Propositional logic

Having introduced the basic notions of validity, soundness and logical form, we will begin our exploration of logic by considering the topic of propositional logic, the branch of logic which deals with relations between propositions. A proposition is something which serves as the premise or conclusion of an argument. In (2) above, Koko is a primate, All primates like daytime television, and Koko likes daytime television are all propositions. Propositions are either true or false. In English, we may think of propositions as roughly like positive or negative factual sentences. The parallel between sentences and propositions is not absolute, however. A sentence like (9) expresses an infinite number of different propositions, depending on the values of the deictic expressions I (my), you (your) and this afternoon:

For each assignment of referents to the deictic expressions, a different proposition results. Similarly, ‘Koko likes daytime television’ can only be considered a proposition as long as the referent of the noun ‘Koko’ has been fixed. Only if we know who ‘Koko’ refers to can we know whether a proposition in which she is mentioned is true or not.

Strictly, the notion of a proposition belongs to logic. We can, however, see it in mental terms. A series of experiments by psychologists has shown that people are very bad at remembering the actual words of utterances. About twenty seconds after hearing or reading an utterance, all people remember is its content or gist: the actual words used usually can’t be remembered accurately. Given this, the propositions discussed here would be one possible representation of this remembered content or gist (see Barsalou et al. 1993 for discussion).

Natural language is not a collection of brute propositional statements without any mutual interrelations: a single statement like (10a) or (10b) can serve as the basis for a whole series of additional statements, depending on the additional linguistic elements added to it. Some examples of these additional statements are given in (10c–h):

It is the italicized elements in (10c–h) which chiefly serve to insert the original propositions (10a–b) into a new, longer one. Among these elements, propositional logic attaches special importance to the four found in (10 e–h). In English, these four elements are expressed by the words and, or, not and if . . . then. We will refer to these as the propositional connectives or logical operators (already mentioned in 4.3.1). These four differ from others, such as those in (10c–d), in that they are truth-functional. This means that whether the larger propositions they are part of are true or not depends solely on the truth of the original basic propositions to which they have been added: the logical operators do not add anything true or false to the basic propositions themselves; all they do is generate additional propositions from the basic ones.

Let’s demonstrate truth-functionality by considering the operator not. Let’s grant that (10a) ‘Daryl Tarte grew up to publish a raunchy family saga in 1988’ is true. Then, (10e) ‘Daryl Tarte did not grow up to publish a raunchy family saga in 1988’ cannot be true: the two propositions are contradictory, and we cannot imagine a world in which they could be simultaneously possible. Conversely, if (10e) is true, then (10a) must be false. We can deduce the truth or falsity of one proposition from the other: if one is true, the other can only be false. Similarly, if (10a–b) are true, then (10f) must also be true. But if one or both of (10a–b) are false, then (10f) as a whole must likewise be false.

As we will see, the other two connectives are also truth-functional. Before showing this, however, we need to abstract away from the English words which express them. Observe that (10e) is not the only way in which negation is expressed in English. The following sentences all involve negations or denials, but unlike (10e), they do not use the grammatical means of do/did + not to express this:

Intuitively, however, it seems obvious that all these sentences contain a denial or a negation, but under different grammatical guises. Examples like these show that language makes differing means available to express what is, intuitively, a single logical operation, negation. Let’s further assume that propositions like those in (11a–e) can be expressed in every language. Let’s assume, in other words, that there is no reason that the propositions have to be stated in English: speakers of any language can negate propositions in a way that is semantically identical to the English negations in (11a–e).

QUESTION What are some alternative ways in which the other operators could be expressed in English?

 Considerations like these mean that we need to find some other way of symbolizing the operators which abstracts away from their translations into any single natural language. To do this, we will adopt a set of symbols for negation and the other operators. Negation, for example, will be symbolized with the symbol ¬. We will introduce the other symbols in the rest of this section. Note that the symbols apply uniquely to entire propositions. If the small letters p, q, r. . . stand for given propositions, ¬p, ¬q and ¬r stand for their negations. We cannot use ¬ to symbolize negations of non-propositional elements like not tomorrow, not again, etc.

