Q-Reading: Formal Logic: Introducing the Symbolic Language of SL

Q-reading: Formal Logic: Introducing the Symbolic Language of SL

(Guttenplan, 40-67)

The analogizing method in the last section was very impractical but revealed something about the structure of valid arguments. To repeat that finding, it showed that the validity of an argument can be assessed by analyzing the structure of how the premises fit the conclusion. In analogizing, we created an invalid counterexample using the same basic form as the original argument. In this section we will begin a whole new approach that uses this same idea. We will invent a new symbolic language that can be used to translate English arguments to isolate their structure. Then, in later sections will show how the validity of these symbolic translations can be tested, in terms of their structure, quite easily and very accurately. We will begin with a very simple version of a symbolic language, called “Primitive” and then proceed by adding to Primitive until we reach our full, symbolic language.

The Sentences of Primitive

Primitive has two sentences which form the basis for the whole language. Each of these is completely unstructured. That is, these sentences do not contain what we would recognize as words. They are written like this:

P

Q

Each of them says something about the weather. Speakers of Primitive use 'P' to describe the weather when it is raining but not when it is sunny. They use 'Q' to describe the weather when it is windy but not when it is calm. Given this, it is reasonable to think of 'P' as meaning that it is raining, and 'Q' as meaning that it is windy. Primitive contains only twobasic sentences, but these can be used in combination to make more complicated, non-basic sentences. These combinations are best explained one at a time. Speakers of Primitive sometimes use sentences like:

P&Q

which is made by using the '&' symbol and two basic sentences. The symbol is called conjunction. It applies in the case where the weather is rainy and windy but not when it is rainy but not windy, nor when it is sunny and windy, nor when it is sunny and not windy. This range of application of 'P & Q' makes it seem reasonable toregard the '&' as functioning something like the word “and” in English, though we will not pause to speculate further about this. Our concern here is, above all, with the Primitive language, not with its relation to natural languages.

True and False

'P & Q' applies to the weather situation where it is raining and windy but not to the situations where it is not both rainy and windy. In other words, ‘P&Q’ is true when ‘P’ is true and 'Q' is true, and'P & Q' is false when 'P' is false and 'Q' is true and when 'P' is true and 'Q' is false and when 'P' is false and 'Q' is false.

Given that you understand what each basic sentence says, you knowprecisely what circumstances must be like when someone asserts thatone or other of them is either true or false. The described situations were helpful forgetting started. They can now be replaced by the use of the words 'true'

and 'false'. Aside from saving me a lot of space, this change removessome of the artificiality that my described situations imposed. For example, if I now tell you that 'P' is true, you know I have said that it is raining. No descriptions are required to convey this to you.

Let us rewrite the explanation of 'P & Q':

1. When 'P' is true, and 'Q' is true, then 'P & Q' is true.
2. When 'P' is true, and 'Q' is false, then 'P & Q' is false.
3. When 'P' is false, and 'Q' is true, then 'P & Q' is false.
4. When 'P' is false, and 'Q' is false, then 'P & Q' is false.

I have used the same numbers here as I did in the explanation given earlier so that you can compare the two, line by line. We can improve this explanation still further by writing it in table form:

P / Q / P&Q
True / True / True
True / False / False
False / True / False
False / False / False

Let us place the truth-values of ‘P’ and ‘Q’ directly under the letters in the sentence for greater clarity:

P / Q
True / True / True
True / False / False
False / False / True
False / False / False

Truth Tables and Truth Functionality

Many lines of print have been used to introduce a language that has so far been shown to contain only a very few sentences. Nonetheless, as I go on to describe further sentences in Primitive, the ways of talking about the language that have been used above will make explanations clearer and shorter.

The tables used above are calledtruth tables.The first row consists of a truth value for 'P', a truth value for 'Q', and a truth value for 'P & Q', written in two different formats. This combined truth value is just what you would expect given the values for 'P' and 'Q' and the meaning of the '&'. Each row gives different possible truth values for the combination of 'P' and 'Q'. The four rows display all the possible ways in which these basic sentences could be true or false. As we have discussed, the truth value of the sentence 'P & Q' is fixed by the truth values of its constituent sentences. Each row describes a way the weather could be at some time, and the first row shows the weather situation in which the sentence ‘P&Q’ is a true statement of the weather conditions.

The next sort of non-basic sentence of Primitive looks like this:

P۷Q

and has the following truth table:

P / ۷ / Q
True / True / True
True / True / False
False / True / True
False / False / False

The '۷' symbol with which it is constructed is called disjunction. Can you get some idea of what P۷Q conveys to a user of Primitive?

