Tuesday, November 02, 2004
Sense and reference
Frege’s work centres on developing a logo-centric version of language, so that it might be applied to the uncovering of knowledge without the danger of bias. His motivation for such work stemmed from his perceived need for a logical language so that he could more easily deal with logic in maths. At first glance, it may appear that he is, in fact, trying to work in the boundaries of psychologism, the theory that logic is a description of the natural and regular thinking pattern of persons. He is not, however, trying to perpetuate this at all, and his views, upon later inspection, reveal themselves to be anti-psychologistic. Rather, he seeks to impose a system of logic on a system of language he otherwise views as being illogical.
Frege makes the division between assertible and ontological truth, drawing the line, respectively, between those truths, which can be proven from feelings and experience, and those, which can be proven to be true simply by means of logic. His focus is on those truths that can be reached via means of logical reasoning, as these are the ones he focuses on, saying that it is these truths which are independent of the human mind, and exist regardless of whether or not people take it to be true. So here we see Frege’s anti-psychologism. If it were the case that logic was a description of the regularities of human thought, then if the entire world were to claim to conclude logically that donkeys were able to fly, then, on a psychologistic account, this would be the case. Frege dismisses assertible truths as being unconducive to the search for truth.
Frege speaks of sense and reference, as well as object and concept. Frege notes that in referring to objects, quite often it is possible to refer to the same object in many different manners. Thus, he splits meaning into two composites. An American might say ‘trashcan’, whilst an Australian would say ‘rubbish bin’ in the same conversation, whilst both referring to the object in which they would like to throw a used napkin. This Frege describes mathematically as ‘a=b’. He proposes that this is in a sense, like saying ‘1+4 = 2+2’. The values - ‘truth value’, i.e., ‘reference’ - of both sides of the equation are in fact, the same, but the angle of presentation (the ‘sense’) is different. Thus, it can be stated that ‘the trashcan is full’ is a ‘proposition’ which we are analysing. The sense of this proposition is different than that of ‘the rubbish bin is full’, but since ‘trashcan = rubbish bin’, and we can logically conclude that the ‘trashcan’ is, indeed, full, we can conclude that the reference of this proposition, is indeed correct, and thus, the sentence is true. Were this not to be the case, and the ‘trashcan’ were actually empty, the proposition ‘the trashcan is full’, while still possessing a discernible ‘sense’, would have the truth value of ‘false’, for the reference is does not, in fact, exist.
Frege also analyses the composites of the proposition, breaking them down into object and concept. In a proposition, say, ‘the butterfly is black’. The butterfly is the object of the sentence, and ‘is black’ is the concept. Frege introduces the idea of an unsaturated sentence, which is the object and verb combined, pending on a noun to complete it. So ‘the butterfly is’ on its own is not yet a proposition on it’s own.
Frege’s account then, seems to draw its basis on the Aristotelian view of language. Indeed, it has many close ties to Aristotle’s work, and at this point still sounds like what Aristotle says, albeit embellished, about sentences being true if the nouns and verbs combine in a way in which they express what is true, and sentences being false, if this is not the case.
However, there is a subtle, yet significant difference to the manner in which Aristotle formulates ‘true’ sentences to that which Frege does. Aristotle’s method is that of syntactic combination – he sees sentences in terms of subject and predicate, or noun phrase and verb phrase, combining to form sentences which are either true or false. Frege, as pointed out earlier, sees sentences in terms of object and concept, as mentioned earlier.
In Aristotle’s view, a sentence can refer to a universal or a particular. Universal concepts can be used to refer to an image or thought that is used to represent all things or beings falling under a single category, such as the word ‘planet’. This can, on the Aristotelian view, be used to signify Venus, Mars, a planet in a distant galaxy, an imaginary one, etc. The word ‘planet’ is thus used to refer to a general category of ‘planethood’. The properties of this universality are properties that would pertain to all items in this category. This universal term can be used to make generalities about, and represent, all members of this category, without referring specifically to any particular. Most importantly, in this view, the word ‘planet’ by itself has properties that can be ascribed to it. A particular is, on the other hand, a concept that can only refer to one item, therefore, most commonly, these words are proper names, such as ‘Foucault’.
