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10 Apr 2010
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Symbolic Logic
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Combining Operations

CHARACTERISTICS OF OPERATIONS | ORDERING MULTIPLE OPERATIONS | COMMON COMBINATIONS | OTHER COMBINATIONS

The issue of complex lines of reasoning brings us to the matter of combining multiple logical operations.  After all, it's rare to find a conclusive argument on any matter of substance that contains just a single logical operation.  To understand how different operations can be used together (as well as how they cannot), we need to become acquainted with some basic factors affecting how logical operations interface and interact.


CHARACTERISTICS OF OPERATIONS | ORDERING MULTIPLE OPERATIONS | COMMON COMBINATIONS | OTHER COMBINATIONS

 

GENERAL CHARACTERISTICS OF LOGICAL OPERATIONS

 Grouping  Just as in mathematics, the order in which multiple operations are performed and evaluated affects the outcome.  In mathematics, there's a default hierarchy of operations, with negations being rated first, then computation of powers and roots, then multiplications and divisions, and finally additions and subtractions.  In logic, only negation enjoys a presumed precedence over other operations.  For any arrangements which don't have a default sequence, or which must be reckoned outside the default sequence, we need some way of identifying which operations take priority.  Like mathematicians, logicians have adopted parentheses as a means of showing which operations must have priority in order to get consistent results.  (They also have alternative methods for doing this, but we'll stick with parentheses.)  Without explicit grouping of operations, the symbolic logic expression p * q  ~r ˅ s is ambiguous gibberish.  It could be interpreted different ways, including:

  • "p is true, and if q is true, then either r is false or s is true"—symbolized: p * (q (~r ˅ s));

  • or "either if both p and q are true then r is false, or else s is true"—((p * q) ~r) ˅ s;

  • or "if both p and q are true, then either r is false or s is true"—(p * q) (~r ˅ s).

In any case, the first operations to be evaluated are those enclosed in the greatest number of parentheses, and the last to be evaluated are those not enclosed at all.  We'll return to this matter in greater detail, but first we need to consider other factors affecting how different types of operations can and cannot interact.  

 Association  An operation is said to be associative if two or more successive instances of the same kind of operation can be grouped and evaluated as one multiple operation.

Conjunction is an associative operation: (p * q) * (r * s) is logically equivalent to p * q * r * s, which is essentially a conjunction with four conjuncts.  It is evaluated simply by adapting the rule we apply to two conjuncts to apply to as many conjuncts as necessary: "A conjunction is true only if every one of its conjuncts is true."  Thus, p * q * r * s is true only if statements p, q, r, and s are all simultaneously true; if any one (or more) of the conjuncts is false, than the conjunction as a whole is false.

Disjunction is also associative: (˅ q) ˅ (˅ s) is logically identical to ˅ q ˅ r ˅ s.  Again, we adapt the rule for a two-element disjunction to apply to any number of disjuncts: "A disjunction is true whenever at least one of its disjuncts is true."  Thus, the disjunction ˅ q ˅ r ˅ s is true whenever statement p is true, even if q, r, and s are all false.  Likewise, this disjunction is true whenever q is true, even if p, r, and s are false; and it's true if either r or s is true, regardless of whether any of the other statements are true or false; the disjunction is false only if all of its disjuncts are false.

Condition and bi-condition are not associative.  They cannot be strung together, but must be evaluated separately.  The symbolic expression  q  r is not capable of logical analysis as it stands.  Its operations must be grouped,
either as (
 q)  r ("If and only if it's the case that if p is true then q must be true, then r must also be true"),
or as
 ( r) ("if p is true, then if and only if q is true then r must be true").
Which grouping we choose depends on precisely what the claim—whatever it may be—represents in plain language.

