ࡱ> 5@ 0bjbj22 &XXr. %%%8.&&D =b*'*'"L'L'L''(+|U-<<<<<<<$>?A<0'('(00<b4(L'L'=9990L'L'<90<9999L'' թ-%_79o<D=0=9A8A90t txt2A9.Zo.@9.4....<< D$ 9 Lecture 1 Logic and Truth in Practice Gary Ebbs University of Illinois at Urbana-Champaign garyebbs@uiuc.edu May 12, 2004 1. Introduction My goal in these lectures is to propose a new way of thinking about the relationship between formal definitions of truth, on the one hand, and our practical applications of those definitions, on the other. There is a long tradition in Western philosophy of trying to clarify existing methods of rational inquiry and of proposing new ones. This tradition was radically transformed in the late 19th and early 20th centuries by Gottlob Frege, Bertrand Russell, and Ludwig Wittgenstein, who developed new mathematical and logical techniques for clarifying rational inquiry. In the 1930's, Rudolf Carnap proposed a new way of doing philosophy that combines insights of Frege, Russell, and early Wittgenstein. Carnap believed that traditional philosophical thinking about knowledge generate unclear questions and fruitless controversies. What is distinctive of the development of science, according to Carnap, is that fruitless controversies are avoided. Scientists work together to formulate questions and test answers to them according to criteria that they all agree on. Carnaps idea was that the methods of scienceespecially of mathematics and mathematical logic--can be used to construct precise rules for evaluating assertions and adjudicating disputes. Starting in the 1950's, W. V. Quine and Hilary Putnam criticized Carnap's proposed clarifications of rational inquiry on the grounds that the rules Carnap articulated do not reflect the ways in which investigations actually unfold. My own approach to clarifying rational inquiry is an attempt to preserve what I see as valuable in Carnap's approach while still acknowledging the force of Quines and Putnam's criticisms of it. I agree with Carnap that a central task of philosophy is to propose new ways of clarifying and facilitating rational inquiry. I also agree with Carnap that one cannot evaluate a given proposal for clarifying rational inquiry unless one is clear about ones motivations for seeking such clarifications. Motivations themselves are not correct or incorrect, but they do have important consequences for our understanding of the task of philosophy, and we must therefore be clear about them from the beginning. Following Carnap, I take for granted that our interest in clarifying rational inquiry is motivated in part by our desire for fruitful collaborations with other inquirers. I start my lecture today by explaining how our desire for fruitful collaborations with other inquirers motivates us to clarify our agreements and disagreements, to paraphrase our words and sentences in the notation of modern logic, and to formulate logical laws by using a truth predicate that sheds light on the logical structure of our sentences. I will then explain the central problem that I want to address in these lectures: it is unclear how the truth predicates that we can define formally are related to our practical identifications of agreements and disagreements, discoveries and proofs. I will briefly explain how standard attempts to solve this problem fail to fit with our actual practices, and I will sketch my own strategy for solving the problem. In lectures 2 and 3 I will examine the problem and explain my solution to it in more detail. 2. Agreement and disagreement To collaborate, we must together keep track of our agreements and disagreements. How do we do this? Suppose you and I are out for a walk, and we are trying to identify the birds we see. Some of them are hidden in the leaves of nearby trees and therefore difficult to identify. If in these circumstances you say, That is a robin, pointing to an ill-glimpsed but contextually salient bird, then unless I see some special reason in the context for not doing so, I will take you to have said that the bird is a robin and, in doing so, I will take your word robin be the same as my word robin. This illustrates our practice of taking words that are spelled or pronounced in the same way to be the same, unless we have reason not to do so in a particular contexta practice that is integral to our identifications of agreement and disagreement. If I believe that the bird is robin, then when I take you to have said that the bird is a robin, I will take myself to agree with you. If I believe that the bird is not a robin, then when I take you to have said that the bird is a robin, either I will take myself to disagree with you, or I will defer to your judgment, revise my belief, and take myself to have learned from you that the bird is a robin. This way of identifying agreements, disagreements, and cases in which we learn from what other say, extends across time, from moment to moment, and in some cases for centuries. For instance, if I discover that in 1650 a jeweler showed John Locke a ring and said, This ring is gold, Ill take the jeweler to have asserted that the ring was gold. Just as in the first case, my practice of taking words that are spelled or pronounced in the same way to be the same, together with my own beliefs and the beliefs I attribute to the jeweler, settles (for the moment, at least) whether I take myself to agree with him, disagree with him, or to learn from him that the