My independent study on the nature of truth is taking up much of my time these days. Its due in April. I'm going to put working drafts of it on the blog, probably in the comments section of this post, for you to peruse.
Much can be said about the rocks at the bottom of the sea. They are hard, some of them are perhaps round. They are all wet. One thing that seems awkward to say about the rocks at the bottom of the sea is that they are true. Being true doesn't seem to be a property that can apply to the rocks at the bottom of the sea. Maybe, at times, individuals have described the rocks at the bottom of the sea as being true. However, in those cases the individual is, more often, using the word “true” as a synonym for another word. For instance, when one says “the rocks at the bottom of the sea are true” one could mean that the rocks are real or that they exist. In this case the word “true” is used to mean exists. Another possible way “true” is used, when applied to the rocks at the bottom of the sea, is as a synonym for “consistent”. The rocks are true in that they persevere in their rockness. In this case, saying “the rocks are true” describes their consistency through time or that we can always rely on the rocks to do what they do and be what they are. Both of these examples illustrate reasonable ways in which the word “true” can be used. However, the word “true” is normally taken to have a meaning exclusive to itself. “True” seems to indicate a property other than existing or being consistent. We are unable to call the rocks at the bottom of the sea “true” because being true is not a property that rocks, of any kind, can have. The property of truth is not applicable to rocks. To explain how it is possible for some properties to not be applicable to certain objects take a beach ball as an object, for example. A beach ball can have many properties they can be round, brightly colored or inflatable. However, no beach ball can be an orphan. Being an orphan is not a property that is available to beach balls. The reason for this, is that in order to be an orphan an object must have lost it's parents. Beach balls, of course, have no parents to lose. It is, therefore, impossible for a beach ball to be an orphan. The property of being an orphan is not applicable to beach balls except for maybe in a metaphorical way. At this point it is important to make a distinction between an object not possessing a property and a property not being applicable to an object. Once again take the beach ball as an example. Beach balls do not posses the property of being salty. Therefore they are not salty. Beach balls do not posses the property of being orphans. However, it does not make sense to say that, because they do not posses the property of being an orphan, that beach balls are not orphans. To say a beach ball is not an orphan would imply that the beach ball could be an orphan. Beach balls are not salty because they do not have the property of being salty. Whereas, beach balls are unable to either be orphans or not. Just as a beach ball can't be an orphan, the rocks at the bottom of the sea can't be true. The property of truth is inapplicable to the rocks of any kind. What then can the property of truth be applied to? To start things off, we seem to have no problem saying that it is true all of the rocks at the bottom of the sea are wet, or that it is untrue that the rocks at the bottom of the sea are soft. It must be determined what, in these cases, is being described as true or not true. Is it the wetness or the softness? We can imagine conditions wherein all the rocks are square or where rocks are soft. Under those conditions it would be untrue that some of the rocks at the bottom of the sea where round and true that the rocks were soft. We see here that in these examples that the word “true” is not being used to describe the rocks themselves, nor roundness or softness or even to the rocks themselves. What is described as true, in these cases, are the sentences. It is the sentence, “All the rocks at the bottom of sea are wet” that is subject to being true or untrue. Truth, therefore, is, at least, a property of sentences. Truth, however, cannot be applied to all sentences, nor does it seem that sentences are the only objects to which truth properly applies. The sentences that can be true or untrue are the sentences that are representations of other objects or states. For example, the sentence “the rocks at the bottom of the sea are wet” is an object that describes the rocks at the bottom of the sea and can be true or untrue. It is the descriptive nature of the sentence that makes the sentence an object to which the property of truth can be applied. Therefore, sentences that make no description are not objects to which the property of truth can be applied. For example, the sentence “Don't go in there.”normally can't be considered as either true or false as it does not postulate any any sort of description. This sentence could only be considered as true or untrue if is was meant to as a way of expressing some sort of description, such as that it is dangerous in there, for example. The truth property can only be applied to affirmative sentences. Also, since it is by virtue of representation that makes the truth property applicable to sentences, other objects that make descriptions are also subject to being true or untrue. Therefore, the set of objects to which the property of truth