Validity, soundness and logical form

Logic may be defined as the study of valid reasoning and inference. On this definition, logic investigates the properties of valid chains of reasoning, and specifies the conditions which these chains must meet in order to be valid, in order to work as arguments. Consider the following exchange:

Initially, B is unable to follow A’s line of thought, and as a result A is forced to state the general principle on which her conclusion rests. This allows us to reconstruct A’s original train of thought as the following argument or syllogism:

Argument (2) thus reveals the explicit logical structure of A’s comment in (1). As Kneale and Kneale explain (1962: 12), the ‘first tentative steps towards logical thinking are taken when men try to generalize about valid arguments and to extract from some particular valid argument a form or principle which is common to a whole class of valid arguments’. Given the meanings of the words all, like and is, the conclusion Koko likes daytime television just has to be true as long as we accept the truth of the proposition All primates like daytime television. It seems likely that it was in domains like mathematics, especially geometry, that the need to make the principles of valid reasoning explicit first arose (Kneale and Kneale 1962: 2); in modern times, the study of logic has been particularly undertaken in the attempt to symbolize the types of reasoning that underlie mathematical arguments.

Logic is important to linguistics for at least three reasons. First, the study of logic is one of the oldest comprehensive treatments of questions of meaning. When people first began to think systematically about the meanings of language and the relations between these meanings, it was logical concepts to which they often appealed for explanations. As a result, the tradition of logical analysis, which we can trace as far back as Aristotle, provides a rich body of reflection on meaning, and most scholars who have studied meaning in the Western tradition have had at least some knowledge of logical principles. The relevance of logic to linguistics is far from simply historical, however. Logical concepts inform a wide range of modern formal theories of semantics, and are also crucial in research in computational theories of language and meaning. We will not be exploring formal theories in themselves here, but our exposition of some fundamental logical ideas will provide some background for those wanting to do so. Lastly, logical concepts provide an enlightening point of contrast with natural language. The basic logical concepts are accessible to practically anyone; indeed, many philosophers have seen in logical principles the universal ‘laws of thought’ which constitute the basic grounds of human rationality: for Immanuel Kant, for example, ‘logic is the science that exhaustively presents and strictly proves nothing but the formal rules of all thinking’ (1998 [1787]: 106). Yet, as we will see, logical meanings often differ strikingly from the types of meaning found in natural language. Studying logic therefore provides a window onto a body of apparently universal concepts with strikingly different behaviour from natural language, which provide a rigorous and enlightening way of disambiguating certain types of natural language expression.

Formal theories

A formal theory is one which offers an analysis of meaning in a technical, usually symbolic, metalanguage, according to principles which can be expressed in mathematical terms. A formal representation of meaning avoids the ambiguities contained in natural language by enforcing a strict correspondence between symbols and meanings: whereas natural languages always contain ambiguous or polysemous terms, in which a single form stands for several meanings (think of English step, match or get), a formal language has a strictly one-to-one relation with its meanings, so that each symbol of the formalism has one and only one interpretation.

As the above quotation from Kant suggests, the principles of valid argument have typically been taken, in the logical tradition, as the very principles governing rational human thought. Logic can be seen, from this perspective, as the science of the laws of rational thought. On this view, logic is the science which tries to specify all the conclusions that can validly be reached from a given set of propositions. It is logical principles which thus describe the process of valid reasoning.

The first two propositions in (2) are called the premises. An argument’s premise may be defined as its starting-point, one of the propositions from which the conclusion follows. In (2), the last proposition is the conclusion. Note that the validity of arguments or of chains of reasoning has a special relationship to the words in which the premises and conclusion may be stated: substitute different words, and the argument may not be valid. None of the following arguments, for instance, is valid:

QUESTION For each of these arguments, explain why it is not valid.