ring was gold. Our practice of taking words that are spelled or pronounced in the same way to be the same, and hence true of all and only the same things, is also integral to our confidence in a given case that we have made an empirical discovery, and not simply stipulated a new use for an old term. For instance, when English-speaking chemists inquire into the nature of gold, they presuppose our largely unreflective history of taking words that are spelled or pronounced in the same way to be the same from moment to moment, in a chain that reaches far into the past. They take this history for granted when they regard a 19th century chemists utterance of the sentence Gold is the element with atomic number 79 as the announcement of a discovery, not the invention of a new use for the English word gold. Similarly, when English-speaking mathematicians prove theorems about ellipses, for instance, they presuppose their largely unreflective history of identifying other English-speaking mathematicians uses of the word ellipse across time. They take this history for granted when they take their proof that a plane intersects a cylinder in an ellipse to justify a new belief about an old topic, ellipses, that mathematicians have identified and studied for centuries. In short, we typically keep track of our agreements and disagreements, empirical discoveries, and deductive proofs by taking words that are spelled or pronounced in the same way to be the same and evaluating our utterances accordingly. 3. How our desire for fruitful collaborations motivates us to regiment our language Fruitful collaboration requires that we have some relatively clear and neutral ways of identifying agreements and disagreements between us. In some situations, our desire to collaborate motivates us to clarify our shared assumptions about how our sentences are related, including our assumptions about which sentences are equivalent, and in this way to sharpen our understanding of when we agree or disagree. For instance, suppose you and I are discussing a house you recently visited, and you say, "A tree is visible from every window of the house." Without hesitating, I take you to have said and to believe that a tree is visible from every window of the house. Suppose also that I know that from every window of the house, some tree or other is visible, but I believe that no tree is visible from every window of the house, and I express this belief by saying, No tree is visible from every window of the house. Then I will conclude that what you said is incompatible with what I said. I take "A tree is visible from every window of the house" to be equivalent to "There is some tree that is visible from every window of the house," and No tree is visible from every window of the house to be equivalent to It is not the case that there is some tree that is visible from every window of the house, so I regard the conjunction of the sentence that expresses your belief with sentence that expresses my belief to have the form S and not S, where S is A tree is visible from every window of the house." If you do not accept the equivalence, you will not notice that we disagree. The equivalence is not difficult to see, but linguistic competence alone is not enough to guarantee that a person will see it. Suppose that you don't at first take the belief you expressed by saying, "A tree is visible from every window of the house," to be incompatible with the belief I expressed by saying, No tree is visible from every window of the house. Discussion in ordinary language may help with this, but in some circumstances it would be quicker and easier to replace the two sentences with sentences written in the standard notation of first-order logic. If we agree to use the symbols for quantifiers and truth functions in the usual way, and agree on how to construct predicates, such as x is a tree and x is visible from y, that we are willing to use in place of ordinary English phrases, we can agree that in place of "A tree is visible from every window of the house," for instance, we will now use (x((x is a tree) & (y((y is a window in the house) ( (x is visible from y)) and in place of From every window of the house, a tree is visible, we will now use (y((y is a window in the house) ( (x((x is a tree) & (x is visible from y))). We can then use uncontroversial logical methods to determine that the two regimented sentences are not equivalent. Hence once we express our beliefs in this new notation, it is easier for us to settle whether we agree or disagree. 4. How our desire for fruitful collaborations motivates us to state logical laws If our desire to collaborate fruitfully motivates us to regiment our sentences, then it also motivates us to state logical laws. To see this, consider the role of logical inconsistency in our evaluations of whether we agree or disagree. Recall that on our walk, when you say, "That is a robin, pointing to an ill-glimpsed but contextually salient bird, then unless I see some special reason in the context for not doing so, I will take you to have said that the bird is a robin. Suppose that I believe it is not a robin, and I express this belief by saying, "That is not a robin. And suppose, further, that we agree to regiment our utterances, using "the bird" to denote the bird you pointed to. Then I will take you to have said that the bird is a robin, and you will take me to have