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 two basic 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 to regard 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 know precisely what circumstances
must be like when someone asserts that one or other of them is either true or false. The described
situations were helpful for getting started. They can now be replaced by the use of the words 'true'
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and 'false'. Aside from saving me a lot of space, this change removes some 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.
2.
3.
4.
When 'P' is true, and 'Q' is true, then 'P & Q' is true.
When 'P' is true, and 'Q' is false, then 'P & Q' is false.
When 'P' is false, and 'Q' is true, then 'P & Q' is false.
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
True True
True False
False True
False False
P&Q
True
False
False
False
Let us place the truth-values of ‘P’ and ‘Q’ directly under the letters in the sentence for greater
clarity:
P
True
True
False
False
&
Q
True True
False False
False True
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 called truth 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
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and has the following truth table:
P
True
True
False
False
۷
Q
True True
True False
True True
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 of the basic sentences. Doing this reveals that
P۷Q is true in each of the first three rows and is only false 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
best could you convey the meaning of '۷' as it occurs in P۷Q using the resources 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 true that it is windy). This might be taken to show that 'or' is not
a good translation 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 is raining 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 did not 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
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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 the effect of reversing the truth value of the basic sentence
it contains. Since this non-basic sentence is formed from just one basic sentence, its truth table is
constructed using only the appropriate sentence. Because P only has two possible truth values, true
or false, the truth table for ~P need only have two rows.
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 truth table for ~P which casts doubt
on this translation.
The symbols '&', '۷' and '~' are called connectives of Primitive. There are two more to come. You
may not think that 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
True
True
False
False
→
Q
True True
False False
True True
True False
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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 of the conditional sentence (P in the above) is
called the antecedent; the second 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.
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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
True
True
False
False
↔
Q
True True
False False
False True
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 as Primitive. In this section, we are
going to enlarge Primitive in two ways. 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 to understand. Let us review the main features of Primitive.
a) There are two basic sentences: P, Q.
b) These sentences are, in any given circumstance, definitely either true or false.
c) There are five ways in which non-basic sentences can be constructed from basic sentences.
Each of these uses a different connective. They are listed below:
 Conjunction: &
 Disjunction: ۷
 Negation: ~
 Conditionalization: →
 Equivalence: ↔
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Each of these is truth-functional. That is, in each case, the truth value of a non-basic sentence
constructed with the connective is fully determined by the truth values of its basic constituents. The
truth table which shows how this determination is made for each connective can be considered 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 to form non-basic sentences.
Sentential uses precisely the same five devices, but it allows non-basic sentences to contain other
non-basic sentences as parts. Thus, the most complex sentence of Primitive had three elements - for
example:
P&Q
whereas, in Sentential, each of the following are acceptable, non-basic sentences:
(H & Y) ۷ F
(~K ↔ J) → R
~(G → C) & (N ۷ S)
Such complexity brings with it the possibility of certain sorts of misunderstanding, and the
parentheses are there to prevent it. Let me spell this out. The above examples of Sentential use
parentheses to indicate the grouping of basic sentences and connectives. Consider the first of the
sentences, and compare it to the one below:
H & (Y ۷ F)
The basic sentence is the same, and so are the type and order of the connectives. Do they say the
same thing? The answer is 'No', though I can only expect you to guess this for the present. You know
that the meaning of these sentences will be given by the meaning of their basic constituents and by a
table that assigns them truth values on the basis of the truth values of these constituents. If you
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guessed correctly about the difference between these two sentences, then this was because you
suspected that the differences in grouping would make a difference to the assignment of such truth
values. (If you did not guess correctly, don't worry. You will see precisely why these differ in the
next section. Continue reading this section on the understanding that they do differ.) In view of the
difference between the above two sentences, what would you make of the following string of
symbols?
X→T&D
Written without any parentheses, this is an ambiguous string. To say that it is ambiguous is to say
that it could be interpreted in more than one way. Given that Sentential contains only truthfunctional connectives, this has the following consequence: if an ambiguous string were allowed to
count as a sentence, then it would have different truth value assignments depending upon which
interpretation was employed. This is unacceptable to speakers of Sentential. For this reason, they do
not regard this as a sentence of their language; they see it as merely a string of symbols. This raises
the question of how to tell whether a given string is a sentence.
Grammar
What is needed is a method of sorting strings of symbols into two groups: those that speakers of the
language recognize as well-formed, and those that count as ill-formed. If this is done properly, then
we would expect strings like this last one to be placed in the second category. This would not qualify
as a genuine sentence of Sentential.
Here are the rules of syntax for Sentential:
Rule 1: Basic sentences are sentences.
Rule 2: If '~' is used to the left of a sentence, then the negated sentence is a sentence.
Rule 3: If either '&', '۷', '→' or '↔' is used between sentences, then the result is a sentence.
Rule 4: Any string of symbols is a sentence if and only if it is constructed according to Rules 1-3.
(Parentheses should be used to clarify meaning and priority for complex sentences.)
It should be clear that you can apply these rules without looking beyond the most superficial feature
of any string - its form. You do not have to know what each symbol means or what any basic
sentence says. It is for this reason that they are called rules of syntax. You can use the rules either to
construct sentences, or to see whether some given string is a sentence.
Here is an example of a string which may or may not be a grammatical sentence:
~ ~ (H→(G&C))
You can begin anywhere, but it is easiest to start with the smallest sentences and work your way out.
If we begin with the “&”, we know that to be sentence, this “&” must have a sentence on either side.
“G” and “C” are basic sentences, so the “&” part is a well-formed sentence. Next, if we look at the
“→” we are looking for the same thing. “H” is a basic sentence, and we just found that “G&C” is a
well-formed sentence. The parentheses separate these sentences in a way that makes them
unambiguous, so everything within the parentheses is a sentence. So far so good. Now for the
negations. The “~” closest to the parenthesis clearly follows Rule 2, so that is included in our wellformed sentence. Well, though it may sound a little odd (or redundant) the left-most “~” also
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follows clearly from Rule 2, because if the inner negation is part of a well-formed sentence, then a
negation can be added to the left of it, legally.
Scope and Main Connective
The negation in the sentence:
~G → K
affects only the basic sentence G, whereas the negation in:
~(G → K)
affects the whole sentence G→K. I take this to be obvious enough, but we cannot leave matters
there. The idea of a connective 'affecting' certain sentences is too vague. In place of this vague
notion, we can use the rules of grammar to help us define a sharp one: the scope of a connective. The
scope of a connective is the shortest sentence in which it occurs. To see what counts as a sentence
we appeal to the rules of syntax. Using this definition, you can see that the scope of the '~' in the
first sentence above is G while the scope of the ‘~’ in the second is G→K.
The definition of the scope of a connective has this consequence: connectives can occur within the
scope of other connectives. This allows us to speak of one connective in a sentence having wider or
narrower scope than another. One connective has wider scope than another if the first includes the
second within its scope.
“Main Connective”
Also, given some long non-basic sentence, we can speak about its main connective as the connective
with the widest scope, the scope that takes in the whole sentence. For example, the main connective
in:
(B ۷ S) → (L & R)
Is ‘→’ since it has the whole of the sentence as its scope. The idea of the main connective of a
sentence is useful for allowing us to talk about the structure of sentences of Sentential. This above
sentence can be described as having the structure of a conditional sentence in virtue of having '→' as
its main connective.
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