Here, Frege makes a break from the views held by Aristotle. Frege rejects the idea of universality. The problem he encounters with it is that while we can imagine particulars, say, ‘Jupiter’, we cannot conjure up an image of a generic ‘planet’. Indeed, the concept seems somewhat cloudy, and this ideal of a ‘planet’ escapes us. The view that Frege is driven to – nominalism – is that there are no universal ‘ideas’ that can be justified. These universals do not have a meaning. They do not possess a reference, so therefore cannot be used to signify the true, as Aristotle does. Frege, instead, elects to say that they have no meaning independently as ideals, but only insofar as they are linked to certain sets of particulars. So, unlike Aristotle, Frege does not view sentences as universals relating to particulars, but rather as objects stating a concept relating to the object.
Frege views the concepts in a sentence as a kind of function, and describes it as a mathematical formula F(x,y), where F states the relationship between x and y. In this way, x and y are not independent of each other, as are the subject and predicate in the Aristotelian view. If x is to refer to ‘chilli’ and y to refer to ‘Calicles’, and the function to be ‘is loved by’, we have the sentence ‘chilli is loved by Calicles.’ The method that Frege uses prevents the problem of syntactical difficulties arising, as it does in Aristotle’s view, by appealing directly to the deep structure of a sentence. By stating the proposition in terms of the relation between x and y, Frege shows mathematically that we are in fact saying the same thing if we are to state ‘Calicles loves chilli’.
Expanding and deepening from where Aristotle left off, Frege introduces the notion of sentential connectives and quantifiers into his theory. Sentential connectives are words such as ‘and’ or ‘yet’, which serve to unify two propositions, forming a single sentence out of them.
Frege confronted the problem presented by propositions such as ‘transparent politicians do not exist’ and ‘nothing is free’ by creating a mathematical function that can be applied to language. The first sentence takes a sentential operator, or a negator and a concept and melds them together to form a ‘not-concept’, and attributing this to something that does not exist. The second is attributing a concept to a non-existent object. Frege breaks down these concepts first to the positive form, so from the first sentence, we have ‘transparent politicians exist’. In front of this claim, Frege places a horizontal line, to represent ‘the case’, or the proposition, so that now we have ‘the case that transparent politicians exist’. Since this is a false claim, Frege breaks up this horizontal line with a vertical line representing ‘not’ extending downwards from the middle of the horizontal. Now we have ‘not the case that transparent politicians exist’. Since we now have a true claim, Frege places what he refers to as a ‘judgement stroke’ vertically at the beginning of the line, and this represents the true. So, now we have ‘It is not the case that transparent politicians exist.’ Frege goes into much more minute detail on the functional representation of truth, but this, I feel is the core of it. From this we can derive a much more practical formula $x Fx, meaning ‘there is something which is F’. To indicate that there is not something which is F, then we add ‘~’, thus ‘~$ transparent politician x’. From the second case we get ‘~$ free x’ – ‘it is not the case that there is something which is free’.
The question arises as to whether Frege is too prescriptive in his theory, and in constructing his view into as many steps as he has made a theory that can only be applied to certain families of languages, including Germanic and Romantic languages. In Finnish, the sentence ‘nothing is free’ is expressed as ‘ei ole mitaan ilmaista’ is literally, ‘there is not something free’ In Thai, the word ‘mee’ means containing a property’, thus ‘mai mee arai tee mai mee rakaa’, is roughly translated as ‘there is not in the property of something that which does not have the property of price.’ Both languages seem to roughly overstep the need to reduce the sense of the sentence into positives from which a truth-value can be discerned. This undermines Frege’s system of three steps ‘proposition’ > ‘sense’ > ‘reference’. Indeed, the speakers of Thai and Finnish show no evidence that they are able to think any faster than the speakers of English or German, and are not congratulating themselves on their logical superiority.