said that (the bird is a robin). If we conjoin the sentences that express our respective beliefs, the result is  (the bird is a robin) & ( the bird is a robin) . Normally we don t bother to conjoin the sentenceswe wouldnt even think of accepting them both simultaneously, because we take for granted that the two sentences are logically incompatible. How should we understand this? Here is one compelling answer: no matter what sentences we substitute for The bird is a robin in the sentence ((the bird is a robin) & ( the bird is a robin)) , the result will be another true sentence: every sentence of the form  (s & s) is true. There are well-known techniques for settling whether or not every sentence of a given logical form is true. In particular, truth-value analysis can be used to show that every sentence of the form  (s & s) is true. And from this generalization, we can infer any particular sentence of that form, regardless of what it is about and regardless of our opinions about the truth values of its component sentences. You and I disagree about whether the bird is a robin, but we may nevertheless both accept  ((the bird is a robin) & ( the bird is a robin)) for the same reason: we see by truth-value analysis that every sentence of the form  (s & s) is true. This logical law makes explicit that we cannot both be right if we did not both at least tacitly accept it, we would not take ourselves to disagree about whether the bird is a robin. By stating and establishing this and other logical laws, we clarify our practical identifications of our agreements and disagreements. 5. Why we need a truth predicate To collaborate fruitfully, I've argued, we need to clarify our practical identifications of agreements and disagreements, and this leads us to state and establish logical laws. But to state logical laws in a practical and illuminating way, a truth predicate is indispensible. To see why, let us begin with the following pattern: (T) ______ is true if and only if _______. This pattern is called disquotational, because words that we put in its blanks will appear within quotes in their first occurrence in the resulting sentence and without quotes in their second occurrence. For instance, if we write  ((the bird is a robin) & ( the bird is a robin)) in the blanks of (T) the result is (t)  ((the bird is a robin) & ( the bird is a robin)) is true if and only if ((the bird is a robin) & ( the bird is a robin)). Truth predicates such as this one partially defined by (t) have two evident virtues. First, they are transparent--clear and neutral about controversial metaphysical questions. Second, and equally important, they are locally eliminablethey allow us to eliminate applications of the truth predicate to particular sentences we are a position to identify and use. For instance, (t) licenses us to replace ((the bird is a robin) & (the bird is a robin)) is true by  ((the bird is a robin) & ( the bird is a robin)) . We need a truth predicate that functions in this way at the level of particular sentences if wish to infer  ((the bird is a robin) & ( the bird is a robin)) , for instance, from  every sentence of the form  (s & s) is true . In these two respects, then, (T) provides a model of how a truth predicate should be defined. But there is a potential infinity of sentences of the form  (s & s) , and we cannot write them all down. We therefore cannot always use truth predicates defined by (T) to replace logical laws by conjunctions in which no truth predicates occur. We need to define a truth predicate that is faithful to the pattern (T), but enables us to apply a truth predicate to generalize over a set of sentences that we cannot list. 6. Tarskis method of definiting of truth Alfred Tarski proposed that to capture and clarify what is appealing about the disquotational truth predicates that result when we write sentences of the object language in the blanks of (T), it is sufficient to define truth predicates that meet the following material condition of adequacy: Convention (T): a formally correct definition of the symbol 'true', formulated in the metalanguage, will be called an adequate definition of truth if it implies all sentences which are obtained from the expression 'X is true if and only if p' by substituting for the symbol 'X' a structural-descriptive name of any sentence of the object language and for the symbol 'p' the expression which forms the translation of this sentence into the metalanguage. To understand this condition of adequacy, we must understand the notion of a structural-descriptive name of a sentence of the object language and when a sentence of the metalanguage translates a sentence of the object language. The easiest way to form a structural-descriptive name of a sentence is put it within single quotation marks; this method is illustrated by the left-hand sides of biconditionals that result from writing sentences in the blanks of (T). The term "translation" is not as clear, however; one might wonder how to determine whether a sentence of the metalanguage translates a sentence in the object language. I propose that we clarify translation, as it occurs in Convention T, by reflecting on the role of a metalanguage in introducing regimented sentences that can be used in place of sentences of the metalanguage. When we agree to regiment ordinary English sentences in first-order predicate logic, we are taking ordinary