According to Galileo Galilei “all truths are easy to understand once they are discovered; the point is to discover them.” Proving the truth values of sentences has been of peculiar interest for thousands of years and philosophers as well as mathematicians worldwide have tried to grasp this enormously complex matter. The intricacy of truth even begins with the definition of the object. Even Alfred Tarski, a Polish-American mathematician and one of the greatest logicians of the twentieth century, stated that the main problem is a satisfactory definition of truth. Obviously, discovering the truth of statements is a rather difficult task to undertake. Howsoever, in this paper we illustrate various semantic
relations and theories as well as logical tools which help to establish the truth.
Table of contents
1 Introduction
2 Sentence Semantics
3 Propositions in General
4 Logic
4.1 The Aristotelian Logic
5 Truth
5.1 The Correspondence Theory
5.2 The Coherence Theory
5.3 Tarski’s Definition of Truth
6 Truth-Conditional Semantics
6.1 The Epistemological Distinction of Truth
6.1.1 A Posteriori Truth
6.1.2 A Priori Truth
6.2 The Metaphysical Distinction of Truth
6.3 The Semantic Distinction of Truth
7 The Semantic Relation of Entailment
7.1 Truth Table Entailment
7.2 Modus Ponens
7.3 Entailment given by Linguistic Structure
7.4 Entailment given by Syntactic Structure
7.4.1 Active and Passive
7.4.2 Synonymy
7.4.3 Contradiction
8 Presuppositions
8.1 Negation of Presupposition
8.2 Two Approaches of Presupposition
8.2.1 The Semantic Approach
8.2.1.1 Truth Table Semantic Approach
8.2.2 The Pragmatic Approach
8.3 Presupposition Failure
8.4 Presupposition Triggers
8.5 Lexical Triggers
8.6 Presupposition and Context
8.7 Pragmatic Theories of Presupposition
9 Conclusion
10 References
11 Appendix
Introduction
According to Galileo Galilei “all truths are easy to understand once they are discovered; the point is to discover them.”1 Proving the truth values of sentences has been of peculiar interest for thousands of years and philosophers as well as mathematicians worldwide have tried to grasp this enormously complex matter.
The intricacy of truth even begins with the definition of the object. Even Alfred Tarski, a Polish-American mathematician and one of the greatest logicians of the twentieth century, stated that the main problem is a satisfactory definition of truth2. Obviously, discovering the truth of statements is a rather difficult task to undertake. Howsoever, in this paper we illustrate various semantic relations and theories as well as logical tools which help to establish the truth.
2 Sentence Semantics
„Die Bedeutung eines Satzes ist eine Funktion der in ihm enthaltenen Ausdrücke und der Art ihrer Zusammensetzung.“3 Sentence semantics contains two assumptions. According to J. Chur, on the one hand , there is the “wahrheitswertfunktionale Semantik” (J. Chur, 117) whereas a reference of a sentence is its truth value. The reference, in this case, is the set of conditions under which the sentence is expressed. The intention of the speaker is not taken into consideration.
On the other hand, linguistics is about “Situationssemantik”, whereas this second approach is presented as a further development of the first. Sentences are said to be true if the conditions and the context can be explicitly classified as true.
“Die Bedeutung eines Satzes kennen, bedeutet angeben können, ob er wahr oder falsch ist.“4 Here J. Chur explains the reason for doing this process of finding the truth, figuring out if a proposition is true or false.
One of the main goals of Sentence Semantics is to determine the possible relations between two propositions and which are easy to notice by the speaker, as well as the listener.
Sentence semantics tries to explain the truth value of declarative speech acts, not of questions, imperatives or requests. There are always semantic relations whether they are true or false. The relations between sentences as well as propositions will be explained in the following chapters.
3 Propositions in General
Defining the term proposition would be asking what exactly is meant by the things somebody says or thinks. It is necessary to explore whether the utterance is true or false. The sentence “It is raining” can be said in any language (e.g. “Il pleut”, “Es regnet”, “Sta piovendo” etc.) and it always contains the same meaning.
The identical content of different utterances and diverse acts of thinking in different languages was transferred by propositions, which are supposed to be the real bearers of truth and falsity.5
“When a person thinks or believes something, it is always a proposition that he thinks or believes.”6 A proposition is said to be an entity which is timeless and non-linguistic, capable of being believed and disbelieved by any number of different minds. Therefore, propositions are the common idea which several sentences may express.
The following three sentences will express only one proposition, viz. they are talking about Sam Jones. Sam Jones’ brother says “My brother is sick”, the same Sam Jones’ mother says at the same time “My son is sick”; and his son “My father is sick”.7
Propositions are defined as being “vehicles for stating how things are or might be”8. An interrogative sentence, on the one hand, would ask how things are, instead of saying how something is. The imperative, on the other hand, commands to do certain things. Thus, only indicative sentences are capable of being expressed as a proposition. In addition to this, it is to mention that the context determines the proposition in order to avoid any ambiguity.