applies consists not only of affirmative sentences but also of beliefs, mental representations, theories or any object that is a description another object or state. My purpose in going through this obvious exercise is to show that our notion of truth is informed by by the objects to which we consider truth can be appropriately applied. Indeed, because truth is applied only to representational objects it seems that the truth of falsity of an object will be determined by the nature of the representation it makes. A representation is true when it accurately describes its object of representation and false when it does not. This notion of truth, for much of history dominated western thought. It is known as the correspondence theory of truth. Properly formulated the correspondence theory of truth states that a representation is true when it corresponds to facts. For instance, The sentence “snow is white” is true if and only if snow is actually white. Upon first glance, correspondence theory seems to satisfy all our intuitions about how the notion of truth works, however, upon closer inspection the correspondence theory of truth runs into some serious problems. Others working in the field have attempted to either modify or to replace the correspondence theories in order to overcome its perceived shortcomings. Polish Logician, Alfred Tarski, is persuaded by the intuitional correctness of the correspondence theory of truth. He feels that any theory of truth should always hold that something is true when it corresponds with the way the world actually is. However as a mathematician and logician he wants a more precise definition than what the correspondence theory gives. He observes that when a definition of truth of the type given in the correspondence theory is given in a formal language a contradiction can be generated. This contradiction, that will be gone through in detail, is known in the literature as the liars paradox. Tarski's purpose will be to formaly define truth in such a way as to be consistent with the correspondence theory but also avoid contradictions like the liars paradox. Tarskey wants what he called a satisfactory definition of truth. A definition of truth would be satisfactory, for Tarski, if it is both ,what he calls, materially adequate and formally correct. By materially adequate, Tarski means, the definition must be commensurable with our existing conceptions of the notion of truth. He explains in is paper “The Semantic Conception of Truth and the Foundations of Semantics” what a definition of truth should accomplish. “The desired definition does not not aim to specify the meaning of a familiar word used to denote a novel notion; on the contrary, it aims to catch hold of the actual meaning of an old notion.” According to Tarski, in order for a satisfactory definition of truth to be of any use it must, while making the notion more precise, catch the customary purpose the notion of truth is used for. He mentions as well that a definition of truth must also be formally correct. Tarsky explains what he means by formal correctness. “We must specify the words or concepts which which we wish to use in defining the notion of truth; and we must also give the formal rules to which a definition should conform.”.Tarsky wishes to define truth exactly in that the language in which it is defined must be exactly defined in order to avoid any ambiguity. In order to construct a definition of truth that is materially adequate, Tarski, must establish what the meaning of “true” is. He does this by examining both the extension and intension of the word “true”. The extension of the word “true” is the set of objects that the word '”true” picks out. Tarski claims that the set of objects the word “true”picks out consists only of sentences. By sentences, in this case, Tarski is referring to declarative sentences. One could wonder why he does not instead say that the extension of “true” consists only in propositions. He does not make this more general claim because he feels that the definition of “proposition” to be too much in dispute. As we shall see later, Quine actually argues against the use of propositions in truth talk and claims that only sentences can belong the extension of true. Tarsky is not so bold. Tarsky avoids propositions to avoid controversy not out of any conviction. Declarative sentences, then, are the only objects that can be true in Tarsky's deffinition. From the extension of “true” Tarski moves to the intension or the meaning of the word “true”. Tarski is satisfied by the consistency of the correspondence theory of truth with our intuitions about truth. He starts with the classical Aristotelian definition of truth. “To say of what is that it is not, or of what is not that it is, is false, While to say of what is that it is, or what is not that it is not, is true.” There is controversy as whether Aristotle subscribed to the correspondence theory. I will no go into this controversy now . It sufficeth me to say that at least that is how Tarsky reads Aristotle. In simpler terms this definition states that the truth of a statement is dependent on it's correspondence with the reality. A statement is true when it corresponds to reality, false when it does not. Form this conception of the extension and intension of the word “true” Tarski begins to make the criterion for which a definition of truth would be materially adequate. Following our intuitions about truth, in order for the sentence “snow is white” to be true it would have to be the case that that snow is white and not black, green or blue. Thus the condition on which the truth of the sentence “snow is white” depends is the colour of snow. Tarski notes this and concludes that “ if the definition of truth is to conform to our conception, it must imply the following equivalence: the sentence “snow is white” is true if and only if, snow is white.”