English as our metalanguage, and regimented English as our object language. We can agree to use regimented English sentences in place of other sentences without claiming that we are capturing the meaning of those sentences. The crucial point is that we can reach agreement about this, and take it as uncontroversial, and in that sense clear, that a given object language sentence can be used in place of a corresponding metalanguage sentence. Moreover, agreements of this kind go hand-in-hand with agreements about how to define truth for sentences of the object language. When we agree to adopt a particular Tarski-style definition of truth for sentences of the object language, we thereby agree to the correspondence between the object language sentences and the metalanguage sentences that are settled by the definition. To define a truth predicate for an object language using Tarskis method one must specify for every sentence of language the conditions under which that sentence is satisfied by a given sequence of objects in the domain of discourse for the object-language. (A sequence of objects is a function from positive integers to objects.) Suppose the object language L for which we are defining satisfaction has a very simple structure: it comprises all and only sentences constructed in the usual way from a finite list of basic predicates, such as mortal and loves, variables x1, x2, ..., truth-functional symbols '(' and (, and a quantifier symbol '('. To define satisfaction for L, we need structural-descriptive variables in the metalanguage that range over variables of the object language. For this purpose, let the ith variable in the sequence x1, x2, ..., be called var(i). In particular, then, var(2)=x2. We also need a variable that ranges over the objects in a given sequence s of objects. For this purpose, let si be the ith object in sequence s. Now suppose we use words of ordinary Englishour metalanguageto introduce and explain the predicates of the object language. We agree, for instance, that the object language will contain regimentations of the ordinary English predicates is a robin and loves. Then we can codify our regimentation of ordinary English predicates by agreeing on when a given sequence satisfies them. We can agree, for example, that for every sequence s, s satisfies robin followed by var(i) if and only if si is a robin. This follows the disquotational pattern: (S1) For every sequence s, s satisfies ___ followed by var(i) if and only if si is _____. We can accept a similar pattern for two-place predicates. We can agree, for example, that a sequence s satisfies loves followed by var(i) and var(j) if and only if si loves sj. If we abbreviate si loves sj as loves si sj this biconditional can be seen as an instance of (S2) For every sequence s, s satisfies ____ followed by var(i) and var(j) if and only if _____ si sj. In general, for any predicate with a finite number of argument places, there is a corresponding disquotational pattern of this kind. Suppose we have specified the satisfaction conditions for each of the n basic predicates of L in the way just described, and labeled each one with a separate numeral from 1 to n. We can then complete our definition of satisfaction for L by adding the following clauses: (n+1) For all sequences s and sentences S: s satisfies the negation of S if and only if s does not satisfy S. (n+2) For all sequences s and sentences S and S': s satisfies the alternation of S with S' if and only if either s satisfies S or s satisfies S'. (n+3) For all sequences s, sentences S, and numbers i: s satisfies the universal quantification of S with respect to var(i) if and only if every sequence s' that differs from s in at most the ith place satisfies S. Clauses (n+1) and (n+2) are straightforward. The key insight behind clause (n+3) is that if s is any sequence of objects from the domain of discourse D for the universal quantifier of the object language, then the set of all and only the ith things of any sequence s' that differs from s in at most the ith place is identical with D. In this way, clause (n+3) codifies our agreement to use the universal quantifier of the object language in place of such phrases as "for everything in D . . .. Together with the satisfaction clauses for the n simple predicates of the language, clauses (n+1), (n+2), and (n+3) define satisfaction for all sentences of the language. Using this definition of satisfaction, we can then define truth for this language L as follows: (*) A sentence S of L is true-in-L if and only if S is satisfied by all sequences. To see that (*) satisfies Convention T, consider the following derivation of a statement of conditions under which the sentence (x1((x1 is a robin) ( ((x1 is a robin)) of L is true-in-L: (1) (x1((x1 is a robin) ( ((x1 is a robin)) is true-in-L if and only if (x1((x1 is a robin) ( ((x1 is a robin)) is satisfied by every sequence. [By (*)] (2) (x1((x1 is a robin) ( ((x1 is a robin)) is satisfied by every sequence if and only if for every sequence s, every sequence s' that differs from s in at most the 1th place satisfies (x1 is a robin) ( ((x1 is a robin). [By (n+3)] (3) For every sequence s, s satisfies (x1 is a robin) ( ((x1 is a robin) if and only if s satisfies (x1 is a robin) or s satisfies ((x1 is a robin). [By (n+2)] (4) For every sequence s, s satisfies ((x1 is a robin) if and only if s does not satisfy (x1 is a