The following example underlines the statement: “Cows do not like grass9.” On the one hand, one could understand it in a way of negating the fact that cows like grass, on the other hand, one could try to find the different meanings of the term “grass” and come to the conclusion that cows do not like marihuana. The technical term for finding different meanings and therefore different contexts of propositions is called “semantic ambiguity”10.
Moreover, the semantic view as well as the syntax can be ambiguous. The sentence “Everyone loves a sailor” could be interpreted like this: Each person loves at least one sailor or that everyone loves every sailor.
Thus, that conclusion is too general and not really comparable to what the first sentence wanted to express. Therefore, the context determines what the speaker intends to say.
Consequently, we need to have a look at the demonstratives, personal pronouns or words like adverbs of time and place. Here the speaker’s perspective is important, including the personal point of view, the location and the time, involving the so-called "indexical/ deictic" (or "token reflexive") terms such as "I" and "here" and "now”. In addition to that, one may also have a look to what the words refer to.
Every proposition has exactly one truth-value. The proposition (P: The sun is shining.) was true yesterday or rather that day the speaker uttered it but the utterance is always connected to the speaker’s point of view.
The same proposition can be false regarding to another day, when the sun isn’t shining and it is pouring outside.
Therefore, the three Laws of Thought need to be mentioned. The first Law of Thought is the Law of Identity which says that 'Everything is what it is and not another thing', or 'If the proposition P then P'. The second is called the Law of the Excluded Middle, saying that a proposition can either be true or false and is always connected to the different presuppositions and entailments which need to be taken into account.
Furthermore, a proposition cannot be true and false at the same time, mentioned by the Law of Non-contradiction11. Coming back to the example of P (P = The sun is shining) which was true at that day and probably false on another day, it cannot be true and false at the same time the speaker is expressing it. It has to be clear whether the sun is shining or not, there is no status in between.
4 Logic
First of all, a definition of logic according to Newton-Smith seems to be adequate. Here, logic turns out to be the study of valid arguments and its distinction from valid and invalid arguments.12 Arguments usually consist of one or more premises and a conclusion. The most famous example of an argument would be:
Socrates is a man.
All men are mortal.
Therefore, Socrates is mortal.
Thus, it is obvious that if the premises are true, the conclusion needs to be true as well. Since the premise is valid, the conclusion is valid and therefore true.
The sky is blue and the grass is green. Therefore, the sky is blue.
The following example will show that the lack of validity can change the truth of the conclusion when the circumstances in the real world are not satisfied anymore. The truth-value plays an important role in logic.
The sky is blue or the grass is orange. Therefore, the grass is orange.
If the premises are true there will be no guarantee that the conclusion is true. Arguments that support the conclusion are valid and they are called deductive arguments13.
According to Hurford, the definition of logic is like this:
Logic deals with meanings in a language system, not with actual behaviour of any sort. Logic deals most centrally with PROPOSITIONS. (.) It tells us nothing about goals or assumptions, or actions in themselves. It simply provides rules for calculation which may be used to get a rational being from goals and assumptions to action. 14
What is important to mention in this case is that one always tries to describe the whole system of logical reference which means that one is trying to build up a comprehensive account of all reasoning. The truth conditions of the individual examples would then follow automatically.15 One of the goals of logic is to describe explicitly the fundamental Laws of Thought which I’ve mentioned earlier on.16
4.1 The Aristotelian Logic
The Aristotelian language forces the speaker to make crisp statements (precise statements which are either true or false) and short statements instead of extended ones. Aristotle’s logic acknowledges only a black and white world, everything is or is not.
His logic is dominated by the Law of the Excluded Middle and it accepts only the extreme edges of reality, excluding the grey areas in between17. Rough statements such as “The room is too hot” offer the listener too many possibilities to interpret the statement, even though they are truer to life. Since accuracy and the wish to express whether black or white seem to be almost impossible in the human communication, the Aristotelian logic does not seem to solve all problems of defining the truth. The structure of language needs to match reality, and a crisp description can rarely do this. 18
What logic can do in semantics and why things are called logical will be proven in the following chapters of this paper.
5 Truth
Philosophers explain the term as a correspondence between a sentence and some components in reality. The truth of the sentence stands or falls with the conditions under which it is expressed. “A theory of truth is a theory of what truth is or, alternatively, of what something’s being true consists of.”19
5.1 The Correspondence Theory
The classical theory of truth is the Correspondence Theory. First proposed in a vague form by Plato and by Aristotle in Metaphysics, this realistic theory says that truth is what propositions consist of by corresponding to a fact in the real world.
The theory says that a proposition is true if there is a fact corresponding to it. In other words, any proposition P is true if and only if P corresponds to a fact.
The challenge to the correspondence theorist will be to give a non-trivial characterization of what is being said and its agreement with or correspondence to reality.
To answer the question to what the sentence corresponds to, these are the facts in reality. According to Collin and Guldmann, the fact for the sentence “It is raining” would be that it is in fact raining. This means in general that a true sentence “P” corresponds to the fact P.
5.2 The Coherence Theory
Truth in this theory, on the contrary to the latter, is seen as a relation between sentences that are actually believed as being true. Stating the fact that a sentence is true would mean that this sentence fits with the largest set of sentences between which agreement obtains20.