Negation is ruled to be not associative, because otherwise it could be misinterpreted and produce unpredictable results.  Negation can be applied only to individual statements, or to operations enclosed in parentheses.  For example, ~(p * q) means "it's not the case that p is true and q is true."  In other words, according to this claim p and q can both be false, or either p or q or can be true, but p and q can't both be true at the same time.
Obviously, ~(p * q) means something quite different from ~p * ~q, which states clearly that neither p nor q is true, that both p and q are false, with no wiggle room.
There's also the matter of double negation.  Although it's considered a no-no in Standard English (e.g., "We aren't never supposed to do it"), it's permissible to a degree in logic.  The effect is that one negation cancels out the other: ~~p (NOT NOT p) is to say "it's false that p is false."  In other words, "p is true."  It's possible to encounter double negation in logic; but when we do, it's customary to cancel out any NOTs that occur in pairs, so as not to be needlessly confusing.  In other words, if a statement is negated an even number of times, then the statement is true; if it's negated an odd number of times, then it's false, because the odd NOT isn't canceled: ~~p is the same as p, but ~~~p is the same as ~p.

 Commutation  An operation is said to be commutative if its two statements can be interchanged without affecting the logical meaning of the operation.

The operations that are commutative are:
conjunction: p * q is logically equivalent to q * p;
disjunction:  p
˅ q is logically equivalent to q ˅ p;
bi-condition:  p
q is logically equivalent to q p.

The condition operation is not commutative, because the relationship of antecedent to consequent is not the same as the relationship of consequent to antecedent: p q is not logically equivalent to q p (though, as we've seen, p  q is equivalent to ~q  ~p, which is hardly the same thing).

And negation also is not commutative, simply because it's unary.  Because negation applies to only one element at a time, there's nothing to interchange.

CHARACTERISTICS OF OPERATIONS | ORDERING MULTIPLE OPERATIONS | COMMON COMBINATIONS | OTHER COMBINATIONS

 

ORDERING MULTIPLE OPERATIONS  

Whenever we combine operations, we need some way to tell in what sequence the various operations must be performed.  In plain language, the sequence and content of the ideas usually provides enough of a clue to get it right.  But sometimes the chronological sequence of operations in plain language doesn't indicate their logical priority.  And when we switch to symbolic logic, we lose any clues that might have been provided by the content.  As we've already mentioned, grouping statements and operations with parentheses is an effective way to make the proper sequence explicit.

Suppose we plan to visit a friend in another town, with the aim of doing a few fun things together.  Being the host, she proposes the following: "We'll have breakfast and play tennis or go swimming."  There are two ways we might interpret this.  On the one hand, it seems we'll either have breakfast and play tennis, or else we'll go swimming.  On the other, we'll have breakfast, and then we'll either play tennis or go swimming.  The meaning of the claim depends on the order in which its AND and OR operations are grouped.  Now, we could doubtless assess the possibilities of this rather simple claim in our heads.  But for a more complex problem involving several statements and a variety of operations, we might not be able to get an accurate assessment of the plausible outcomes without some pencil-and-paper work, such as a truth table.  So, to demonstrate how truth-table analysis works using this easy example, we begin by breaking the proposal into its three individual statements, symbolizing each with a letter of the alphabet:

  • b = "we have breakfast"

  • t = "we play tennis"

  • s = "we go swimming"

Now, if both of the logical operations in the claim were conjunctions (AND), it wouldn't matter in which order we grouped them.  Conjunctions are commutative and associative, so they can be arranged in any order or even be analyzed as a single unit.  It doesn't matter, because we'd be doing all three in any case.

b * (t * s)

 

"We have breakfast, and we play tennis and we go swimming"

 

...is equivalent to...

(b * t) * s

 

"We have breakfast and we play tennis, and we go swimming"

 

...is equivalent to...

b * t * s

 

"We have breakfast and we play tennis and we have lunch."

 

 

Likewise, if both of the operations were disjunctions (OR), the order wouldn't matter.  Since they are also both commutative and associative, they can be arranged in any order or be considered as a unit.  It doesn't matter, because each action is an option independent of the others.

b ˅ (t ˅ s)

 

"We have breakfast, or we play tennis or we go swimming"

 

...is equivalent to...

(b ˅ t) ˅ s

 

"We have breakfast or we play tennis, or we go swimming"

 

...is equivalent to...

b ˅ t ˅ s

 

"We have breakfast or we play tennis or we go swimming."

 

 

But in our friend's actual claim, "We'll have breakfast and play tennis or go swimming," one operation is a conjunction and the other a disjunction.  Conjunctions and disjunctions are commutative and associative among themselves, but not with each other.  They must be taken in a certain order; and for the meaning to be clear, the order must be explicit.

b * (t ˅ s)

 

"We have breakfast, and we (either) play tennis or we go swimming"

 

...is clear but not equivalent to...