robin). [By (n+1)] (5) For every sequence s, s satisfies x1 is a robin if and only if si is a robin. [By the satisfaction clause for robin] (6) For every sequence s, s satisfies (x1 is a robin) ( ((x1 is a robin) if and only if si is a robin or si is not a robin. [By (3)-(5)] (7) For every sequence s, every sequence s' that differs from s in at most the ith place satisfies (x1 is a robin) ( ((x1 is a robin) if and only if everything in D is either a robin or not a robin. [From (6) and the fact that the set of all and only the ith objects of any sequence s' that differs from any sequence s in at most the ith place is identical with D.] (8) (x1((x1 is a robin) ( ((x1 is a robin)) is true-in-L if and only if everything in D is either a robin or not a robin. [From (2) and (7)] From (*) we can derive a biconditional of this form for each sentence of L. I proposed earlier that we take our Tarski-style definitions of truth to settle a "translation" of sentences of the object language by sentences of the metalanguage. Seen in this way, (*) settles that the right-hand side of (8), for instance, "translates" (x1((x1 is a robin) ( ((x1 is a robin)). More generally, for each sentence of L we can derive a biconditional similar to (8), and for each of these biconditionals, the metalanguage sentence that occurs on its right hand side "translates" the object language sentence quoted on its left-hand side. If we understand "translation" in the way that I propose, therefore, we can see that (*) satisfies Convention T. 7. Truth and logical laws Let us now consider how the truth predicate enables us to formulate logical laws for sentences of L. For instance, consider the law that every L-sentence of the form (x1(Fx1 ( ( Fx1) is true-in-L. To understand this law, we need to know what counts in L as a sentence of the form (x1(Fx1 ( ( Fx1). The schematic letter F indicates the place in the sentences of that form where one-place predicates of L occur. Taking __ is a robin as a one-place predicate of L, we can see that (x1((x1 is a robin) ( ((x1 is a robin)) is a sentence of the form (x1(Fx1( ( Fx1). But it is not always easy to tell whether a sentence has this form. Is (x1(((y1(x1 loves y1) ( (( ((y1(x1 loves y1)) an L-sentence of the form (x1(Fx1( ( Fx1)? What about (x1(((x1(x1 loves x1) ( (( ((x1(x1 loves x1))? To answer such questions (the answers are, respectively, Yes, and No) we need a precise characterization of what counts as an L-sentence of one of these forms (x1(Fx1( ( Fx1). This need is met by well-known syntactical criteria for admissible substitutions of regimented English sentences and predicates for schematic letters. The next question is whether the truth predicate we defined, true-in-L, enables us to generalize on sentences of L in the appropriate way. Given the grammatical structure of L, we know that there are infinitely many sentences of the form (x1(Fx1( ( Fx1). We cannot write them all out, so we need a truth predicate to say that they are all true. As I explained above, however, we want particular instances of our truth predicate to be eliminable, so we can get, for instance, from (x1((x1 is a robin) ( ((x1 is a robin)) is true-in-L to (x1((x1 is a robin) ( ((x1 is a robin)). But so far we have only shown that our definition of true-in-L satisfies Convention Tin particular, we have only established (8), which states that (x1((x1 is a robin) ( ((x1 is a robin)) is true-in-L if and only if everything is either a robin or not a robin. This claim does not by itself show that we can replace (x1((x1 is a robin) ( ((x1 is a robin)) is true-in-L by (x1((x1 is a robin) ( ((x1 is a robin)). To make this final step, we need only suppose that our metalanguage contains the sentences of the object language, and that in the metalanguage we can use sentences of the object language interchangeably with those sentences of the metalanguage that "translate" them in the way described above. When we define truth for L in the way described above, we thereby agree, in effect, to use (x1((x1 is a robin) ( ((x1 is a robin)) in place of everything is either a robin or not a robin; given this agreement, and our derivation of (8), we can replace (x1((x1 is a robin) ( ((x1 is a robin)) is true-in-L by (x1((x1 is a robin) ( ((x1 is a robin)). Similar equivalences hold for all of the other sentences of L. In this way, our definition of true-in-L enables us to state logical laws for sentences of L. 8. Idiolects and practical judgments of sameness of denotation At any given time, a person can define satisfaction and truth in the way described above only for words and sentences of her own current idiolect--the set of words and sentences she can directly use at that time. Moreover, to use the method, we must suppose that our metalanguage contains the object language, and that we can identify and distinguish between sentences of the object language solely on the basis of their spelling. If our current idiolect contains ambiguous words, we must replace them with unambiguous words before we can define truth for sentences of the object language that contain those words. No one is ever in a position to guarantee that she has met this requirement. Instead, the requirement functions as a regulative ideal: if one becomes aware of ambiguities in one's own words, one must further regiment ones idiolect before one can define a truth predicate for sentences that contain