of (I) valid. Thus, Drai's claim that: \begin{quote} Anyone who understands the notions of truth value and True, knows that to say [that the truth-value of $\phi$ = 1, R.U.] is no more and no less than to say [that $\phi$]. (Drai 2002: 202) \end{quote} is misleading: my commitment when I believe that $\phi$ is different from the commitment I make when I believe that the truth-value of $\phi$ =1. Of course, there is a reading of `the truth-value of $\phi$ is 1' which allows also a nominalist to believe it. It is the sense in which `the truth value of $\phi$' and `1' are not interpreted objectually as in fact referring to objects, and the whole sentence is treated just as a fancy and ontologically misleading way of saying that $\phi$ is true. Indeed, in that case to say that $\phi$ may be no more and no less than to say that the truth-value of $\phi$ is 1. But in this reading the slingshot does not work, because there is no real singular term one can substitute for.\footnote{This kind of reading is based on a suggestion by Tadeusz Kotarbi\' nski, a representative of Lvow-Warsaw school, who was a radical nominalist. A very basic claim of his {\em semantic nominalism} was that noun phrases that on the face of it are taken to name something, but do not name concrete individuals only do not name anything in fact, and their role, insofar as their use is meaningful, is to abbreviate or reword some expressions that do not contain such `abstract' noun phrases. A very basic example would be that in `whiteness is a colour.' neither `whiteness' nor `colour' names anything and the whole sentence works as an abbreviation for `everything that is white is coloured'. Abstract names of that sort, because they `pretend' to be names but they do not name anything were called by Kotarbi\' nski {\em onomatoids}.} Since there is nothing contradictory or impossible in the fact that there exists a nominalist who has such beliefs (it does not matter how debatable his beliefs may be), it seems that \rref{Drai8} and \rref{Drai9} are false. Note that the above argument has nothing to do with the assumption about the existence of truth-values. Even if we do assume that there are truth values and even if we believe that the truth-value of $\phi$ =1, and we take all singular terms in this sentence to refer to their respective unique objects, it has no bearing upon the fact that there may exist a nominalist who concedes $\phi$ and does not concede that the truth value of $\phi =1$. This fact alone, no matter what our position about truth-values is, falsifies \rref{Drai8} and \rref{Drai9}. AU - Drai, Dalia PY - 2002/// UR - http://dx.doi.org/10.1111/1467-9329.00184 ER -