Just as for conjunction, the truth table tells you all you need to know about the meaning of '۷'. To understand disjunction you should read each of the rows of the truth table, seeing what truth value is assigned to the non-basic sentence given the values ofthe basic sentences. Doingthis reveals that P۷Q is true in each of the first three rows and is onlyfalse when both ‘P’ and ‘Q’ are both false. Strictly speaking, this should suffice to give you an understanding of P۷Q. Armed with this information, you would be able to go among speakers of Primitive and use sentences constructed with '۷' correctly. You would know that you could use such a sentence truly in every case except when both of its constituent basic sentences were false. In the specific case of P۷Q, you would know that it was false only when it was false that it is raining and false that it is windy.

It is often helpful in understanding a foreign language sentence to be able to translate it, perhaps only approximately, into your own. So, granting that there might be no exact equivalent in English, how bestcould you convey the meaning of '۷' as it occurs in P۷Q using theresources of English?

It is raining or it is windy.

The problems arise in connection with row 1, and might be put as follows: P۷Q is true in row 1 (when it is true that it is raining and truethat it is windy). This might be taken to show that 'or' is not a goodtranslation of '۷'. Someone might argue: the use of 'or' in a sentence conveys the idea that one or the other constituent sentence is true, not when both are true. So, if it is raining and it is windy, it is false that: it is raining or it is windy.

Is this consideration compelling? Think about the following dialogue:

Teacher {disgruntled}: Every time we have the school picnic it israining or windy.

Pupil: No, last year we had the picnic and it was raining and windy.

The pupil's remark is (mildly) amusing. Why? Clearly, the teacher didnot intend to rule out the possibility of rain and wind by his remark. He wanted to make an assertion that included that worst eventuality, but also made provision for the fact that rain alone (or wind alone) had adverse effects on the school picnic. … Using 'or' seemed the proper way of expressing the thought. The pupil's remark draws attention to a somewhat different use of'or' as illustrated in this dialogue:

A: What are your holiday plans?

B: I am going for one week to Paris or Vienna.

It is implicit in this exchange that B has not spoken accurately if he ends up going to Paris and to Vienna. In essence, then, the pupil's remark is a play on the word 'or': the teacher intended it one way, while the pupil interpreted it in another. Does this mean that 'or' is ambiguous - that it has more than one meaning in English? We cannot fully answer this question now, but we can venture this much: 'or' can be used in different ways, and the use that matches '۷' most closely is that in the teacher's lament about the weather. The teacher used 'or' in an inclusive way, whereas it was used in an exclusive way in B's remark about his holiday. Speakers of Primitive do not have these different uses - the student's joke could not be made in it. This is because '۷' is given an exhaustive and precise meaning by its truth table. Primitive disjunction is always inclusive.

Here is another non-basic sentence of Primitive:

~P

The '~' in it is called negation, and the explanation of its use is simple. The truth table for the above sentence is.

~ / P
False / True
True / False

As you can see from the truth table, it has theeffect of reversing the truth value of the basic sentence it contains. Sincethis non-basic sentence is formed from just one basic sentence, its truthtable is constructed using only the appropriate sentence. Because P onlyhas two possible truth values, true or false, the truth table for ~P need only have tworows.

The most natural translation of ~P in English is:

It is not raining.

and I think you can satisfy yourself that there is nothing in the truthtable for ~P which casts doubt on this translation.

The symbols '&', '۷' and '~' are called connectivesof Primitive.There are two more to come. You may not thinkthat this is the most appropriate word, since '~' doesn't appear to do much connecting. However, logicians have been used to calling symbols such as '&' two-place connectives (for obvious reasons), and they do not therefore find it odd to think of '~' as a one-place connective.

The next sort of non-basic sentence of Primitive looks like this:

P→Q

and has this truth table:

P / → / Q
True / True / True
True / False / False
False / True / True
False / True / False

The '→' symbol used in its construction is called the conditional. As before, it is quite easy to use the truth table to master the use of '→' in Primitive. A sentence P→Q constructed with it can be truly asserted in all cases except where P is true and Q is false. In discussing the conditional, it is helpful to use the following terminology: the first element ofthe conditional sentence (P in the above) is called the antecedent; thesecond element (Q) is called the consequent.

Speakers of Primitive think it correct to use the above example of a conditional sentence in all cases except when, as we would put it, it is raining and it is not windy. Can you think of any close approximation to P→Q in English? That is, can you think of some form of words in English that translates '→' in the way that 'and' was used in connection with ''? You will come to appreciate that these questions lead to the unfolding of a complex and fascinating story. Here we can only begin the story.The most plausible translation in English of P→Q is:

If it is raining then it is windy.