those words. In short, to define truth and satisfaction for ones own words and sentences at any given time one must regard sameness of spelling as a criterion for sameness of the words of one's own idiolect at that time. When two or more speakers together define a truth predicate for sentences of a language they agree to use, they take for granted that they share an idiolect. In such local contexts of agreement, sameness of spelling can serve as an intersubjective criterion of sameness of word. For instance, two speakers of American English may agree to include among the satisfaction clauses for L a clause for the American English word robin. The clause will state satisfaction conditions for the word, identified solely by its spelling. After their agreement, they can treat the sameness of spelling of their word robin as an intersubjective criterion for sameness of word, and hence sameness of satisfaction conditions. But sameness of spelling is not in general a criterion for sameness of satisfaction conditions or of truth conditions. For instance, in British English the word spelled r-o-b-i-n is true of birds that look in some respects like robins, but arent robins. A speaker of American English who knows this will not take a British speaker's utterance of a sentence containing a word spelled r-o-b-i-n as an utterance of the sentence containing the word of her idiolect that is spelled in the same way. Such examples show that in the sense of word that matters to us when we identify our agreements and disagreements with other speakers, a word is not individuated by its spelling alone. Yet it is unclear how to fit this fact with Tarskis method of defining truth. The method identifies and distinguishes between sentences of the object language solely on the basis of their spelling. It is tempting to take this to imply that Tarskis method of defining truth requires that words be individuated by their spelling. In fact, however, to use Tarski's method, we need not suppose that words are individuated by their spelling, only that we can identify and distinguish between words of our current idiolect by spelling alone. Understood in this way, Tarski's method is silent about when a truth predicate defined for sentences of one idiolect applies to the sentences of another. It seems that one could use Tarski's formal method to define a truth predicate for sentences of one's own idiolect without making any commitments whatever to its application to sentences of other idiolects. Seen in this way, truth predicates defined by Tarski's method are unrelated to our practical identifications of agreement, disagreement, discoveries, and proofs. To see how this affects our understanding of how to apply truth predicates defined by Tarski's method, recall the following pattern: (S1) For every sequence s, s satisfies ___ followed by var(i) if and only if _____ si. The analogous disquotation pattern for denotation is: (D1) For every sequence s, ____ followed by var(i) denotes si if and only if _____ si. In principle, at least, each speaker can apply these disquotational patterns to his own words. For instance, if I affirm the results of writing is a robin in the blanks of (S1) and (D1), I assert that sequence s satisfies (my predicate) is a robin followed by var(i) if and only if si is a robin, and (my predicate) is a robin followed by var(i) denotes an object si if and only if si is a robin. Suppose you and I have together constructed a truth predicate for the regimented language L in the way described above. Then in effect we share an idiolect in which we define truth for L. The truth definition itself, considered formally, does not illuminate this aspect of our linguistic interactions. We can begin to clarify this aspect of our linguistic interactions, however, by describing the relationship between our definitions of denotation (satisfaction) and truth for L and our unreflective practice of identifying words. For instance, if I affirm the result of writing my predicate is a robin in the blanks of (D1), I can see that when I take your predicate is a robin to be the same as my predicate is a robin, I in effect take for granted that your predicate is a robin denotes si just in case si is a robin, and so your predicate is a robin has the same denotation as my predicate is a robin. This is what I call a practical judgment of sameness of denotation. If I take your predicate is a robin to be the same as my predicate is a robin while I am talking to you, I make what I call a practical judgment of sameness of denotation at a given time. If I take another English speakers predicate is a robin to be the same as my predicate is a robin while I am reading a sentence he wrote some time ago, I make what I call a practical judgment of sameness of denotation across time. The problem is that Tarski's method the defining truth does not by itself license our practical judgments of sameness of denotation. It seems that one could use Tarski's formal method to define a truth predicate for sentences of one's own idiolect without accepting any of our practical judgments of sameness of denotation. Moreover, if we do accept them, it seems that our acceptance of them requires some explanation or justification that is independent of Tarski's formal method of defining truth. 