What it suggests, instead, is that ‘sense’ is to some extent implicit in the ‘reference’ presented by any given sentence, and that the jump can therefore be made directly from the ‘proposition’ to the ‘reference’. Since it is impossible to always discern that a ‘reference’ may have more than one ‘sense’, recipients of a proposition treat these propositions on a case-by-case basis, with ‘sense’ factored into the ‘reference’. The speakers of a language do not remove the ‘reference’ from the context of the sentence. So, in asking ‘how is Mary’s partner?’ and ‘how is Jane’s father?’ perhaps, objectively speaking, both cases are referring to ‘Jonathan’. However, when the speaker replies ‘he is fine’, he is not referring to ‘Jonathan’, but rather to either ‘Mary’s partner’ or ‘Jane’s father’. The intention of the one making the interrogative has to kept in mind, and the intention when asking about ‘Jane’s father’ is different from the intention of inquiring about ‘Mary’s partner’. These two questions are referring to different roles of the same person, and the answer to these two questions may be radically different: ‘he is a good father’ or ‘he hasn’t been talking to Mary much recently’
Frege describes a doctrine of ontological truths combining to uncover more truths. He seems to have created a language that is free from bias in its scientific aims. However, for whom has he designed this system of notation? Frege is openly derisive about the way people have the tendency to think, claiming, “[there is] a widespread inclination to acknowledge as existing only what can be perceived by the senses”, opting, instead to take the path of logic. Logic, however, can be seen as too theoretical. As much as it is possible to logically conclude that time travel is possible, this does not mean that it exists.
As comprehensive as Frege’s account may be, he does not make any room for value judgements other than true or false in formulas. As far as describing human language goes, this seems to be a large inadequacy. Words such as ‘good’, ‘bad’, ‘courageous’, ‘beautiful’ are words that Frege is forced reject as being universals. Frege’s nominalist view would have to, controversially, reject the idea of there being an objective good. Socrates, while basing much of his argument on logic, as does Frege, would hardly agree here. Of course, their accounts of logic do not agree. They may both agree that in logic resides the truth, but Socrates seems to lean towards a more psychologistic approach to logical argument, appealing to his opponent’s logic to reinforce his arguments. Frege, on the other hand, presupposes an independently existing logical system. In not factoring in any, in his view ‘subjective’ values, such as ‘good’, ‘bad’ or ‘beautiful’, Frege makes his study of logic even drier than mathematics, for mathematics does deal with unpredictable variables.
Frege says in the preface of ‘Begriffsschrift’ “it is not the psychological genesis, but the best method of proof that is at the basis of the classification”. Insofar as science is concerned, perhaps this is the case, but Frege is attempting to describe the machinations of language, something that is an invention of humans, and prescribe a logic by which it should function. His rationality is somewhat too concise, and in the end, it turns on itself. Frege does not take, into his calculations, account that people do not generally think in terms of mathematical functions. Any system, no matter how complete and comprehensive, only has value so long it is functional. The way Frege describes language does not truly represent the way in which logic functions in a person, but is a highly idealised, impractical account.
Bibliography
Aristotle, De Interpretatione, chapters 1-7 from The Works of Aristotle vol 1., ed. Jonathan Barnes. Princeton: Princeton University Press, 1984, pp. 25–7.
Plato, Gorgias, trans. Walter Hamilton and Chris Emlyn-Jones. Penguin Books, England, 2004
Gottlob Frege, Preface to Begriffsschrift from Frege and Gödel: Two Fundamental Texts in Mathematical Logic, ed. Jean van Heijenoort. Cambridge, Mass.: Harvard, 1970, pp. 5-7
Gottlob Frege from the Preface to Grundgesetze der Arithmetic, from The Basic Laws of Arithmetic, trans. Montgomery Furth, Berkeley: University of California Press, 1964, pp. 10-25
Gottlob Frege, ‘Function and Concept’, from Michael Beaney ed. The Frege Reader. Oxford: Blackwell, 1997, pp. 130-48
Gottlob Frege, ‘Letter to Husserl’ from Michael Beaney, ed. The Frege Reader. Oxford: Blackwell, 1997, pp. 149-50
Roy Harris, ‘Frege on Sense and Reference’ from Landmarks in Linguistic Thought, 2nd edn. vol. 1, London and New York: Routeledge, 1997, pp. 196-208
Karen Green, ‘PHL2120 Language Truth and Power’. Clayton: Monash University Arts, Semester 2, 2004
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