The values or meanings of the operators can be specified in the form of diagrams called truth tables. Truth tables display the way in which logical operators affect the truth of the propositions to which they are added. (The use of truth tables is a fairly recent innovation in logic: they are implicitly present in Frege, but first overtly used by Wittgenstein.) The truth table for ¬ is very simple, and is given in Table 6.1:

All this says, reading left to right and top to bottom, is that if p is true, ¬p (‘not p’) is false, and that if p is false, ¬p is true. Let’s say that p is the proposition ‘Marie Bashir is governor of New South Wales’. If this is true, then ¬p, ‘Marie Bashir is not governor of NSW’ must be false; conversely, if it is false, then ¬p must be true. The truth table can be read in either direction. It is equally true, then, that if ¬p is false, then p is true, and if ¬p is true, then p is false.

 The next logical operator is conjunction. As its name implies, this denotes the conjunction or union of two propositions. The conjoined propositions are called conjuncts. The symbol for conjunction is the ampersand, &. The lexical realizations of conjunction are quite various. In particular, the logical operator translates English and and but, as well as other contrastive conjunctions like in spite of and although. If p stands for the proposition ‘The Emperor has no money’ and q for ‘he has 400 000 soldiers’, then p & q can stand for any one of the following complex propositions:

If two propositions are both true, then their conjunction is also true (case (a) in Table 6.2). If the proposition apricots are fruit and the proposition beans are vegetables are both true (as, indeed, they are), then the compound proposition apricots are fruit and beans are vegetables must also be true. But if one of the conjoined propositions (conjuncts) is false, then the entire con junction is also false (cases (b) and (c)). For example, let’s take the two propositions apricots are fruit (which is true) and beans are fruit (which is false); their conjunction, apricots are fruit and beans are fruit, is false since the second conjunct is false. The fact that one of the conjuncts is true makes no difference: we could have a conjunction made up of ninety-nine true propositions and a single false proposition, but as a whole the conjunction would be false (apricots are fruit, and peaches are fruit, and apples are fruit, and pomegranates are fruit, and tamarinds are fruit . . . and beans are fruit). Finally, if both conjuncts are false then the conjunction is clearly also false. If apricots are vegetables and beans are fruit are individually false, their conjunction apricots are vegetables and beans are fruit can only be false.

Note that ordinary language and does not always correspond to logical &. As we have seen, & serves purely to join propositions. In natural language, however, and frequently links nominals:

Conjunction joins two propositions together. Propositions may also, however, be dissociated. This is accomplished by the operation of disjunction. The two propositions in a disjunction are called the disjuncts. There are two types of disjunction. Exclusive disjunction says that just one of the disjuncts applies, but not both. Exclusive disjunction is not usually given a special symbol; we shall refer to it simply as ‘X-OR’. Its truth table is shown in Table 6.3.

Only in cases (b) and (c) is the disjunction true.

 In many respects, exclusive disjunction is like English either . . . or, but with one important difference. In English, either . . . or can be used inclusively, that is, even if both disjuncts are affirmed:

This type of disjunction is known as inclusive disjunction. We will symbolize it with the sign. Note that inclusive disjunction is an even more likely interpretation of simple English or (without either), in contexts like the following:

Clearly, (16) leaves open the possibility that some people were wearing both hats and sunscreen: to enforce an exclusive reading, we would have to add the phrase but not both.