9. Subjective or objective? There are apparently two possibilities. Either our practical judgments of sameness of denotation are determined by independently specifiable facts, or they are not. If the former, they are objective, or factual; if the latter, they are subjective. In the past 40 years philosophers have embraced one or the other of these alternatives. W. V. Quine, for instance, argued that our practical judgments of sameness of denotation are scientifically arbitrary, and therefore neither true or false. In Quines view, our practical judgments of sameness of denotation merely reflect our subjective preferences about how to translate other speaker's words. (Quine sharply distinguish between such practical judgments of sameness of denotation and the results of defining truth for one's own words by using Tarski's methods, which he regarded as part of logic, hence objective.) But many philosophers are unconvinced by Quines arguments, and seek theories that justify or explain our practical judgments of sameness of denotation. According to Saul Kripke and Hilary Putnam, for instance, facts about the objects to which we apply our words and facts about the relationships that speakers bear to each other in their linguistic communities together determine that our practical judgments of sameness of denotation are true or false, hence objective. Neither of these alternatives enables us to accept our practical judgments of sameness of denotation as integral to our understanding of truth and rational inquiry, however. Even worse, neither of these alternatives actually fits with the practical judgments of sameness of denotation that we trust in the midst of our inquiries. Quines view implies that truth is not an objective property of other people's utterances, and therefore implies that ordinary cases in which we take ourselves to disagree with others are based our arbitrary psychological inclinations about how to translate their words. This conflicts with our conviction that agreement and disagreement are not merely psychological phenomena, but concern truth and falsity. On the other hand, for reasons I will explain in lecture 2, the theory proposed by Kripke and Putnam fails to justify or explain our own best judgments of when speakers are using the same words with the same denotations, and no other attempts to justify or explain such judgments have yet succeeded. There's a growing pessimism among philosophers about the prospects of explaining or justifyng our practical judgments of sameness of denotation. 10. A Proposal To overcome these problems with these standard ways of understanding Tarskis method of definiting truth, I propose that we combine Tarskis formal method of defining denotation and truth for words and sentences of our own idiolect with our ordinary intersubjective identifications of other speakers words. These identifications are often, but not always, based on sameness of spelling or pronunciation. By incorporating our ordinary criteria for sameness and difference of words into our understanding of truth predicates defined in the way described above, we can apply truth predicates defined by Tarskis disquotational method to words and sentences of other English speakers idiolects in a way that fits with the fact that we typically take words that are spelled or pronounced in the same way to be true of the same things, unless we see good reason in a given context for not doing so. This way of relating truth predicates defined by Tarskis disquotational method to words and sentences of other English speakers idiolects suggests that our understanding of truth is inextricable from our practical identifications of words. We make practical judgments of sameness of denotation without appealing to or providing evidence for them. We nevertheless take ourselves to be entitled to accept them unless we see good reason not to do so. We frequently revise our practical judgments of sameness of denotation in light of what we learn, and this shows that we do not think their true or false is determined by our subjective inclinations about how to take other speakers words. It is less obvious but equally important, and a central theme of these lectures, that we have no understanding of denotation apart from our practical judgments of sameness of denotation. The heart of my proposal is that we view our practical judgments of sameness of denotation as objective, hence factual, even though they are not determined by independently specifiable facts. This proposal requires that we reject the standard view that we cannot see our practical judgments of sameness of denotation as objective unless there are determined by independently specifiable facts. Unfortunately, there is a deeply entrenched conception of words that makes it difficult to accept my proposal. In lecture 2 I will identify the standard conception of words, and explain how it creates the illusion that either our practical judgments of sameness of denotation are subjective or they are determined by independently specifiable facts. I will also explain why neither of these alternatives is acceptable. In lecture 3 I will present a practical alternative to the standard conception of words, and explain how this alternative allows us to see practical judgments of sameness of denotation as integral to our understanding of satisfaction, denotation and truth. PAGE  PAGE 22 Our awareness of deductively valid arguments also shapes our understanding of what is compatible with what. If you say, "Every human is a robin, and Achilles is human, I will take you to have said that Every human is a robin and Achilles is human. Since I know that Every human is a robin and Achilles is human implies Achilles is a robin, I will take your claim to be incompatible with the claim that Achilles is not mortal. This elementary example suggests a recipe for describing situations in which two speakers might disagree about whether they disagree: if my conclusion that I disagree with another speaker depends on my confidence that a p %&quz{|` a d f g i k l \ k % 4 L v  ϿhIh mh=h<hhh/h(h#h]h3\hC@hTNhe>hqh)[h;hqA.hIahXrh0N`hY*[h=th=th,CJaJh=thm'CJaJh=th?CJaJh,2 &0[mz{| *1234T!!!!!!!!!!!8P!!:!;-@!Y!Y!Y!Y!4 d`gd#dgd% d`gd d`gd}_dgdqgdXrgd,$a$gd=t$a$gdm'$a$gdqA.