(b * t˅ s

 

"We (either) have breakfast and we play tennis, or we go swimming"

 

...which is also clear but not equivalent to...

b * t ˅ s

 

"We have breakfast and we play tennis or we go swimming"

 

...which is ambiguous—not clear at all.

Now we can set up a truth table to plot the plausible interpretations of the possible groupings.  Since we have three statements, we must provide eight (23, or 2 × 2 × 2) rows to accommodate all possible combinations of true and false values for each.

 

b

t

s

 

(b * t) ˅ s

b * (t ˅ s)

1

T

T

T

 

 

 

2

T

T

F

 

 

 

3

T

F

T

 

 

 

4

T

F

F

 

 

 

5

F

T

T

 

 

 

6

F

T

F

 

 

 

7

F

F

T

 

 

 

8

F

F

F

 

 

 

Bear in mind that our objective here isn't to memorize particular outcomes.  We just want to acquire an understanding of the importance of the sequence in reasoning, and in the process to get a hands-on feel for (1) manipulating ideas expressed abstractly as symbols, and (2) the truth-table method of evaluating the consistency of relatively simple lines of thought.

Now let's consider what happens to the two interpretations that contain both AND and OR operators, depending on how we group them.  First, we'll consider the situation when the AND operation is isolated from the rest of the statement, so let's temporarily insert a column for the * t operation only.  Considering the AND operation by itself, the conjunction can be true only if both conjuncts are true.  Statements b and t are both assumed true only on lines 1 and 2, so we can mark the * t conjunction as true on those lines.  On all other lines, either one or both of statements b and t are assumed false, so the conjunct must be marked false on those lines.

 

b

t

s

 

b * t

 

(b * t) ˅ s

b * (t ˅ s)

1

T

T

T

 

T

 

 

 

2

T

T

F

 

T

 

 

 

3

T

F

T

 

F

 

 

 

4

T

F

F

 

F

 

 

 

5

F

T

T

 

F

 

 

 

6

F

T

F

 

F

 

 

 

7

F

F

T

 

F

 

 

 

8

F

F

F

 

F

 

 

 

Having evaluated the AND relationship as a unit, we can now consider whether the OR portion of the claim is true or false, using statement s as one disjunct and the result of the AND relationship as the other.  A disjunction is true whenever any of its disjuncts is true.  So, for any line in which s is assumed true (lines 1, 3, 5, and 7), we can mark T for the truth value of the (b * t) ˅ s claim as a whole.  Likewise, for any line in which we've found * t true (lines 1 and 2), we can mark T for the truth value of the whole claim. 

 

b

t

s

 

b * t

 

(b * t) ˅ s

b * (t ˅ s)

1

T

T

T

 

T

 

T

 

2

T

T

F

 

T

 

T

 

3

T

F

T

 

F

 

T

 

4

T

F

F

 

F

 

 

 

5

F

T

T

 

F

 

T

 

6

F

T

F

 

F

 

 

 

7

F

F

T

 

F

 

T

 

8

F

F

F

 

F

 

 

 

For all remaining lines, in which both statement s and conjunction * t are false (lines 4, 6, and 8), the truth value of the (b * t) ˅ s claim is false, so we mark it F.

 

b

t

s

 

b * t

 

(b * t) ˅ s

b * (t ˅ s)

1

T

T

T

 

T

 

T

 

2

T

T

F

 

T

 

T

 

3

T

F

T

 

F

 

T

 

4

T

F

F

 

F

 

F

 

5

F

T

T

 

F

 

T

 

6

F

T

F

 

F

 

F

 

7

F

F

T

 

F

 

T

 

8

F

F

F

 

F

 

F

 

Now let's do the same for the other interpretation of the statement, in which the OR relationship t ˅ s is isolated and considered as a unit, while statement b is independent.  Again, we'll evaluate the isolated unit—in this case, the disjunction—first, and place its truth values in a temporary t ˅ s column.  Since it's a disjunction, we can evaluate it as true whenever at least one disjunct (t or s) is true.  Statement t is assumed true in lines 1, 2, 5, and 6, and statement s is true in lines 1, 3, 5, and 7.  So on these lines we mark T for the truth value of the ˅ s disjunction.  On lines 4 and 8, both statement t and statement s are deemed false, so the disjunction is false.