. This equivalence has the form “X is true if, and only if p” where X is the name of the sentence, “snow is white”, and p is the content or meaning of that sentence. It is important that X stand for the name of the sentence and not the sentence itself for two reasons. First, X as the subject of the sentence “X is true if, and only if p” must be a name in order for the sentence to make sense grammatically. This becomes apparent if we try to replace X by something other than a name like an Adverb, for instance. The sentence “joyfully is true if, and only if p” makes no sense at all. Second, as Tarski claims “According to the conventions of language we must use the name of the object to refer not the object itself.” The reason for this is clear when we realize how impossible it would be to use the actual man, Alfred Tarski, each time I referred to him rather than simply using the word “Tarski”. Thus X stands for “snow is white” as a name of the sentence “snow is white”. It is possible to imagine other names for this sentence such as “The three word sentence that asserts that white is the color of snow.” or even more Abstract names like “Henry” or “X”. Tarski refers to the sentence schema “X is true if, and only if p” as (T). He uses (T) to give his conditions for material Adequacy of a definition of truth. “we wish to use to use the term “true” in such a way that all equivalences of the form (T) can be asserted.” Thus a definition of truth is materially adequate if it implies all equivalence of the form (T). Now turn to the condition of formal correctness. First of all, in order for a definition of truth to be formally correct, the language in which we give it must be exactly specified. Tarski states seven elements of an exactly specified language. First, all the words and expressions considered meaningful must be unambiguously classified. Second, all the words to be used without being defined must be indicated. Third, the rules of definition must be given. Fourth, there must be a criteria set up for distinguishing within the meaningful expressions those which are sentences. Fifth, conditions must be given whereby a sentence of the language can be asserted. Sixth, the rules of inference must be given. Tarski claims “The problem of the definition of truth obtains a precise meaning and can be solved in a rigorous way only for those languages whose structure has been exactly specified.” Only in Languages that are exactly specified is it possible to give a satisfactory definition of truth. Also as mentioned above, once truth is defined in languages that are exactly specified contradictions appear. In languages that that are not exactly specified, such as the natural language of English, not only can no satisfactory definition be given but the associated problems with the commonly held notion of truth can only be approximately addressed. Tarski felt that if one considered truth as a semantic concept then one could generate of definition of truth that could overcome difficulties found in other definitions. Also a semantic definition of truth would begin the ground work of a systemics theoretical semantics. Semantics according to Tarski, “is a discipline which, speaking loosely, deals with certain relations between expressions of a language and the objects (or “states of affair”) referred to by those expressions.” Tarski uses the words “designates”, “satisfies” and “defines” as examples of semantic concepts. These words all express a relations, namely that between certain expressions and the objects referred to by those expressions. For example the expression “the vessel from which I'm drinking” designates my mug. The word “designates” in this case expresses the relationship of my mug with the expression “the vessel from which I'm drinking”. Tarski felt that Truth could be thought of as a semantic concept. “Is true” express a relationship between a given sentence and a certain state of affairs. Once a definition of truth is both materially adequate and formally correct one runs the contradiction known as the liars paradox. The liars paradox emerges when trying to ascribe truth values to sentences that make assertions about their own truth value. Take the following sentence. (1) “Sentence(1) is not true”. Let us now name (1) “x”. Using the (T) schema, Tarsky proposes, we assert the following equivalence “x is true if and only if sentence (1) is not true”. Since x is identical with (1) we should be able to interchange them to obtain “x is true if and only if x is not true”. Thus is made the contradiction. Contradictions like this, for Tarski present real problems and are not merely tricks or “sophistries”. Understanding this contradiction and resolving it represents, to him, a significant advance in the field of truth theory. Because a contradiction has been generated the premises that were accepted have to be re-examined and at least one of them must be rejected. He gives three premises upon which this the contradiction has been generated.
1. The language in which the antimony is constructed contains, in addition to its expressions, also the names of these expressions, as well as semantic terms such as the term “true” referring to sentences of this language; we have also assumed that all sentences which determine the adequate usage of this term can be asserted in the language. A language with these properties will be called “semantically closed”. 2. In this language the ordinary laws of logic hold. 3. We can formulate and assert in our language an empirical premise such as the statement [x is identical with with the sentence “this sentence is not true”]
Because they lead to contradiction one of these premises must be rejected or revised. Tarski claims that even if we don't assert the third premise it is still possible the generate the contradiction. Therefore we should look to other premises for a possible resolution. The consequences of rejecting the second premise are too grave to consider dropping it. Tarski, therefore, is left only with the first premise as a possible culprit. Tarski claims that all languages that are semantically closed will generate the Liars paradox. In fact, he takes this claim even further to say that similar contradictions will form using any semantic concept in a semantically closed language. Consequently, Tarski asserts that it is impossible for a language to contain its own truth predicate without this contradiction popping up. His solution is to not use semantically closed languages. It is therefore necessary when speaking of a definition of truth to use two languages. The first language is the language where are found the sentences we wish to define as true or false. This language is known as the object language because it is the object of our truth predicate. The second language is the language in which we ascribe truth to the object language. This second language is the metalanguage. The metalanguage is the language in which is found the truth predicate for the object language. Thus is made a hiearchy of languages where sentences that make reference to truth values are semantically higher. The metalanguage must, as well, satisfy all the conditions of a satisfactory definition of truth in that the definition must be able imply all equivalences of the form “x is true only iff p” The equivalence must be stated in the metalanguage. “p”, however, is an arbitrary sentence of which we are concerned in the object language. X is the name of p in the metalanguage and it is for this reason that the metalanguage must be able to name all the sentences of the object language. In order for a satisfactory definition of truth be given in terms of of an object and metalanguage Tarsky gives conditions that the each language must satisfy. Tarski wishes that semantic terms, such as the truth predicate, be introduced into the metalanguage by definition. He wants this so that the metalanguage “will explain the meaning of the term being defined in terms whose meaning appears completely clear and unequivocal.” It is important to note that the relationship between the object and metalanguage is relative. That is, if we wished to define truth for what we are using as a metalanguage that language would become the object language and we would require a new metalanguage with which to to talk about that language. Another condition of a satisfactory definition of truth is that the metalanguage must be “essentially richer” than the object language. Put simply, this condition states that it cannot be possible for a complete interpretation of the metalanguage in the object language. Were this possible then all assertible sentences in one language could be asserted in the other. It would then be possible to reconstruct the Liar's Paradox in the metalanguage and we would run into the same problems as before. Tarski claims that this condition of essential richness is a necessary and sufficient condition for a satisfactory definition of truth wherein semantic concepts are introduced by definition. To review, in order for a definition of truth to be satisfactory for Tarsky it must satisfy several conditions. It must be materially adequate, that is it must imply all equivalences of the form ““x” is true if and only if x”. A definition must also be formally correct, that is it must avoid problems such as the liar's paradox. To do this, a definition must be stated in a language that is exactly specified. Truth for a given language must be defined in another language that is essentially richer than the one for which we are defining truth. Now that all these conditions are clear Tarski can begin to construct his definition.
WOW Dave I didn't expect anyone to read it I posted it more as a joke than anything. But since you asked the languages tarsky talks about are not natural languages like english, or French. Those languages are not exactly specified. In other words their meanings and rules are too ambiguous to either clearly apprehend or solve the liars paradox problem. Tasky is using a hypothetical language that is exactly specified (I've noted the properties he gives for such a language in the previous section). His point is that in this language its imposible to have a truth operator without running into contradiction. A richer language is required. To answer your question, a an essentialy richer laguage would be one in which every possible sentence of the first language would be expressible, as well this language would contain semantic operators such is is true, or is defined by that are applicable to the fist hypothetical language. hope that helps.