Understanding Primitive speakers as using P→Q with the sense of “if, then” goes some way to showing why they assign the truth values they do to the conditional sentences. There are problems, however, so it is best to discuss this in a little detail.

Row #1: It is true that it is raining, and true that it is windy.

It seems fairly obvious why this situation would be a true conditional, for it fit the sense of “if, then” quite exactly.

Row #2: It is raining, and it is not windy.

Do we agree that this means that “if it is raining, then it is windy” is false? Here matters are less complicated. This situation seems to flatly contradict the statement, so false is clear.

Row #3: It is not raining, and it is windy.

In this situation, does it seem natural that “if it is raining, then it is windy” should be true? This question seems most peculiar. One is very tempted to protest: what possible sense to speak of an “If, then” when the antecedent is false? The question before us is whether “if it is raining, then it is windy” is true in the circumstances given in row 3. We are not interested in whether you would actually use “if it is raining, then it is windy”if you knew it was not raining and it was windy.

It is fairly clear that “if it is raining, then it is windy”is not falsified by having a false antecedent. After all, there could be many other weather conditions that happen to go with its being windy. “If it is raining, then it is windy” does no more than claim that rain is one such condition.

Consider the following dialogue which makes the point using a different example of an English sentence with 'if, then'.

(On Wednesday)

Smith: If England wins the toss then they will win the Test Match.

(A week later)

Jones: You were wrong. They lost the toss and won the Test Match.

Smith: No. I was perfectly right. I never said that winning the toss was the only way they would win the Test. I said that if they won the toss, they would win the Match. In saying this, I spoke the truth even as things turned out.

Row #4: It is not raining and it is not windy.

Does this entail that “if it is raining, then it is windy” is true? This case shares with row 3 the somewhat puzzling fact that the antecedent is false, but it differs in that the consequent is also false. Does this change things? Not really. Imagine that someone has offered you “if it is raining, then it is windy”as a bit of lore about the weather. While out for a walk on a sunny and still day, he says: I told you, if it is raining then it is windy. Isn't what he says true in those circumstances? I expect that many would agree that it is, and certainly would not think it false.

The last sort of non-basic sentence of Primitive is constructed with '↔'.

An example of such a sentence is:

P↔Q

The new symbol is called equivalence, and has the following truth table.

P / ↔ / Q
True / True / True
True / False / False
False / False / True
False / True / False

P↔Q is true when both basic sentences are true, and when both basic sentences are false. In the other two cases it is false. A first attempt to express P↔Q in English might come out as:

it is raining if and only if it is windy.

The phrase 'if and only if' is called the biconditional. All the problems that arose in connection with the conditional '→', and its translation into English, come up again in regard to ‘↔’.

Enlarging Primitive (The language of Sentential Logic: “SL”)

There isn't much that you can do with a language as limited asPrimitive. In this section, we are going to enlarge Primitive in twoways. I will call the new language Sentential.Sentential will prove to be a considerably more powerful language,but the enlargements to Primitive that it requires are not difficult tounderstand. Let us review the main features of Primitive.

a)There are two basic sentences: P, Q.

b)These sentences are, in any given circumstance, definitelyeither true or false.

c)There are five ways in which non-basic sentences can be constructed from basic sentences. Each of these uses a differentconnective. They are listed below:

• Conjunction:
• Disjunction: ۷
• Negation: ~
• Conditionalization: →
• Equivalence: ↔

Each of these is truth-functional. That is, in each case, the truth valueof a non-basic sentence constructed with the connective is fully determined by the truth values of its basic constituents. The truth tablewhich shows how this determination is made for each connective can beconsidered a complete explanation of the meaning of the connective. The move from Primitive to Sentential requires us to look at features(a) and (c).

The Stock of Basic Sentences

Sentential has many more basic sentences than Primitive. This is brought about by our simply stipulating that Sentential has the following basic sentences: A, B, C, D, …etc.

When you learned Primitive, it was possible for me to give you some idea of what the basic sentences of the language are used to say. I was able both to use English language sentences, to give you an idea of the content of two of them. Sentential has an indefinitely large number of basic sentences, so neither of these methods is at all practical. This may lead you to wonder how it would be possible to use Sentential as a language.

Unlike speakers of Primitive, speakers of Sentential are very casual about their use of basic sentences. If one of them wants to say that it's a lovely day, he might come out with the sentence “L”. On another occasion, he may use “L”mean that it is very cold. It is thus very important for them to define their basic sentences at the outset.

More Non-basic Sentences

Primitive offered five ways in which its basic sentences could be used toform non-basic sentences. Sentential uses precisely the same five devices,but it allows non-basic sentences to contain other non-basic sentencesas parts. Thus, the most complex sentence of Primitive had threeelements - for example: