12. Truth Functional Compounds
Our theme is still argument. What is unique about this section is that here we concentrate on the place of compound statements in argument. We are now going to cover the definition, the types, the connectives, the translation, the truth functional aspect, and the procedure with respect to compound statements.
1A. Definition
As usual we will build up the definition of a compound statement. By first reviewing the definition of a statement, we establish a reference point for defining a statement in the new context of truth functional compound statements. The building stages will be: statement, simple statement, and finally the compound statement.
1B. Statement
A statement is an informational expression that is either true or false. Thus, "I hope Noreiga will be captured" is not a statement as far as logic is concerned even though it is a sentence. A hope does not convey information that is either true or false (unless you are thinking that the reality of this hope itself is true or false).
2B. Simple Statement
A simple statement is an information giving expression (T or F) that has one positive component. Some examples are:
John is Tall
Tricia got married
Clint Eastwood is eight feet tall.
Do not confuse simple statements with simple arguments. Simple arguments are sets of statements that have unity on both sides (the premise side and the conclusion side).
3B. Compound Statement
A compound statement is an informational expression that has more than one component as in the following examples:
John is tall and Bill is tall.
Either Joe will go or Tim will go.
If I go to the mall, then I spend money.
Clinton is not eight feet tall.
2A. Types
There are four types of compound statement: conjunction, disjunction, conditional, and negation. Thus, if a statement is conjunctive, disjunctive, conditional, or negative, then it is a compound statement.
Consider how the negative statement is compound. This can be shown by noting first that the expression "Clinton is not eight feet tall" is built up from the simple statement, "Clinton is eight feet tall." When the statement is negated it is as if we insert the phrase "it is not the case that" before the simple statement:
it is not the case that [Clinton is eight feet tall]
Put this way, we can see that the negative statement (Clinton is not eight feet tall) is compound because it has more than one component. It has 1) the simple statement component, and 2) the negative component. To be sure, we do not usually say, "it is not the case that the lights are on" when we wish to tell someone that the lights are not on. But in logic we use such cumbersome grammar for analysis purposes. The "it is not the case" phrase reveals the additional negative component, shows the compound nature of the negative statement, and highlights the fact that compound statements are built up from simple "root" statements.
It might be helpful to think of simple statements as root statements because they hold the truth functional tree in place and everything else grows up from them. All compound statements grow out of simple statements; they are all built up from this foundation.
3A. Connectives
1B. Definition
Connectives are words that make simple statements into compound statements.
2B. Identification
There are four specific words that can be identified as connectives and which correspond to the four types of compound statement. They are: not, and, or ("either...or"), and if ("if...then" or "if and only if...then").
4A. Translation
Logic has its own technical language as we have seen. With compounds, we now enter the domain of symbolic logic, which is the study of arguments containing compound statements. These are studied by means of a rigorous set of symbols. The symbols make up a language. An example from algebra should clarify the point: 124 says 12x12x12x12. Likewise, technical symbols in logic will express simple and compound statements. Therefore, we must translate from English into the technical language of logic.
The areas of translation are the simple statement, connectives, compound statements, punctuation, and the implication indicator. We will now examine how each of these are translated into symbols.
1B. Simple Statements
The translation of simple statements is accomplished through the use of either variables or constants. Variables are lower case letters beginning with p which represent any simple statement. Thus, we will use variables to represent statement forms and in turn argument forms. Constants are upper case letters (A, B, C..) used to represent particular simple statements.
Therefore, constants will be used to represent particular arguments with specific content. The contrast between variables and constants can be illustrated thus:
Variables
John is tall - can be translated: p
Mary is short - can be translated: p
Any simple statement - can be translated: p
Constants
John is tall - can be translated: T
John is tall - can be translated: J
Mary is short - can be translated: S
When specific content is in view: use constants
The decision as to which constant to use depends on what is being said. You choose what is best for you and for clarity. The only hitch is that you must remain consistent. For example, if you have an argument with two premises (1, 2) which have similar constants for the key words then your choice must insure that you can distinguish the statements symbolically. Here are two premises:
1. Tim is tall.
2. Tony is rough.
Statement 1 would be translated with the letter T and statement 2, with the letter R. If we used T (keying on Tim) for 1 and T (keying on Tony) for 2 we would be unable to distinguish 1 from 2 symbolically. The choice is yours and the only rule is common sense.
2B. Connectives
Logical operators translate connectives. By definition logical operators are symbols that represent connectives. Connectives, of course, are words that make simple statements into compound statements. Connectives are words whereas operators are symbols. The following chart reveals the correspondence between connectives and operators:
Connective Logical Operator
not tilde: ~
and dot: .
or (either/or) wedge: v
if…then (sufficient conditional) horseshoe: >
if and only if (biconditional) equal sign: =
3B. Compound Statements
Compound statements are built up from simple statements, or root statements, through the use of constants and logical operators. Some examples will get us started (note the location of the logical operators in relation to the constants):
1) "John will go or Mary will go" becomes: J v M
2) "John will drive and Mary will deliver" becomes: J . M
3) "If Bill leaves, Tim will stay" becomes: B > T [read middle (if), left (B), middle (then), right (T), if B then T]
4) "Clinton is not eight feet tall" becomes: ~ C
4B. Punctuation, statement parts, and the dominant operator
Punctuation is indicated by parentheses (), brackets [], or braces {}. They have a grammatical function and serve to avoid confusion. For example, J . M v B could mean either (J. M) v B or J . (M v B). These symbolic punctuation marks reflect the grammar of the statement.
Statement parts are the expressions, however complex, that are connected by a logical operator. These expressions are specifically identified according to the function they have because of the operator. The parts of a conjunction located on each side of the dot are called "conjuncts." The parts of a disjunction located on each side of the wedge are called "disjuncts." The parts of a conditional are "antecedent" (the "if" part of a conditional) and consequent (the "then" part of a conditional). The main part of a negation is simply the constant.
At this point, we can introduce the notion of the dominant operator. A dominant operator is the operator that is positioned outside a parentheses and that governs or controls everything within it. Thus, in J . (M v B) the dot is the dominant operator and the statement as a whole is a conjunction (and one of its conjuncts is a disjunction). There may be more than one dominant operator (or dominating operator) but one will be the most dominant. Try to keep the statement as a whole in focus (no matter how complex it is) by keying in on the most dominant operator.
Governance with respect to a dominant operator refers to what the compound statement becomes, as a whole, because of the controlling operator. What was once a disjunctive statement, T v Q, becomes a negation: ~(T v Q). Likewise, the disjunction becomes an antecedent when dominated by the horseshoe: (T v Q) > T (now the disjunctive compound has an antecedent function being governed by the horseshoe).
5B. The implication indicator
The implication indicator for the reasoning indicator therefore is two dots: .. (sometimes with a slash: /.. ). The slash indicates a separation of the premises before it from the conclusion following it.
5A. The truth functional aspect
1B. The meaning of "truth functional"
Calling compound statements truth functional means that the whole compound, however formed, is true or false. So the expression, "John is not tall," not only says that he is not tall but it also asserts that it is true that John is not tall. The translation of this assertion into ~J expresses the same two things: 1) it says that John is not tall, and 2) it says that it is true that John is not tall. Therefore, the claim that ~J is true is either true or false. Note the way this builds up from the simple statement:
First: "J"
Then: "it is not the case that J" which is ~J
Finally: we note that the claim, ~J, is either T or F
2B. When is a compound true?
Under what conditions is a compound statement, as a whole, true? We will answer this question by reference to negatives, conjunctives, disjunctives, and conditionals.
1C. Negation
A negative statement is true when what it claims corresponds to the facts. If I say that a logic book does not have 634 pages, you can check the correspondence between my claim and the actual number of pages by referring to the page numbers at the end or by counting each page throughout. The same principle applies to a simple statement, which is said to be true when what it claims corresponds to the facts.
2C. Conjunction
A conjunction is true when both conjuncts are true.
3C. Disjunction
A disjunction is true when at least one disjunct is true.
4C. Conditionals (both regular and biconditionals)
A conditional is always true except when the "if" statement is true and the "then" statement is false. Note the following diagram:
The truth of each part The truth of the unit
p, q p > q
1. T F F
2. T T T
3. F T T
4. F F T
The last three cases, rows two through four, are such that the compound unit, p > q, is true. How this is the case can be seen when we define the conditional by a "can't have" principle. In other words, the regular conditional means that you can't have p without q. Of the four possibilities, only the first violates the "can't have principle" (there you have p and not q). The second row is obvious. In the third and fourth rows there is no violation of the "can't have p without q" principle since neither one of them has p to begin with. The notion of truth for conditional units therefore means conformity to the "can't have" principle.