6 comments:
Much can be said about the rocks at the bottom of the sea. They are hard, some of them are perhaps round. They are all wet. One thing that seems awkward to say about the rocks at the bottom of the sea is that they are true. Being true doesn't seem to be a property that can apply to the rocks at the bottom of the sea. Maybe, at times, individuals have described the rocks at the bottom of the sea as being true. However, in those cases the individual is, more often, using the word “true” as a synonym for another word. For instance, when one says “the rocks at the bottom of the sea are true” one could mean that the rocks are real or that they exist. In this case the word “true” is used to mean exists. Another possible way “true” is used, when applied to the rocks at the bottom of the sea, is as a synonym for “consistent”. The rocks are true in that they persevere in their rockness. In this case, saying “the rocks are true” describes their consistency through time or that we can always rely on the rocks to do what they do and be what they are. Both of these examples illustrate reasonable ways in which the word “true” can be used. However, the word “true” is normally taken to have a meaning exclusive to itself. “True” seems to indicate a property other than existing or being consistent. We are unable to call the rocks at the bottom of the sea “true” because being true is not a property that rocks, of any kind, can have. The property of truth is not applicable to rocks.
To explain how it is possible for some properties to not be applicable to certain objects take a beach ball as an object, for example. A beach ball can have many properties they can be round, brightly colored or inflatable. However, no beach ball can be an orphan. Being an orphan is not a property that is available to beach balls. The reason for this, is that in order to be an orphan an object must have lost it's parents. Beach balls, of course, have no parents to lose. It is, therefore, impossible for a beach ball to be an orphan. The property of being an orphan is not applicable to beach balls except for maybe in a metaphorical way.
At this point it is important to make a distinction between an object not possessing a property and a property not being applicable to an object. Once again take the beach ball as an example. Beach balls do not posses the property of being salty. Therefore they are not salty. Beach balls do not posses the property of being orphans. However, it does not make sense to say that, because they do not posses the property of being an orphan, that beach balls are not orphans. To say a beach ball is not an orphan would imply that the beach ball could be an orphan. Beach balls are not salty because they do not have the property of being salty. Whereas, beach balls are unable to either be orphans or not.
Just as a beach ball can't be an orphan, the rocks at the bottom of the sea can't be true. The property of truth is inapplicable to the rocks of any kind. What then can the property of truth be applied to? To start things off, we seem to have no problem saying that it is true all of the rocks at the bottom of the sea are wet, or that it is untrue that the rocks at the bottom of the sea are soft. It must be determined what, in these cases, is being described as true or not true. Is it the wetness or the softness? We can imagine conditions wherein all the rocks are square or where rocks are soft. Under those conditions it would be untrue that some of the rocks at the bottom of the sea where round and true that the rocks were soft. We see here that in these examples that the word “true” is not being used to describe the rocks themselves, nor roundness or softness or even to the rocks themselves. What is described as true, in these cases, are the sentences. It is the sentence, “All the rocks at the bottom of sea are wet” that is subject to being true or untrue. Truth, therefore, is, at least, a property of sentences.
Truth, however, cannot be applied to all sentences, nor does it seem that sentences are the only objects to which truth properly applies. The sentences that can be true or untrue are the sentences that are representations of other objects or states. For example, the sentence “the rocks at the bottom of the sea are wet” is an object that describes the rocks at the bottom of the sea and can be true or untrue. It is the descriptive nature of the sentence that makes the sentence an object to which the property of truth can be applied. Therefore, sentences that make no description are not objects to which the property of truth can be applied. For example, the sentence “Don't go in there.”normally can't be considered as either true or false as it does not postulate any any sort of description. This sentence could only be considered as true or untrue if is was meant to as a way of expressing some sort of description, such as that it is dangerous in there, for example. The truth property can only be applied to affirmative sentences. Also, since it is by virtue of representation that makes the truth property applicable to sentences, other objects that make descriptions are also subject to being true or untrue. Therefore, the set of objects to which the property of truth applies consists not only of affirmative sentences but also of beliefs, mental representations, theories or any object that is a description another object or state.