 

b

t

s

 

t ˅ s

 

(b * t) ˅ s

b * (t ˅ s)

1

T

T

T

 

T

 

T

 

2

T

T

F

 

T

 

T

 

3

T

F

T

 

T

 

T

 

4

T

F

F

 

F

 

F

 

5

F

T

T

 

T

 

T

 

6

F

T

F

 

T

 

F

 

7

F

F

T

 

T

 

T

 

8

F

F

F

 

F

 

F

 

With the t ˅ s disjunction evaluated, we can now consider its truth value as a unit as one of the conjuncts of the AND relationship in the claim, with statement b as the other conjunct.  A conjunction, we recall, is true only if every one of its conjuncts is true.  So we scan the b and t ˅ s columns of each line for the truth values of these two conjuncts.  If we find that both b and t ˅ s are true (as on lines 1, 2, and 3), then we can mark a truth value of T for the claim on that line in the  b*(t ˅ s) column.  But on any line on which we find either that b is false (lines 5 through 8) or that t ˅ s is false (lines 4 and 8), the claim as a whole is false.

 

b

t

s

 

t ˅ s

 

(b * t) ˅ s

b * (t ˅ s)

1

T

T

T

 

T

 

T

T

2

T

T

F

 

T

 

T

T

3

T

F

T

 

T

 

T

T

4

T

F

F

 

F

 

F

F

5

F

T

T

 

T

 

T

F

6

F

T

F

 

T

 

F

F

7

F

F

T

 

T

 

T

F

8

F

F

F

 

F

 

F

F

Remember, the point of this exercise is not to memorize specific patterns.  That would be impossibly difficult, for the number of all possible combinations of unlimited logical operations is beyond counting.  Rather, we simply want to learn whether the sequence in which multiple operations are handled makes a difference.  As we can see in the direct comparison below, it does indeed matter.  With this particular sample of two different interpretations of the same collection of AND and OR statements, there are different results in two (lines 5 and 7) out of the eight instances possible among three variables.

 

b

t

s

 

(b * t) ˅ s

b * (t ˅ s)

1

T

T

T

 

T

T

2

T

T

F

 

T

T

3

T

F

T

 

T

T

4

T

F

F

 

F

F

5

F

T

T

 

T

F

6

F

T

F

 

F

F

7

F

F

T

 

T

F

8

F

F

F

 

F

F

Even just one difference is enough to demonstrate the need for our attention to the matter, because it shows that, without proper grouping, the claim is ambiguous.  If (as on line 5) we do not have breakfast and we play tennis and we go swimming, we get different truth values depending on which grouping pattern we choose, true in one case and false in the other.  Or if (as on line 7) we do not have breakfast and we do not play tennis, and we do go swimming, we get conflicting truth values here as well.

Now let's translate our symbols back into plain language, and see what our analysis tells us.  Both interpretations evaluate as true in three instances:

  • line 1: we have breakfast, we play tennis, we go swimming;

  • line 2: we have breakfast, we play tennis, we don't go swimming;

  • line 3: we have breakfast, we don't play tennis, we go swimming.

Both interpretations evaluate as false in these three instances:

  • line 4: we have breakfast, we don't play tennis, we don't go swimming;

  • line 6: we don't have breakfast, we play tennis, we don't go swimming;

  • line 8: we don't have breakfast, we don't play tennis, we don't go swimming.

Conflicts between the two interpretations arise in lines 5 and 7.  To see why, we'll insert plain-language clarifiers in place of parentheses, to indicate differences in grouping that lead to differences in interpretation, with true and false premises highlighted in color for added clarity.

  • line 5: we don't have breakfast, we play tennis, we go swimming
    Under these conditions...

    • (b * t) ˅ s:

      • "Either we have breakfast and also we play tennis, or else we go swimming."
        Here the conjunction "we have breakfast AND we play tennis," being grouped within parentheses as a unit (b * t), is evaluated first.  The conjunction contains at least one false conjunct, "we have breakfast," so the whole conjunction evaluates as false:

      • "Either we have breakfast AND also we play tennis, or else we go swimming."
        The disjunction is evaluated last (since it's outside the parentheses), with the already evaluated false conjunction as one of its disjuncts and the true premise "we go swimming" as the other disjunct.  Since the independent disjunct is true, the disjunction itself, and hence the entire claim, is true.

      • "Either we have breakfast and also we play tennis, OR else we go swimming."
         

    • b * (t ˅ s):

      • "We have breakfast, and either we play tennis or we go swimming."
        In this interpretation, it's the disjunction "we play tennis or we go swimming" (t
        ˅ s) that's in parentheses, so the disjunction is evaluated first.  Because at least one disjunct "we play tennis" or "we go swimming" is true, the disjunction is true:

      • "We have breakfast, and either we play tennis OR we go swimming."
        The conjunction is evaluated last; the one false conjunct "we have breakfast" renders the conjunction, and thus the entire claim, false.

      • b * (t ˅ s): "We have breakfast, AND either we play tennis or we go swimming."
         

  • line 7: we don't have breakfast, we don't play tennis, we go swimming
    Under these conditions...

    • (b * t) ˅ s:

      • "Either we have breakfast and also we play tennis, or else we go swimming."
        Here the conjunction is grouped as a unit and so is evaluated first.  The conjunct is false because at least one of the conjuncts "we have breakfast" or "we play tennis" is false.

      • "Either we have breakfast AND also we play tennis, or else we go swimming."
        The disjunction is evaluated last, and with one true disjunct (we go swimming) confirms both the disjunction and the claim.

      • "Either we have breakfast and also we play tennis, OR else we go swimming."
         

    • b * (t ˅ s):

      • "We have breakfast, and either we play tennis or we go swimming."
        This time the disjunction is grouped as a unit, and so is evaluated first.  At least one of the disjuncts "we go swimming" is true, so the disjunction is true.

      • "We have breakfast, and either we play tennis OR we go swimming."  Because one of the conjuncts (we have breakfast) is false, the conjunction itself, and hence the claim as a whole, are false.

      • "We have breakfast, and either we play tennis or we go swimming."

Note that it isn't the relative numbers of true and false premises that determines the outcome.  Truth isn't a "majority rules" game; it is the state of being in accord with what is real.  Truth is investigated, not by a vote, but by the critical examination of facts and the logical relationships among them.  Sometimes it turns out that a majority of true premises results in a true claim, or a majority of false premises in a false claim.  But as we've seen here, sometimes this is not the case, and one term can override all the others.  Only logic can explain how and why, and only a clear understanding of how premises are ordered and grouped can produce a logically reliable result.

CHARACTERISTICS OF OPERATIONS | ORDERING MULTIPLE OPERATIONS | COMMON COMBINATIONS | OTHER COMBINATIONS

 

COMMON COMBINATIONS  

There are many useful combinations of logical operations.  Some are so handy and are used so often that they've acquired standard names.  (We needn't commit the names to permanent memory, but it's very much to our advantage to understand the concepts.)


 Exclusive Disjunction (XOR)  As we noted when logical operations were being introduced, the logical OR operation is always considered in the inclusive sense of the word—i.e., "either one option or the other, or both."  The exclusive sense of OR, "either one or the other, but not both," has no symbol of its own, despite that it's fairly common in non-symbolic reasoning.  However, it can be expressed with various combinations of the five available operators.  Perhaps among the first that spring to mind is (p * ~q)
 ˅ (q * ~p), "either p and not q, or else q and not p."  The possibility of an "or both" option in this instance is ruled out by the fact that the two disjuncts, (p * ~q) and (q * ~p), are mutually contradictory.  However, there's another version that's both more concise and less subject to faulty manipulation: p ~q.

Recall from our truth-table analysis that a bi-condition evaluates as true whenever the truth values of both of its component statements are the same, either both true or both false.  Thus, p ~q evaluates as true if both p and ~q are true (i.e., p is true and q is false), or if both p and ~q are false (i.e., p is false and q is true), and evaluates as false whenever p and q are either both true or both false.  Moreover, the behavior of a bi-condition with a negation is such that it functions exactly the same whether we express it...

p ~q     ...or...    ~p  q     ...or...     ~(p  q).