Finally, when is a biconditional true as a unit? To figure this out we need to examine the notion of "bi" in the biconditional. It says we have two regular conditionals. This means that we can read the biconditional from left to right as one regular conditional and then from right to left as a second. Thus, "if and only if A, then B" becomes "if A then B and if B then A." Thus the biconditional has a conjunction of two regular conditionals as its equivalent:
A = B is equivalent to the conjunction: (A > B) . (B > A)
To determine the truth value of the biconditional as a unit, think of it as a conjunction of two conditionals. Next, determine the truth-value of each conditional unit that makes up the biconditional. If each conditional unit is true then the conjunction is true and the equivalent biconditional unit is true. A short way to determine the truth value of the biconditional is by the following principle: if both sides of a biconditional are true or if both sides are false, then the biconditional unit is true; but if there is any mix of true and false, then the biconditional unit is false.
Is "if and only if a person believes in Christ, then he/she is a Christian" true? It is true as a unit (a whole) given that both of the following are true: "a believer in Christ is a Christian" and "a Christian is a believer in Christ."
What about "if and only if one hears the gospel then one is a Christian"? C > H is a sufficient conditional but H > C is not; so one conjunct is false and thus the whole conjunction is false (and in turn the biconditional is false).
6A. Goal
What do we want to do with these compound statements? We want to determine deductive validity or invalidity with respect to arguments that contain them. We want to see how compound patterns and forms bear on truth and validity. Two things are at stake here: awareness of the language of compounds and logical skill in handling this language.
7A. Procedure and a look at short truth tables
How do we determine DV? We do so by 1) truth tables and by 2) proofs using inference rules and laws.
Truth tables are exhaustive lists that show all the possible truth-values of an argument including each simple statement, each compound as a whole, and the conclusion. They are procedures that demonstrate the validity of the inference rules that we will assume to be correct and which we will put to thoughtful use in doing proofs (what we did with the conditional we can do with an argument).
The basis for this procedure is the DV principle that if the premises are true then it is impossible to have a false conclusion. So if we show all the possibilities and a true premise with false conclusion pattern emerges then the argument is invalid. Two examples will illustrate this fact.
1B. Argument: A > B, ~A /.. ~B (here the comma separates premises)
What we are trying to do is show all the possible scenarios to see if any true premise leads to a false conclusion. If even one occurs, no matter how many possibilities are considered, then the argument is not deductively valid (in virtue of the definition of DV, that a DV argument with true premises must lead to a true conclusion).
How do we set up the truth table? It is made up of columns and rows. For the simplest argument there will be two columns, one for each simple statement, and it will take four rows to exhaust the possibilities. In the first column, standard truth tables place T's in the first two rows and F's in rows 3 and 4:
Column 1
Row 1 T
Row 2 T
Row 3 F
Row 4 F
This same pattern cannot be repeated in Column 2 because then we would not show every possible scenario. Therefore, column 2 will have a T, F, T, F pattern down the rows:
Column 2
Row 1 T
Row 2 F
Row 3 T
Row 4 F
Together we have the beginning pattern to be used with arguments having two simple statements. Again, there are two columns, one for each statement and four rows set up to exhaust all possible combinations.
T T
T F
F T
F F
We might call these the normative or reference columns because it is by reference to them that the truth value of compounds is determined (cf. what we did in 4C above with respect to conditionals). Now let's apply this normative pattern to an example argument, A > B, ~A /.. ~B, to exhaust the T/F possibilities:
premise 1 premise 2 conclusion
1 2 3 4 5 6 7
A > B ~ A ~ B
T T T T
T F T F
F T F T
F F F F
Observe that the columns under the simple statements (A, B) are the normative patterns and they are placed under every occurrence of these statements (columns 1, 5 for A and columns 3, 7 for B). For the moment we have no truth value for the compounds in which they are found. Thus there are no truth values assigned in columns under the logical operators (2, 4, 6). The next graph shows the truth values of each compound row by row in bold type. Consider row 1 . The conditional is true as a unit so T is placed in row 1 column 2. Premise two (the compound ~A) is false because claim A is true. The conclusion (~B) is false because the simple statement B is true.
premise premise two conclusion
1 2 3 4 5 6 7
A > B ~ A ~ B
row 1 T T T F T F T
T F F F T T F
F T T T F F T
F T F T F T F
Now we are ready to ask the question of validity. Is there any premise, premise, conclusion row that has true premises with a false conclusion? Yes, such exists in row 3 where both premises are true but the conclusion is false. What does this tell us? It tells us that the argument A > B, ~A /.. ~B is not deductively valid.
2B. Argument: A v B, ~A / .. B
Initially we have the base line or reference pattern:
premise premise conclusion
A v B, ~A B
T T T T
T F T F
F T F T
F F F F
The filled out pattern looks like this:
premise premise conclusion
A v B, ~A /.. B
T T T F T T
T T F F T F
F T T T F T
F F F T F F
Where do we have both premises true, which row?
What is the conclusion of that row where both premises are true?
What then is the validity status?
There is a lesson to learn from the truth table application of truth functions. Namely, truth tables give another look at the DV principle that if the premises are true then, for the argument to be DV, the conclusion must also be true. Thus, it is impossible to have a false scenario in the possible patterns of a DV argument that has true premises. Therefore, if you accept the premises of an argument and it is DV, then you must accept the conclusion of that argument (however it goes against the grain of your framework of thought). For example, consider this argument: If God exists then evil cannot exist. Evil does exist so God does not exist. If you accept the premises as true and if the argument is DV, then you must accept the conclusion as true. Where you have difficulty with a conclusion, you have two places to go for evaluation: form and content, validity and truth. This argument is valid. Rejection has to show unsoundness in the premises (for example, why should we accept the first premise?). Note the validity on a truth table (where there are true premises assumed there is also a true conclusion; both premises could be true without leading to a false conclusion):
G > ~ E, E ~ G
T F F T T F T
T T T F F F T
F T F T T T F
F T T F F T F
Truth Functional Compounds ws1
A. Be sure you can define all relevant terms in this section like the following:
1. Simple Statement
2. Symbolic Logic
3. Compound Statement
4. Logical operator
5. Constant
6. Connective
7. Statement part
8. Antecedent
B. Truth Values
1. t f "John is not tall" is a compound statement.
2. t f A disjunction can be true even if one disjunct is certainly false.
3. t f If a conditional has a true consequent and a false antecedent, then its unit truth value is true.
4. t f Variables are letters that stand for particular statements.
C. Identify all dominant operators in the following:
Note: there may be more than one per example.
1. ~(A . B) > ~T
2. (T . Z) v (Y . R)
3. ~O v (M > N)
4. [ (A v B) . T ] > (M . N)
5. ~{ K v [ (A. B) > ~U ] }
6. ~(A v B)
D. What is meant by root expressions and how do they function?
E. Translate the following statement into symbolic language with proper symbolic punctuation.
If the President is an honorable man and we can trust the health care reform bill, then white water is an unjust problem if Hilary is not implicated in any wrong doing.
F. Translate into symbols
For 1-4 use: W, the wine is plentiful; B, the band stays; P, the party continues.
1. If the wine runs out or the band leaves, then the party will end.
2. If there is sufficient wine but the band leaves, the party will continue.
3. The band plays on but there is no wine nor is there a party.
4. If there is no wine or no party, then there is no band playing on.
For #5 use: R, there a resurrection of the dead; C, Christ is risen; S, we are yet in our sins.
5. If there be no resurrection of the dead, then Christ is not risen and we are yet in our sins.
6. For this problem use only two letters: D and L.
"if by the transgression of the one, death reigned through the one, much more those who receive the abundance of grace and of the gift of righteousness will reign in life through the One, Jesus Christ" (Rom. 5:17).
7. Translate this paraphrase of Romans 5:17 (Translate using, G, D, L).
The second Adam is much greater than the first Adam as a representative of the human race (enthymeme). If the first Adam certainly brought death to the human family, then it is even more certain that the second Adam brought life if the second Adam is much greater than the first. The first Adam certainly did bring death (context, especially v. 12). So, it is even more certain that the second Adam brought life.
Truth Functional Compounds ws2
The Truth Functional Aspect
Determine whether or not the statements below are true; note that each statement must be viewed as a unit. Be sure to keep the dominant operators in focus; as you do this keep the most dominant operator in clear view. Assume the following truth values of these four simple statements A, B, C, and D:
A is true
B is false
C is false
D is true
What then is the truth value of the compound units below?
1. C . D
2. ~C
3. A v B
4. ~D
5. (B . D) v A
6. (A v C) . D
7. B > D
8. C v ~A
9. (~D v B) > (A . ~C)
10. B = C
11. (C = D) > A
12. ~[ (B = A) > C ] v D
13. ~{ [ (A > C) v D ] > B } v B
Truth Functional Compounds ws3
Short Truth Tables
Determine the validity of the following arguments by means of a truth table.
1. p . q /.. p
2. ~q > ~p, p /.. q
3. p v ~q, ~p /.. q
4. p > q, ~p / .. ~ q