This is another of those ideas that ive had independently, and that it turned out that others had thought of before me, by thousands of years in this case. The idea is that longer expressions of language as made out of smaller parts of language, and that the meaning of the whole is determined by the parts and their structure. This is rather close to the formulation used on SEP. Heres the introduction on SEP:<\/p>\n
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Anything that deserves to be called a language must contain meaningful expressions built up from other meaningful expressions. How are their complexity and meaning related? The traditional view is that the relationship is fairly tight: the meaning of a complex expression is fully determined by its structure and the meanings of its constituents\u2014once we fix what the parts mean and how they are put together we have no more leeway regarding the meaning of the whole. This is the principle of compositionality, a fundamental presupposition of most contemporary work in semantics.<\/span><\/p>\n Proponents of compositionality typically emphasize the productivity and systematicity of our linguistic understanding. We can understand a large\u2014perhaps infinitely large\u2014collection of complex expressions the first time we encounter them, and if we understand some complex expressions we tend to understand others that can be obtained by recombining their constituents. Compositionality is supposed to feature in the best explanation of these phenomena. Opponents of compositionality typically point to cases when meanings of larger expressions seem to depend on the intentions of the speaker, on the linguistic environment, or on the setting in which the utterance takes place without their parts displaying a similar dependence. They try to respond to the arguments from productivity and systematicity by insisting that the phenomena are limited, and by suggesting alternative explanations.<\/span><\/p>\n <\/p>\n SEP goes on to discuss some more formal versions of the general idea:<\/p>\n <\/p>\n (C) The meaning of a complex expression is determined by its structure and the meanings of its constituents.<\/span><\/p>\n <\/p>\n and<\/p>\n (C\u2032) For every complex expression e in L, the meaning of e in L is determined by the structure of e in L and the meanings of the constituents of e in L.<\/span><\/p>\n <\/p>\n SEP goes on to disguish between a lot of different versions of this. See the article for details.<\/p>\n The thing i wanted to discuss was the counterexamples offered. I found none of them to be rather compelling. Based mostly on intuition pumps<\/a> as far as i can tell, and im rather wary of such (cf. Every Thing Must Go<\/em>, amazon<\/a>).<\/p>\n <\/p>\n Heres SEP’s first example, using chess notation (many other game notations wud also work, e.g. Taifho<\/a>):<\/p>\n <\/p>\n Consider the Algebraic notation for chess.<\/span>[<\/a>15]<\/sup><\/span> Here are the basics. The rows of the chessboard are represented by the numerals <\/span>1<\/span><\/em>, <\/span>2<\/span><\/em>, \u2026 , <\/span>8<\/span><\/em>; the columns are represented by the lower case letters <\/span>a<\/span><\/em>, <\/span>b<\/span><\/em>, \u2026 , <\/span>h<\/span><\/em>. The squares are identified by column and row; for example <\/span>b5<\/span><\/em> is at the intersection of the second column and the fifth row. Upper case letters represent the pieces: <\/span>K<\/span><\/em> stands for king, <\/span>Q<\/span><\/em> for queen, <\/span>R<\/span><\/em> for rook, <\/span>B<\/span><\/em> for bishop, and <\/span>N<\/span><\/em> for knight. Moves are typically represented by a triplet consisting of an upper case letter standing for the piece that makes the move and a sign standing for the square where the piece moves. There are five exceptions to this: (i) moves made by pawns lack the upper case letter from the beginning, (ii) when more than one piece of the same type could reach the same square, the sign for the square of departure is placed immediately in front of the sign for the square of arrival, (iii) when a move results in a capture an <\/span>x<\/span><\/em> is placed immediately in front of the sign for the square of arrival, (iv) the symbol <\/span>0-0<\/span><\/em> represents castling on the king’s side, (v) the symbol <\/span>0-0-0<\/span><\/em> represents castling on the queen’s side. <\/span>+<\/span><\/em> stands for check, and ++ for mate. The rest of the notation serves to make commentaries about the moves and is inessential for understanding it.