This makes the bi-condition with negation a versatile, concise, and functionally exact expression of "either p or q, but not both."


 Modus Ponens  Consider the IF...THEN condition: p 
 q.  By itself, it doesn't tell us anything about the actual truth values of p or q.  But suppose we add an assumption that p is in fact true, expressed simply as: p.  Now this tells us something useful about → q, specifically this: IF p is true, THEN q must be true; AND indeed p is true.  Since p is actually true, q must be true as well!  We can express this whole logical relationship symbolically as a group of three statements:

1. 

 

Assumption:

 

pq

 

(if p is true, then q must be true)

2. 

 

Assumption:

 

p

 

(p is true)

3. 

 

Conclusion, from 1 & 2:

 

q

 

(so, q is true)

We can also combine statements, taking one pair at a time and grouping the operations as we go.  Combine the line-1 condition p  q with the line-2 assumption p, setting the condition off with parentheses and using conjunction to indicate that both statements are assumed true:

1. 

 

Assumption:

 

(pq) * p

 

(if p is true, then q must be true; and p is true)

2. 

 

Conclusion, from 1:

 

q

 

(so, q is true)

If we like, we can also graft the conclusion onto the statement of assumptions.  When we do this, the whole "if...and...so" construction takes the form of another condition, with the existing assumptions bundled together as the antecedent, and the conclusion as the consequent:

((pq) * p) → q

 

(if it's true, both that if p is true then q must be true, and also that p is indeed true, then q is true)

Symbolically, it makes perfect sense, and (with a little practice) can be comprehended in little more than the blink of an eye (versus maybe a dozen blinks for a novice).  But the plain-language interpretation is unwieldy, given the imprecision of language in arranging multiple thoughts into clearly defined yet interconnected groups.  That's why logicians have cooked up this method of precisely expressing and analyzing reasoning in the abstract.  Not only is it easier on both the eyes and the brain, but it also does away with writer's cramp!  But we digress...

The matter of true significance from this little exercise is that here we have a very useful and widely used logical combination of a condition and the confirmation of its antecedent...
(pq) * p
...to derive the truth of its consequent...
\q

Or to illustrate the abstraction with a vaguely familiar concrete example:
"If Paisley goes to prison, then Quigley can enjoy quiet; and Paisley goes to prison.  So, Quigley can enjoy quiet."

Or: "If Grandma picks apples, then she'll bake an apple pie; and Grandma is picking apples.  So, she'll bake an apple pie."

Or substitute any similarly related statements for p and q, and the results should be consistent.


 Modus Tollens  Now recall that, in a conditional relationship, we can also use a false consequent to draw an inference about an antecedent with an unknown truth value.  Specifically, for any condition where if p is true then q must be true, it follows (by a logical inference known as contraposition) that if q is false, then p cannot be true—and therefore p must be false.  Or, to put it symbolically, p
q (IF p THEN q) is logically equivalent to ~q ~p (IF NOT q, THEN NOT p).  So we can combine the conditional relationship with other statements thus...

1. 

 

Assumption:

 

pq

 

(if p is true, then q must be true)

2.

 

Contraposition, from 1:

 

~q → ~p

 

(if q is false, then p must be false)

3. 

 

Assumption:

 

~q

 

(q is false)

4. 

 

Conclusion, from 2 & 3:

 

~p

 

(so, p is false)

Again, we can combine the assumptions into a single statement (and we can omit the contraposition step, since, as we noted in the IF...THEN truth-table exercise, it's an inherent implication of a condition, included here only for clarification):

1. 

 

Assumption:

 

(pq) * ~q

 

(if p is true, then q must be true; and q is false)

2. 

 

Conclusion, from 1:

 

~p

 

(so, p is false)

This gives us modus tollens, the opposite side of the modus ponens relationship.

For both modus ponens and modus tollens, in each case we use two logical statements taken together to derive a conclusion.  We do this by integrating the logical implications of each statement into an overall conclusion not implied by either of the individual statements taken alone.  The condition states only a hypothetical relationship, making no claim about the actual truth values of its component statements.  The other assumption establishes either that the condition's antecedent is true (modus ponens) or that its consequent is false (modus tollens).  Linking these two pieces of information by conjunction, we deduce the truth value of the condition's other statement.