We may therefore say that the validity of this argument type is a function of the meaning of the terms in the syllogism, in particular the meanings of the determiner all, and of the verbs like and is. In (4), if we retain the ‘logical’ words all and are, but substitute words whose meaning is unknown, the resulting argument is still valid: if the premises are true, the conclusion must be true:

Any argument which conforms to the pattern of (4) will be valid. All the arguments below contain the same pattern of reasoning as (4):

This allows us to see that the pattern of valid inference exemplified in (4) and (5) is systematic and does not depend on the details of the material inserted into the logical formula. The validity of these arguments is a function purely of the meaning of the predicates and of the term all and of the argument’s logical form, its underlying logical structure.

The fact, however, that an argument conforms to the pattern of (4) may make it valid, but this does not make its conclusion true. Logical validity and truth are quite different properties: validity is a property of arguments, truth a property of sentences. The following argument is valid in logic, but its first premise is not true.

This means that (6) is valid in logic, but it is not true. Valid arguments whose premises are true are referred to as sound. Arguments like (6) which are valid, but which do not have true premises, are thus unsound. We can tell that (6) is valid, since if the first premise were true, then the conclusion would also necessarily be true. (Note that whether Bogomil is, as a matter of fact, unhappy, has nothing to do with the soundness of (6). Bogomil may well actually be unhappy, but this is not proven by the argument in (6).)

As we have observed, the properties of sentences which make them true are linguistic properties. This suggests that logic and semantics are closely related. Some scholars, indeed, such as McCawley (1981: 2), have assumed that logic and semantics share an identical subject matter: the meanings of natural language sentences. As we will see, not everyone would agree with this: the degree of correspondence between logic and natural language has often been questioned, and with good reason. Nevertheless, as McCawley (1981: 2) notes, logic requires semantic analysis: the meanings of sentences must be identified before their logical properties can be dis cussed. If we do not know the meanings of are and all in (4) we are not in a position to determine the validity of the arguments involving them.

The link between logic and semantics is further revealed by the fact that it is meanings, not sentences, that function as the premises and conclusions of arguments. Thus, assuming (perhaps wrongly) that unhappy and discontented are synonyms, we can substitute any of the synonymous expressions in (7) for the premise of (6), and the synonymous expression in (8) for the conclusion of (6):

These variations do not affect the underlying logical form of the argument.

اخر الاخبار

اخبار العتبة العباسية المقدسة

قسم التطوير يقدم دورة عن أخلاقيات المهنة ضمن برنامج تأهيل المنتسبين الجدد

ضمن برنامج (وفاءً للدماء).. المجمع العلمي يقيم محفلًا قرآنيًّا في محافظة المثنى

قسم الشؤون الفكرية يرفد المكتبة المركزية في ديالى بإصدارات معرفية

قسم التطوير يبحث مع ديوان الوقف الشيعي آفاق التعاون في مجال التدريب والتطوير المؤسسي

المزيد

الآخبار الصحية

المركز الاوروبي : كل ركاب سفينة "هانتا" مخالطون معرضون لخطر كبير

ألمانيا.. رئيس الأطباء يدعو للإسراع بتطبيق "ضريبة السكر"

بين الفائدة والضرر.. جدل حول تأثير الشاي والقهوة على العظام

أمراض خطيرة.. أبحاث تكشف مخاطر الجلوس لفترة طويلة

المزيد

الآخبار التكنلوجية

سبع حقائق غير متوقعة عن موجات الراديو

الأجسام الغريبة.. ماذا رأى رواد "أبولو 17" على القمر؟

حرارة المحيطات عند مستويات مقلقة.. وعودة "إل نينيو" تقترب

باحثون يحددون تأثير القهوة على الميكروبيوم والمزاج

المزيد

مواضيع ذات صلة

Polysemy and monosemy as a cline

Tests for polysemy

Polysemy, monosemy and homonymy

Polysemy and meaning division

Componential analysis

Connective Adjuncts

Synonymy

Taxonomy

Hyponymy

Meronymy

Antonymy

Lexical relations