My purpose in going through this obvious exercise is to show that our notion of truth is informed by by the objects to which we consider truth can be appropriately applied. Indeed, because truth is applied only to representational objects it seems that the truth of falsity of an object will be determined by the nature of the representation it makes. A representation is true when it accurately describes its object of representation and false when it does not.
This notion of truth, for much of history dominated western thought. It is known as the correspondence theory of truth. Properly formulated the correspondence theory of truth states that a representation is true when it corresponds to facts. For instance, The sentence “snow is white” is true if and only if snow is actually white. Upon first glance, correspondence theory seems to satisfy all our intuitions about how the notion of truth works, however, upon closer inspection the correspondence theory of truth runs into some serious problems. Others working in the field have attempted to either modify or to replace the correspondence theories in order to overcome its perceived shortcomings.
Polish Logician, Alfred Tarski, is persuaded by the intuitional correctness of the correspondence theory of truth. He feels that any theory of truth should always hold that something is true when it corresponds with the way the world actually is. However as a mathematician and logician he wants a more precise definition than what the correspondence theory gives. He observes that when a definition of truth of the type given in the correspondence theory is given in a formal language a contradiction can be generated. This contradiction, that will be gone through in detail, is known in the literature as the liars paradox. Tarski's purpose will be to formaly define truth in such a way as to be consistent with the correspondence theory but also avoid contradictions like the liars paradox.
Tarskey wants what he called a satisfactory definition of truth. A definition of truth would be satisfactory, for Tarski, if it is both ,what he calls, materially adequate and formally correct. By materially adequate, Tarski means, the definition must be commensurable with our existing conceptions of the notion of truth. He explains in is paper “The Semantic Conception of Truth and the Foundations of Semantics” what a definition of truth should accomplish. “The desired definition does not not aim to specify the meaning of a familiar word used to denote a novel notion; on the contrary, it aims to catch hold of the actual meaning of an old notion.” According to Tarski, in order for a satisfactory definition of truth to be of any use it must, while making the notion more precise, catch the customary purpose the notion of truth is used for. He mentions as well that a definition of truth must also be formally correct. Tarsky explains what he means by formal correctness. “We must specify the words or concepts which which we wish to use in defining the notion of truth; and we must also give the formal rules to which a definition should conform.”.Tarsky wishes to define truth exactly in that the language in which it is defined must be exactly defined in order to avoid any ambiguity.
In order to construct a definition of truth that is materially adequate, Tarski, must establish what the meaning of “true” is. He does this by examining both the extension and intension of the word “true”. The extension of the word “true” is the set of objects that the word '”true” picks out. Tarski claims that the set of objects the word “true”picks out consists only of sentences. By sentences, in this case, Tarski is referring to declarative sentences. One could wonder why he does not instead say that the extension of “true” consists only in propositions. He does not make this more general claim because he feels that the definition of “proposition” to be too much in dispute. As we shall see later, Quine actually argues against the use of propositions in truth talk and claims that only sentences can belong the extension of true. Tarsky is not so bold. Tarsky avoids propositions to avoid controversy not out of any conviction. Declarative sentences, then, are the only objects that can be true in Tarsky's deffinition. From the extension of “true” Tarski moves to the intension or the meaning of the word “true”. Tarski is satisfied by the consistency of the correspondence theory of truth with our intuitions about truth. He starts with the classical Aristotelian definition of truth. “To say of what is that it is not, or of what is not that it is, is false, While to say of what is that it is, or what is not that it is not, is true.” There is controversy as whether Aristotle subscribed to the correspondence theory. I will no go into this controversy now . It sufficeth me to say that at least that is how Tarsky reads Aristotle. In simpler terms this definition states that the truth of a statement is dependent on it's correspondence with the reality. A statement is true when it corresponds to reality, false when it does not. Form this conception of the extension and intension of the word “true” Tarski begins to make the criterion for which a definition of truth would be materially adequate.
Following our intuitions about truth, in order for the sentence “snow is white” to be true it would have to be the case that that snow is white and not black, green or blue. Thus the condition on which the truth of the sentence “snow is white” depends is the colour of snow. Tarski notes this and concludes that “ if the definition of truth is to conform to our conception, it must imply the following equivalence: the sentence “snow is white” is true if and only if, snow is white.”.
