There are two kinds of statements used in logic: simple statements and compound statements. Compound statements are those that contain two or more simple statements. Thus “My dog has fleas” is a simple statement and “My dog has fleas and he barks at the mailman" is a compound statement, consisting of the simple statement "My dog has fleas" and another simple statement, "he barks at the mailman". There are several types of compound statements - this one is a conjunction. Conjunction
In ordinary language, there are several ways to conjoin simple statements into conjunctions. In some cases, the choice is merely among conjunctions - "My dog has fleas but he loves a bath", for instance, doesn't contain the word "and". In such statements as "Jack and Jill went up the hill" are conjunctions, but differ from our first example in that the word "and" does not appear between two simple statements, but nonetheless indicates that two simple statements are conjoined. But in symbolizing any of these compound statements, we must observe certain conventions.
As we have said, statements are either true or false. Another way to say this is that all statements have a truth value. That is, any statement, if true, has the truth value true
and any statement, if false, has the truth value false
. Here we are only introducing the nomenclature truth value
A conjunction is true only if both of its conjuncts
- that is, the simple statements being conjoined - are true. This is merely what we mean when we make a statement that is a conjunction - we are asserting that both (or all) of the simple statements are true, at the same time. So, if even one
of the conjuncts is false, then the conjunction, taken as a whole
, is also false. Conjunctions are called truth-functionally compound
statements for this reason – the truth value of a conjunction is wholly determined by the truth of its component parts.
The reason that symbols are used by logicians is that the correctness of the above paragraph is more easily seen. So, lets call any two statements
“p” and “q”, following convention.
So, given any two statements p and q there are only four possible sets of truth values available. These four sets determine the truth value of their conjunction:
1. p is true and q is true (“p and q” is true)
2. p is true and q is false (“p and q” is false)
3. p is false and q is true (“p and q” is false)
4. p is false and q is false (“p and q” is false)
We’ll use a dot (•) as the symbol for “and”. With this, we’ll now construct a truth table
mirroring the values stated above, which will serve as the definition for that symbol. It is a definition for • because it accounts for the truth value of the conjunction in every possible case.
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p q p • q
1. T T T
2. T F F
3. F T F
4. F F F
Note that the last column provides the truth value for the conjunction taken as a whole.
To illustrate this in our examples, above, we will write
"My dog has fleas and he barks at the mailman" as "My dog has fleas • He barks at the mailman"
"My dog has fleas but he never scratches himself" as "My dog has fleas • My dog loves a bath"
"Jack and Jill went up the hill" as "Jack went up the hill • Jill went up the hill"
If we assign a symbol to each simple statement, and by convention use capital letters to do so,
"My dog has fleas • He barks at the mailman" is written as "F • B"
"My dog has fleas • My dog loves a bath" is written as "F • L"
"Jack went up the hill • Jill went up the hill" is written as "A • I"
The choices of letters used as symbols is rather arbitrary - often the first letter of an important word in the proposition is used - but this is really only a mnemonic device. Negation
Another kind of truth-functionally compound statement is negation
. It is the combination of any statement and its denial. So the negation of “My dog has fleas” can be written “It is not true that my dog has fleas”, “My dog doesn’t have fleas”, “There are no fleas on my dog” etc, but if “My dog has fleas is symbolized as “D” (following again the convention that simple statements are symbolized by upper-case letters), then any of these expressions can be symbolized the same way. We’ll use the tilde (~) as that symbol and we then have ~D. So, following our convention above, for any statement p, its negation is symbolized as ~p.
And we can construct a truth table to define ~ as well.
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Next we consider disjunctions, indicated by the word “or” - also truth-functionally compound (they're all going to be). Here, the simple statements that make up the components of this kind of compound statement are called disjuncts
, or alternatives
. “Or” is a little trickier than “and”, for it has two common meanings – the inclusive and the exclusive. The inclusive, or weak, use of the word “or” is as in “Fleas can get on either my cat or my dog”, meaning that both
of my pets are susceptible to fleas (the meaning is inclusive
of both). But if I say, “My pets never have fleas at the same time: it’s either one or the other” we have the exclusive
, or strong
sense of “or”.
Here’s the difference – the weak sense asserts that at least one
disjunct is true, but the strong sense asserts that at least one is true, but they are not both
true. That at least one disjunct is true is a meaning that is common to both senses - it is what is called the common partial meaning
, and it is this common partial meaning that we will use to define our symbol for disjunction. For this reason, the symbol for "or", or disjunction will account for all
instances of "or" - both the weak and the strong senses. We will use the symbol (v) for disjunction, and define it thusly (please note that the software I am using here doesn't allow for lower-case letters):
Here we see that if at least one disjunct is true, then the disjunction, again taken as a whole, is also true. The only case in which the whole disjunction is false is when both disjuncts are false.
But what of the exclusive sense of “or”? Here, we must use a more complex-looking
(but not more logically complex) expression. For here we mean “either but not both”. We need a little more “punctuation” for this, which may be familiar from mathematics – parentheses.
The statement "I will go to the doctor and I will go to the butcher or I will go to the movies" is ambiguous. It could mean that I am going to go either to both the doctor and the butcher or I will go to neither and will go to the movies instead, or it could mean that I will go to the doctor and then either to the butcher or the movies. Here, the word "either" and its placement in the sentence helps us to determine just what statement I am making. It's not always easy to divine the statement contained in a sentence, but that is a "pre-logical" problem. The presence of and placement of "either" is often, but not always a help. In symbolizing such statements, we need to remove ambiguity, for one formulation is a conjunction and the other a disjunction.
"I will go to the doctor and I will go to the butcher or I will go to the movies" can be rewritten as "I will go to the doctor" • "I will go to the butcher" v "I will go to the movies". But we have no symbol for "either", and this may or may not be a help, anyway.
This statement is just as ambiguous, of course. We will solve this the same way as we would in numeric algebra - with parentheses. We'll assign a capital letter to each of the simple statements we are using here, such that "I will go to the Doctor" is D, "I will go to the Butcher" is B, and "I will go to the Movies is M. We will then have D • B v M. And just like we would in an algebraic expression, we all assign parentheses depending upon what our meaning really is. So we have a choice between (D • B) v M and D • (B v M). And again we will see that one expression is a conjunction and one a disjunction.
So, now that we have introduced parentheses, let's get back to the exclusive, or strong sense or “or”. Some systems do
use a symbol for the strong sense of "or", but we will not, here. We will use symbols that we already have at our disposal, so that we can see how parentheses are used, and so we can see how conjunctions and disjunctions can be combined using them.
So let's again write any conjunction as p v q. If we mean this in the strong sense, we mean "either p v q but not both p and
q". "Either p v q" is symbolized as "p v q" of course. "But" as we have seen, will be symbolized as •. "Both p and q" signifies a conjunction, so we have "p • q". And "not", of course, is ~. To apply this ~, we need to apply it to the conjunction as a whole, so it will be written as ~(p • q).
So our entire expression "p or
q but not (both) p and
q" will be (p v q) • ~(p • q).