<\/span><\/p>\n Someone who understands the Algebraic notation must be able to follow descriptions of particular chess games in it and someone who can do that must be able to tell <\/strong><\/span>which<\/strong><\/span><\/em> move is represented by particular lines within such a description.<\/strong><\/span> Nonetheless, it is clear that when someone sees the line <\/span>Bb5<\/span><\/em> in the middle of such a description, knowing what <\/span>B<\/span><\/em>, <\/span>b<\/span><\/em>, and <\/span>5<\/span><\/em> mean will not be enough to figure out what this move is supposed to be. It must be a move to <\/span>b5<\/span><\/em> made by a bishop, but we don’t know which bishop (not even whether it is white or black) and we don’t know which square it is coming from. All this can be determined by following the description of the game from the beginning, assuming that one knows what the initial configurations of figures are on the chessboard, that white moves first, and that afterwards black and white move one after the other. But staring at <\/span>Bb5<\/span><\/em> itself will not help.<\/span><\/p>\n <\/p>\n It is exacly the bold lines i dont accept. Why must one be able to know that from the meaning alone? Knowing the meaning of expressions does not always make it easy to know what a given noun (or NP) refers to. In this case \u201cB\u201d is a noun refering to a bishop, which one? Well, who knows. There are lots of examples of words refering to differnet things (people usually) when used in diffferent contexts. For instance, the word \u201cme\u201d refers to the source of the expression, but when an expression is used by different speakers, then \u201cme\u201d refers to different people, cf. indexicals (SEP<\/a> and Wiki<\/a>).<\/p>\n <\/p>\n Ofc, my thoughts about are not particularly unique, and SEP mentions the defense that i also thought of:<\/p>\n <\/p>\n The second moral is that\u2014given certain assumptions about meaning in chess notation\u2014we can have productive and systematic understanding of representations even if the system itself is not compositional. The assumptions in question are that (i) the description I gave in the first paragraph of this section fully determines what the simple expressions of chess notation mean and also how they can be combined to form complex expressions, and that (ii) the meaning of a line within a chess notation determines a move. One can reject (i) and argue, for example, that the meaning of <\/span>B<\/span><\/em> in <\/span>Bb5<\/span><\/em> contains an indexical component and within the context of a description, it picks out a particular bishop moving from a particular square. <\/span>One can also reject (ii) and argue, for example, that the meaning of <\/strong><\/span>Bb5<\/strong><\/span><\/em> is nothing more than the meaning of \u2018some bishop moves from somewhere to square <\/strong><\/span>b5<\/strong><\/span><\/em>\u2019\u2014utterances of <\/strong><\/span>Bb5<\/strong><\/span><\/em> might carry extra information but that is of no concern for the semantics of the notation.<\/strong><\/span> Both moves would save compositionality at a price. The first complicates considerably what we have to say about lexical meanings; the second widens the gap between meanings of expressions and meanings of their utterances. Whether saving compositionality is worth either of these costs (or whether there is some other story to be told about our understanding of the Algebraic notation) is by no means clear. For all we know, Algebraic notation might be non-compositional.<\/span><\/p>\n <\/p>\n I also dont agree that it widens the gap between meanings of expressions and meanings of utterances. It has to do with refering to stuff, not meaning in itself.<\/p>\n –<\/p>\n <\/a>4.2.1 Conditionals<\/strong><\/span><\/p>\n Consider the following minimal pair:<\/span><\/p>\n (1) Everyone will succeed if he works hard. A good translation of (1) into a first-order language is (1\u2032). But the analogous translation of (2) would yield (2\u2032), which is inadequate. A good translation for (2) would be (2\u2033) but it is unclear why. We might convert \u2018\u00ac\u2203\u2019 to the equivalent \u2018\u2200\u00ac\u2019 but then we must also inexplicably push the negation into the consequent of the embedded conditional.<\/span><\/p>\n (1\u2032) \u2200<\/span>x<\/span><\/em>(<\/span>x<\/span><\/em> works hard \u2192 <\/span>x<\/span><\/em> will succeed) This gives rise to a problem for the compositionality of English, since is seems rather plausible that the syntactic structure of (1) and (2) is the same and that \u2018if\u2019 contributes some sort of conditional connective\u2014not necessarily a material conditional!\u2014to the meaning of (1). But it seems that it cannot contribute just that to the meaning of (2). More precisely, the interpretation of an embedded conditional clause appears to be sensitive to the nature of the quantifier in the embedding sentence\u2014a violation of compositionality.<\/span>[<\/a>16]<\/sup><\/span><\/p>\n One response might be to claim that \u2018if\u2019 does not contribute a conditional connective to the meaning of either (1) or (2)\u2014rather, it marks a restriction on the domain of the quantifier, as the paraphrases under (1\u2033) and (2\u2033) suggest:<\/span>[<\/a>17]<\/sup><\/span><\/p>\n (1\u2033) Everyone who works hard will succeed. But this simple proposal (however it may be implemented) runs into trouble when it comes to quantifiers like \u2018most\u2019. Unlike (3\u2032), (3) says that those students (in the contextually given domain) who succeed if they work hard are most of the students (in the contextually relevant domain):<\/span><\/p>\n (3) Most students will succeed if they work hard. The debate whether a good semantic analysis of <\/span>if<\/span><\/em>-clauses under quantifiers can obey compositionality is lively and open.<\/span>[<\/a>18]<\/sup><\/span><\/p>\n <\/p>\n Doesnt seem particularly difficult to me. When i look at an \u201cif-then\u201d clause, the first thing i do before formalizing is turning it around so that \u201cif\u201d is first, and i also insert any missing \u201cthen\u201d. With their example:<\/p>\n <\/p>\n (1) Everyone will succeed if he works hard. <\/p>\n this results in:<\/span><\/p>\n <\/p>\n (1)* If he works hard, then everyone will succeed. <\/p>\n Both \u201ceveryone\u201d and \u201cno one\u201d express a universal quantifer, \u2200. The second one has a negation as well. We can translate this to something like \u201call\u201d, and \u201cno\u201d to \u201cnot\u201d. Then we might get:<\/span><\/p>\n <\/p>\n (1)** If he works hard, then all will succeed. <\/p>\n Then, we move the quantifier to the beginning and insert a pronoun, \u201che\u201d, to match. Then we get something like:<\/span><\/p>\n <\/p>\n (1)*** For any person, if he works hard, then he will succeed. <\/p>\n These are equivalent with SEP’s<\/span><\/p>\n <\/p>\n (1\u2033) Everyone who works hard will succeed. <\/p>\n The difference between (3) and (3′) is interesting, not becus of relevance to my method about (i think), but since it deals with something beyond first-order logic. Quantification logic, i suppose? I did a brief Google and Wiki search, but didnt find something like that i was looking for. I also tried Graham Priest’s Introduction to non-classical logic<\/em>, also without luck.<\/p>\n <\/p>\n So here goes some system i just invented to formalize the sentences:<\/p>\n <\/p>\n (3) Most students will succeed if they work hard. <\/p>\n Capital greek letters are set variables. # is a function that returns the cardinality<\/a> a set.<\/p>\n <\/p>\n <\/a> (3)* (\u2203\u0393)(\u2203\u0394)(\u2200x)(\u2200y)(Sx\u2194x\u2208\u0393\u2227\u0394\u2286\u0393\u2227#\u0394>(#\u0393\/2)\u2227(y\u2208\u0394)\u2192(Wy\u2192Uy))<\/p>\n <\/p>\n In english: There is a set, gamma, and there is another set, delta, and for any x, and for any y, x is a student iff x is in gamma, and delta is a subset of gamma, and the cardinality of delta is larger than half the cardinality of gamma, and if y is in delta, then (if y works hard, then y will succeed).<\/p>\n <\/p>\n Quite complicated in writing, but the idea is not that complicated. It shud be possible to find some simplified writing convention for easier expression of this way of formalizing it.<\/p>\n <\/p>\n (3\u2032)* (\u2203\u0393)(\u2203\u0394)(\u2200x)(\u2200y)(((Sx\u2227Wx)\u2194x\u2208\u0393)\u2227\u0394\u2286\u0393\u2227#\u0394>(#\u0393\/2)\u2227(y\u2208\u0394\u2192Uy))<\/p>\n <\/p>\n In english: there is a set, gamma, and there is another set, delta, and for any x, and for any y, (x is a student and x works hard) iff x is in gamma, and delta is a subset of gamma, and the cardinality of delta is larger than half the cardinality of gamma, and if y is in delta, then u will succeed.<\/p>\n <\/p>\n To my logician intuition, these are not equivalent, but proving this is left as an exercise to the reader if he can figure out a way to do so in this set theory+predicate logic system (i might try later).<\/p>\n <\/p>\n –<\/p>\n 4.2.2 Cross-sentential anaphora<\/strong><\/span><\/p>\n Consider the following minimal pair from Barbara Partee:<\/span><\/p>\n <\/p>\n (4) I dropped ten marbles and found all but one of them. It is probably under the sofa.<\/span><\/p>\n (5) I dropped ten marbles and found nine of them. It is probably under the sofa.<\/span><\/p>\n <\/p>\n There is a clear difference between (4) and (5)\u2014the first one is unproblematic, the second markedly odd. This difference is plausibly a matter of meaning, and so (4) and (5) cannot be synonyms. Nonetheless, the first sentences are at least truth-conditionally equivalent. If we adopt a conception of meaning where truth-conditional equivalence is sufficient for synonymy<\/strong>, we have an apparent counterexample to compositionality.<\/span><\/p>\n <\/p>\n I dont accept that premise either. I havent done so since i read Swartz and Bradley years ago. Sentences like<\/span><\/p>\n <\/p>\n \u201cCanada is north of Mexico\u201d<\/span><\/p>\n \u201cMexico is south of Canada\u201d<\/span><\/p>\n <\/p>\n are logically equivalent, but are not synonymous. The concept of being north of, and the concept of being south of are not the same, even tho they stand in a kind reverse relation. That is to say, xR1<\/sub>y\u2194yR2<\/sub>x. Not sure what to call such relations. It’s symmetry<\/a>+substitition of relations.<\/span><\/p>\n <\/p>\n Sentences like<\/span><\/p>\n <\/p>\n \u201cEverything that is round, has a shape.\u201d<\/span><\/p>\n \u201cNothing is not identical to itself.\u201d<\/span><\/p>\n <\/p>\n
\n(2) No one will succeed if he goofs off. <\/span><\/p><\/blockquote>\n
\n(2\u2032) \u00ac\u2203<\/span>x<\/span><\/em> (<\/span>x<\/span><\/em> goofs off \u2192 <\/span>x<\/span><\/em> will succeed)
\n(2\u2033) \u2200<\/span>x<\/span><\/em> (<\/span>x<\/span><\/em> goofs off \u2192 \u00ac(<\/span>x<\/span><\/em> will succeed)) <\/span><\/p><\/blockquote>\n
\n(2\u2033) No one who goofs off will succeed.<\/span><\/p><\/blockquote>\n
\n(3\u2032) Most students who work hard will succeed.<\/span><\/p><\/blockquote>\n
\n(2) No one will succeed if he goofs off. <\/span><\/p>\n
\n(2)* If he goofs off, then no one will succeed.<\/span><\/p>\n
\n(2)** If he goofs off, then all will not succeed.<\/span><\/p>\n
\n(2)*** For any person, if he goofs off, then he will not succeed.<\/span><\/p>\n
\n(2\u2033) No one who goofs off will succeed.<\/p>\n
\n(3\u2032) Most students who work hard will succeed.<\/p>\n