This equivalence has the form “X is true if, and only if p” where X is the name of the sentence,
“snow is white”, and p is the content or meaning of that sentence. It is important that X stand for the name of the sentence and not the sentence itself for two reasons. First, X as the subject of the sentence “X is true if, and only if p” must be a name in order for the sentence to make sense grammatically. This becomes apparent if we try to replace X by something other than a name like an Adverb, for instance. The sentence “joyfully is true if, and only if p” makes no sense at all. Second, as Tarski claims “According to the conventions of language we must use the name of the object to refer not the object itself.” The reason for this is clear when we realize how impossible it would be to use the actual man, Alfred Tarski, each time I referred to him rather than simply using the word “Tarski”. Thus X stands for “snow is white” as a name of the sentence “snow is white”. It is possible to imagine other names for this sentence such as “The three word sentence that asserts that white is the color of snow.” or even more Abstract names like “Henry” or “X”. Tarski refers to the sentence schema “X is true if, and only if p” as (T). He uses (T) to give his conditions for material Adequacy of a definition of truth. “we wish to use to use the term “true” in such a way that all equivalences of the form (T) can be asserted.” Thus a definition of truth is materially adequate if it implies all equivalence of the form (T).
Now turn to the condition of formal correctness. First of all, in order for a definition of truth to be formally correct, the language in which we give it must be exactly specified. Tarski states seven elements of an exactly specified language. First, all the words and expressions considered meaningful must be unambiguously classified. Second, all the words to be used without being defined must be indicated. Third, the rules of definition must be given. Fourth, there must be a criteria set up for distinguishing within the meaningful expressions those which are sentences. Fifth, conditions must be given whereby a sentence of the language can be asserted. Sixth, the rules of inference must be given. Tarski claims “The problem of the definition of truth obtains a precise meaning and can be solved in a rigorous way only for those languages whose structure has been exactly specified.” Only in Languages that are exactly specified is it possible to give a satisfactory definition of truth. Also as mentioned above, once truth is defined in languages that are exactly specified contradictions appear. In languages that that are not exactly specified, such as the natural language of English, not only can no satisfactory definition be given but the associated problems with the commonly held notion of truth can only be approximately addressed.
Tarski felt that if one considered truth as a semantic concept then one could generate of definition of truth that could overcome difficulties found in other definitions. Also a semantic definition of truth would begin the ground work of a systemics theoretical semantics. Semantics according to Tarski, “is a discipline which, speaking loosely, deals with certain relations between expressions of a language and the objects (or “states of affair”) referred to by those expressions.” Tarski uses the words “designates”, “satisfies” and “defines” as examples of semantic concepts. These words all express a relations, namely that between certain expressions and the objects referred to by those expressions. For example the expression “the vessel from which I'm drinking” designates my mug. The word “designates” in this case expresses the relationship of my mug with the expression “the vessel from which I'm drinking”. Tarski felt that Truth could be thought of as a semantic concept. “Is true” express a relationship between a given sentence and a certain state of affairs.
Once a definition of truth is both materially adequate and formally correct one runs the contradiction known as the liars paradox. The liars paradox emerges when trying to ascribe truth values to sentences that make assertions about their own truth value. Take the following sentence. (1) “Sentence(1) is not true”. Let us now name (1) “x”. Using the (T) schema, Tarsky proposes, we assert the following equivalence “x is true if and only if sentence (1) is not true”. Since x is identical with (1) we should be able to interchange them to obtain “x is true if and only if x is not true”. Thus is made the contradiction. Contradictions like this, for Tarski present real problems and are not merely tricks or “sophistries”. Understanding this contradiction and resolving it represents, to him, a significant advance in the field of truth theory.
Because a contradiction has been generated the premises that were accepted have to be re-examined and at least one of them must be rejected. He gives three premises upon which this the contradiction has been generated.
1. The language in which the antimony is constructed contains, in addition to its expressions, also the names of these expressions, as well as semantic terms such as the term “true” referring to sentences of this language; we have also assumed that all sentences which determine the adequate usage of this term can be asserted in the language. A language with these properties will be called “semantically closed”.