Coherence theories are valuable because they help to reveal how we achieve our truth claims, our knowledge. The beliefs should fit into a coherent system. The following example shows what is meant with this theoretical explanation.
We assume a drunk driver saying this sentence: "There are pink elephants dancing on the highway in front of us"21 Since most of the people know that, firstly, elephants are grey and, secondly, the highway is not their habitat and in addition to that there is no zoo or circus anywhere nearby, we determine this expression as being not valid and false. The speaker who is said to be drunk is probably having hallucinations. Thus, the world knowledge helps to find out whether the sentences belong together or not.
5.3 Tarski’s Definition of Truth
The notion of truth is central to logic and Tarski’s definition of truth is a set of rules. Tarski shows how to provide a rule for the application of a certain predicate, “x is T”, to formulae which can be constructed in a symbol system. He provided an argument for the claim that this mystery notion ”T” is tantamount to truth.22
Taking this definition into account, the example “The snow is white.” is true if and only if the snow is in fact white. This means that the person who wants to say that a sentence is true needs to know everything about the facts and references being related to the sentence, viz. the material equivalence and therefore its truth-conditions.
Tarski developed a test (viz. “Tarski’s Test”) in order to determine the truth of a concept. It is assumed that somebody who is characterizing a sentence as true needs to be capable of saying what the sentence actually says23.
According to this, the sentence “Bucephalus is a horse” will show its conditions under which we can call the sentence to be true. The proper name, for instance, specifies the meaning of the sentence. The thing referred to by the name Bucephalus is Bucephalus. The meaning of predicate in this case can also be specified by the notion of satisfaction whereas “x is a horse” is true if and only if it is a horse24. Regarding to this example the sentence is true if and only if Bucaphelus is called Bucaphalus and if it is a horse. A sentence needs to be satisfied by all objects in order to be true, otherwise the sentence is false.
There are numerous sentences which are actually non-compositional in nature. The use of metaphors, like in the sentence “Time is money”, shows that only by understanding the two nouns of the sentence would not transfer the meaning. In addition to that, using idioms, sayings would be another source of it. “Reading between the lines” and interpreting gestures and one’s body language shows a different aspect of sentence semantics.
The works of Kant, Tarski and other philosophical pioneers play a major role for supporting concepts which deal with the meaning of statements. Touching semantics, an excursion into the wide field of logic thus is absolutely essential to represent sentence meaning. Some logical tools will serve as instruments and eventually help to interpret particular truth values. Not to go beyond the scope of our topic, just selected and meaning-based logical tools are used to illustrate our issues.
6 Truth-Conditional Semantics
The Truth-Conditional Semantics is an approach that emphasizes the role of truth within and between sentences/ propositions. The theme would be: understanding a sentence means knowing the conditions under which it would be true. Some propositions are empirically true, some false.
The listener needs to decide whether the sentence is true or false by checking the facts in the real world, viz. the truth conditions.
6.1. The Epistemological Distinction of Truth
According to Kant, the first distinction can be characterized as epistemological because it deals with cognition and in particular with a speaker’s knowledge. Here a distinction is made between a posteriori and a priori truth.
6.1.1 A posteriori truth
A posteriori truth on the one hand is also known as empirical truth as empirical testing is required to know whether a certain statement is true or false. So statements can be declared true from experience and in hindsight. Thus, it is crucial what the speaker knows or needs to know before making a judgement about the truth of any proposition. It is widely discussed whether truth is preserved or lost by putting sentences into different patterns. Truth here is taken to mean a correspondence with facts or, in other words, correct descriptions of states of affairs in the world.25
My father was the first man to visit Mars.
The truth or falsehood of this sentence depends on facts about the speaker’s father’s life. Thus the truth here can only be known on the basis of empirical input and hence is a posteriori. The constituent words alone cannot indicate whether the proposition is true or false. There are different ways in which the truth value of a proposition may be discovered.
[...]
1 http://www.quotationspage.com/quote/937.html
2 http://www.ditext.com/tarski/tarski.html
3 Chur: 2004, 117
4 Chur: 2004, 118
5 Rosenberg: 1971, 225
6 Ibid. 226
7 Ibid. 228
8 Newton-Smith: 1985, 7
9 Ibid. 7
10 Ibid. 7
11 http://www.philosophyprofessor.com/philosophies/excluded-middle-law.php
12 Newton-Smith: 1985, 1
13 Newton-Smith: 1985, 6
14 Hurford: 2007, 142
15 Ibid. 150
16 Ibid. 167
17 http://www.thefreelibrary.com/General+semantics+and+fuzzy+logic.-a0123408744
18 http://www.thefreelibrary.com/General+semantics+and+fuzzy+logic.-a0123408744
19 Finn: 2005, 89
20 Ibid. 98
21 http://www.iep.utm.edu/t/truth.htm
22 Morris: 2007, 180
23 Ibid.180f
24 Ibid. 182
25 Saeed: 2003, 81
- Citar trabajo
- Andreas Nauhardt (Autor), 2009, Logic: Sentence Relations and Truth, Múnich, GRIN Verlag, https://www.grin.com/document/138235
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