The last operator is also the most interesting. It is called the material conditional, and it corresponds (roughly – but only roughly) to the meaning of English if p . . . then q. It is symbolized by the operator). The proposition to the left of) is known as the antecedent; the one to the right is called the consequent. Thus, in (17), the underlined clauses are the ante cedents, and the italicized ones the consequents:

Case (a) of the truth table says that if two propositions are true, then the material conditional in which they are antecedent and consequent is also true. Thus, since each antecedent and consequent in (18a–d) is true, the material conditionals are also true:

The propositions in (18) are, of course, somewhat odd, and we would not normally express them in ordinary language. This is because the meanings of antecedent and consequent are completely unrelated. The fact that snow is white does not usually allow us to draw any conclusions about whether pigs have tails, and whether or not this is a semantics book has nothing to do with everyone’s having birthdays. Recall, however, that like the other operators,) is purely truth-functional: all that matters is whether the antecedent and consequent it relates are true, and whether they have anything to do with one another is completely irrelevant. Thus, the fact that the sentences in (18 a–d) would not normally be uttered in language is irrelevant to the calculation of truth-values for): the truth or falsity of the proposition is the only relevant consideration. And since the antecedents and consequents in (18 a–d) are all true, the material conditionals involving them are also true.

In real language, the material conditional is often found in an argument type traditionally known as modus ponens (Latin: ‘affirming mode’). In this argument type, an antecedent is affirmed (i.e. said to be true) and the truth of the consequent is deduced from it, as in (19):

Note that the order in which the first two propositions are stated makes no difference to the logical form of the argument. Thus, (19) could just as easily be stated as (20):

In case (b) of the truth table, where the antecedent is true and the consequent is false, the conditional is also false. Here are some examples:

In (21a), imagine that there is a recession, but that interest rates have not fallen. The proposition as a whole is clearly false. The same is true for (21b): France did not successfully bid for the 2012 Olympics, so (21b) as a whole is false. (21c) and (d) remind us that all the logical operator cares about are truth-values, and that antecedent and consequent do not have to have anything to do with each other.

Let’s now consider case (c) of the truth table, in which antecedent and consequent are false. In this case, we can see from the truth table that the conditional proposition which links them is still true. This is exemplified by the following conditionals.

These conditionals are all true, even though antecedents and consequents are false. We will explain the reasoning behind this after introducing the last row of the truth table for the material conditional.

Case (d) of the truth table for the material conditional says that if the antecedent is false and the consequent is true, the conditional is also true. This means that the following statements must be true:

There is something slightly peculiar about saying that cases like (23) are true. The rationale for this, however, is that one is entitled to deduce anything from a false premise: in case (c) of the truth table we deduced a false consequent from a false premise; here, we deduce a true one. (The principle that anything follows from a false premise was first enunciated in the Middle Ages.) If we start out with something that is false, we have a basis for any conclusion, whether or not it is true. Given the premise ‘Tolkien wrote War and Peace’, which is false, we can draw true and false conclusions alike: since the initial premise is false, whether or not the consequent is true is simply irrelevant.

اخر الاخبار

اخبار العتبة العباسية المقدسة

قسم رعاية الصحن الشريف ينشر لافتات الحزن بذكرى شهادة الإمام الجواد (عليه السلام)

قسم العلاقات العامة يقيم مجلس عزائه الأسبوعي لاستذكار فضائل أهل البيت (عليهم السلام)

متحف الكفيل يستعرض درعًا هنديًّا يعود إلى القرن التاسع عشر

ضمن محفلها الأسبوعيّ.. دار علوم نهج البلاغة تقدّم محاضرةً عن تحرير الإنسان من قيود المنصب

المزيد

الآخبار الصحية

طريقة مبتكرة للكشف المبكر عن أمراض القلب والشرايين

منظمة الصحة تحذر من خطر إدمان الشباب لأكياس النيكوتين

خبراء يحذرون من عادة شائعة تهدد صحة الملايين

خبير يكشف 5 حقائق طبية صادمة وغير معروفة

المزيد

الآخبار التكنلوجية

الأولى في العالم.. الصين تجري تجربة رائدة في الفضاء باستخدام "جنين اصطناعي"

دراسة حديثة تفند ربط التستوستيرون بالمخاطرة لدى الرجال

ميتا تضيف وضع "المحادثة المخفية" إلى تطبيق "واتساب"

فوبيا المرتفعات ليست من الدماغ.. دراسة تكشف السبب

المزيد

مواضيع ذات صلة

Logic as representation and perfection of meaning

Validity, soundness and logical form

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