ҨӨ +jk ./*,-/0:;]^)*;OTac&<y3ؼ촪윕h h}_h(`h(`h!h!6]h!h}6]h}h"h}_h!hfhf6]hfhA}6]hfhhA}hQkhu!Zh4Xh hPh<h19!,-.01243UYv"LO!=@ KL{Д h{!h%hkhU@hU@h?L6]h?Lh?L6]hR=hNsFhch#h?Lh%hO=?h}hRh}_hb:hhz h"6] h"h"h"h4Xh"h4X6]5=!"" #!#$)..,/-/4g>>>!UE!*!8P!O !Y!Y!!8P!?P!Y!Y!Y!Y!YdgdJ d`gdYdgdO=?dgd^k d`gdwdgd%dgd;C d`gd% d`gdei d`gdNsF""""""#W)X)Y)**!,",3,4,E,F,,,,,------.-/-1-2-C-D-----------------....//*0131:1a1x1111ȹh7h jh%hh%6 j"h% j$h%hh%6]jh;C0JU h;Ch;Ch;Ch>h^kh% h%6D11111L2222r3333333.434\4{4|44445(5666,6466686:6`66666778888::;;;>;F;H;J;L;r;$<&<0<2<==g>AAA? @BC|IfLgLLLM}OMQtV YZ\!id!De !Y!Y!Y!id! ! !U!*+!!o% d`gd d`gd}_hhd]h^hgd d`gd{)dgd d`gdJdgd}20d^`0gddgddgdJgdJBBBBBBBB0C2CdCfChCjCCCCCD4D?DDD*EJEdEFF4F6FFFFFFFFGGGGGGH,HHHHH|IIIIJJlKeLfLLLL{O|OOO鹵 h6jh0JUh{)h1hh;hJOJQJ hh}2h}2OJQJhm hJ6h.Wh}2hJ hhhOJQJh@O3P=PKQLQQQlRwRRRSSSSsVtVwV}VW"W.W6WXXXXXXXXYYYYYYYYYYYYZZ`ZaZZZZZZZZZ[[`\a\c\e\p\u\\\\\\\\\\\\]]$]'] h6H* j"h jh jh hH*hmh{)jh0JU h6hM\1]H^^<_L``La$bd$e{e:ffgmhheiibk!Y!O !4 !4 !id!4 !4 !O ! !id!Y!d!De !^!De ! !4 ! !dgd}Gdgdq'ddgd' d`gd$0d^`0a$gd d`gd0d^`0gddgd']]]]]]]]]]]]]]]^^^^^^^b^c^e^g^^^^^^^^^^^^^^<_______L`M`N`d`e`w`x`````````+a,adgd3rdgd0 d`gd0 d`gdw.dgdw. d`gdqdgdqQZ[hʎˎ !)*+-:;<=@GUly 9i̐Y֑ :;RuɔΔ&ѿѳh|yh+Tsh)h#hwh0hsggh06]h0h{ hq6>* hqH* hq6H* hq6 hw.hw.hqhw.D&*34vޕ%,467<hl–Ŗ˖̘ؗٗ͘ژ "TW_sǙ$1;Лӛ8TؾƾԶƶhb>hzhr&hEhx ~h@3UhShSht06]hpht0hkBhch+Tsh@-hwh\h,h0h2VDE]Ɲȝ&"#ߡuѣأ)W\0;Mf|Φ69HQ~ѷѳ쟛쓛h!>Yh~h;hxxhThhSh)hPhPhq6]hjb hYhq hq6hqh-f,jhA0JU hA6h4ahzhAh0hb>h@3U8Ƨɧѧҧڧ[eƨǨ˨̨ͨΨϨШҨӨԨڨۨܨިߨ|AB} hh> h>6Ujh>0JUh'`I0JmHnHuh> h>0Jjh>0JUh}h7hh%h;Ch1hkDjhq0JUh hq6]h hhjbhzhqh!>Yh;h.Q.ʨ˨̨ͨΨϨШҨӨܨݨިAYX!UE!Y!Y!Y!Y!Y!Y!Y! !v:!v:!v:!gdgd% &`#$gd;Cdgdu!ZdgdO=?dgd;CdgddgdkD d`gd;articular complicated argument is deductively valid, and the other speaker is unaware of the argument, does not know that it is deductively valid, or doubts that it is, then we may disagree about whether we disagree. This difficulty is connected with the one about equivalence, since equivalence is mutual implication. Tarski, "The Concept of Truth in Formalized Languages," from Logic, Semantics, Meta-Mathematics, by Alfred Tarski, pages 187-188.  Another way is to use spelling and concatenation. If the metalanguage contains elementary arithmetic, one can also use Gdel numbers to form structural descriptive names of object language sentences. Hence structural-descriptive names of sentences of the object language can be constructed in the metalanguage in a number of different ways by using formal vocabulary that is unproblematic and clear. L has no separate symbol for the existential quantifier; existential quantifications must be expressed in terms of negation and universal quantification. Other truth functional connectives, such as '(' and '&', can expressed in terms of '(' and '('. The definition is inductive, not explicit. Say what that means, and how to convert it to an explicit definition. See for instance Quine's syntactical criteria for substitution, presented in chapters 26 and 28 of his book Methods of Logic, fourth edition, (Cambridge, Mass.: Harvard University Press, 1982). The orthodox answer to this question is that Tarskis method of defining truth applies in the first instance to languages viewed as abstract objects, independent of particular language users. Once a truth predicate is defined for an abstract language L, we can ask how L is related to languages that are actually in use. The orthodox view is that there must be facts about how a speaker uses her words that determine whether or not her sentences have the same truth conditions as the sentences of L that are spelled in the same way. On this view, Tarskis method of defining must be supplemented by a substantive theory of the conditions under which one speakers definitions of satisfaction and truth for his own words and sentences is also a definition of satisfaction and truth for to the words and sentences of another speaker. If we embrace this standard view, truth predicates defined by Tarskis method for particular idiolects must be supplemented by substantive theory of when two speakers use the same language. Satisfaction, denotation and truth can only be defined for regimented languages. But most speakers do not even know how to regiment their sentences or define truth for them. Strictly speaking, therefore, most speakers do not make judgments of sameness of denotation. But they are always in a position to do so, provided they acquire the necessary logical vocabulary and formal methods. Loosely speaking, they are in principle committed to such judgments by their practice of taking each other's words at face value. 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