2. In this language the ordinary laws of logic hold.
3. We can formulate and assert in our language an empirical premise such as the statement [x is identical with with the sentence “this sentence is not true”]
Because they lead to contradiction one of these premises must be rejected or revised. Tarski claims that even if we don't assert the third premise it is still possible the generate the contradiction. Therefore we should look to other premises for a possible resolution. The consequences of rejecting the second premise are too grave to consider dropping it. Tarski, therefore, is left only with the first premise as a possible culprit. Tarski claims that all languages that are semantically closed will generate the Liars paradox. In fact, he takes this claim even further to say that similar contradictions will form using any semantic concept in a semantically closed language. Consequently, Tarski asserts that it is impossible for a language to contain its own truth predicate without this contradiction popping up. His solution is to not use semantically closed languages.
It is therefore necessary when speaking of a definition of truth to use two languages. The first language is the language where are found the sentences we wish to define as true or false. This language is known as the object language because it is the object of our truth predicate. The second language is the language in which we ascribe truth to the object language. This second language is the metalanguage. The metalanguage is the language in which is found the truth predicate for the object language. Thus is made a hiearchy of languages where sentences that make reference to truth values are semantically higher. The metalanguage must, as well, satisfy all the conditions of a satisfactory definition of truth in that the definition must be able imply all equivalences of the form
“x is true only iff p” The equivalence must be stated in the metalanguage. “p”, however, is an arbitrary sentence of which we are concerned in the object language. X is the name of p in the metalanguage and it is for this reason that the metalanguage must be able to name all the sentences of the object language.
In order for a satisfactory definition of truth be given in terms of of an object and metalanguage Tarsky gives conditions that the each language must satisfy. Tarski wishes that semantic terms, such as the truth predicate, be introduced into the metalanguage by definition. He wants this so that the metalanguage “will explain the meaning of the term being defined in terms whose meaning appears completely clear and unequivocal.” It is important to note that the relationship between the object and metalanguage is relative. That is, if we wished to define truth for what we are using as a metalanguage that language would become the object language and we would require a new metalanguage with which to to talk about that language. Another condition of a satisfactory definition of truth is that the metalanguage must be “essentially richer” than the object language. Put simply, this condition states that it cannot be possible for a complete interpretation of the metalanguage in the object language. Were this possible then all assertible sentences in one language could be asserted in the other. It would then be possible to reconstruct the Liar's Paradox in the metalanguage and we would run into the same problems as before. Tarski claims that this condition of essential richness is a necessary and sufficient condition for a satisfactory definition of truth wherein semantic concepts are introduced by definition.
To review, in order for a definition of truth to be satisfactory for Tarsky it must satisfy several conditions. It must be materially adequate, that is it must imply all equivalences of the form ““x” is true if and only if x”. A definition must also be formally correct, that is it must avoid problems such as the liar's paradox. To do this, a definition must be stated in a language that is exactly specified. Truth for a given language must be defined in another language that is essentially richer than the one for which we are defining truth. Now that all these conditions are clear Tarski can begin to construct his definition.
can you give me an example of a so called richer language?
PS i read the whole thing and my head hurts!
WOW Dave I didn't expect anyone to read it I posted it more as a joke than anything. But since you asked the languages tarsky talks about are not natural languages like english, or French. Those languages are not exactly specified. In other words their meanings and rules are too ambiguous to either clearly apprehend or solve the liars paradox problem. Tasky is using a hypothetical language that is exactly specified (I've noted the properties he gives for such a language in the previous section). His point is that in this language its imposible to have a truth operator without running into contradiction. A richer language is required. To answer your question, a an essentialy richer laguage would be one in which every possible sentence of the first language would be expressible, as well this language would contain semantic operators such is is true, or is defined by that are applicable to the fist hypothetical language. hope that helps.
Caleb, I haven't read it yet, but love the first sentence!
Andi
I think i'll stick to depreciating assets!
Caleb i have alot to say about the rocks at the bottom of the sea, but you've never asked.
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