CAUTION!  These are the only two valid implications of an IF...THEN condition.  Don't make the all too common mistake of getting them backward!  As truth tables of the basic operations clearly demonstrated, it's logically impossible to infer the truth value of a consequent from a false antecedent, or the truth value of an antecedent from a true consequent.  Attempting to do so results in fallacies of the "formal error" category (denying the antecedent and affirming the consequent, respectively), which are responsible for turning countless logical lines of reasoning into illogical nonsense.


 Hypothetical Syllogism  We can use a chain of related IF...THEN operations to determine the relationship between the first and last links of the chain.

1. 

 

Assumption:

 

pq

 

(if p is true, then q must be true)

2.

 

Assumption:

 

q r

 

(if q is true, then r must be true)

3. 

 

Conclusion, from 1 & 2:

 

p r

 

(so, if p is true, then r must be true)

If we know that whenever one thing is true a second must be true, and that whenever the second thing is true a third must also be true, then we can logically conclude that whenever the first thing is true the third will also be true.  This doesn't nullify the intermediate link in the chain, but if the linkage is well established and is understood to work every time without fail, we can skip enumerating intermediate steps for convenience's sake.  In some situations, hypothetical syllogism can bridge several intermediate steps, provided we're well convinced that each step must indeed always follow from the ones before it.

Consider this multi-step example.  A college freshman might speculate thus about his prospects: "If I earn a liberal arts degree, then chances are I'll become broadly knowledgeable, well respected, and sought after.  If I become broadly knowledgeable, well respected, and sought after, then I'll have my choice of lucrative and satisfying occupations.  If I have a lucrative and satisfying occupation, then I should be able to live happily."  So, assuming the freshman's assumptions are realistic and no unexpected calamity overwhelms his plans, he's justified in abbreviating his line of reasoning to, "If I earn a liberal arts degree, then chances are I should be able to live happily."


 Disjunctive Syllogism  The falseness of one disjunct of a disjunction implies the truth of the other disjunct.

1. 

 

Assumption:

 

p ˅ q

 

(either p is true or q is true)

2.

 

Assumption:

 

~p

 

(p is false)

3. 

 

Conclusion, from 1 & 2:

 

q

 

(so, q must be true)

The general claim of a disjunction is that at least one of its disjuncts must be true.  So, if one of the disjuncts is false, then the other disjunct must be true.  If, as in this example, p is false, then q must be true.  Or if we know that q is false, then p must be true.  If it should ever happen that both disjuncts are false, then the disjunction itself is exposed as a false claim.

Here's a familiar (if less than air-tight) example: an ultimatum handed down from each generation to the next.  "You must either go to school or find a job."  If you don't go to school, then you must find a job; if you don't find a job, then you must go to school.  Doing both is also an acceptable option, but doing neither is not.


 Negation of Operations  The negation of an individual statement is easily enough understood; however, the negation of an operation or a combination of operations bears a bit of clarification.

When we negate an operation, the conjunction p * q for example, we first enclose the operation in parentheses, (p * q), and then insert the negation sign before the first parenthesis: ~(p * q).

Now exactly what does this signify?  That p is false and q is false?  No, not necessarily.  The negation of an operation does not imply the negation of every statement in the operation, but just the negation of the operational relationship as a whole.  Recall that the implication of the un-negated operation p * q is that p is true and q is true.  Any false truth value for either p or q renders the p * q conjunction operation as a whole false.  So ~(p * q), the negation of p * q, could have any of three implications: (1) p is false and q is false (~p * ~q); (2) p is true and q is false (p * ~q); or (3) p is false and q is true (~p * q).  The expression ~(p * q) does not tell us which of these conditions is actually the case, but only that it's not the case that both p and q are true.  Thus, while ~p * ~q is one possible implication of ~(p * q), it's not the only possibility; so we can't logically jump to that conclusion.

The situation with a conditional relationship is similar; ~(p → q) might mean ~p → ~q, but it could just as well mean ~p → q or p → ~q.  The only thing it signifies for certain is that the relationship IF p THEN q is not true, so we'd be unwise to draw any other conclusion without further information.

With a bi-conditional relationship, the situation gets a shade more predictable.  Recall that a bi-condition has the interesting characteristic of evaluating as true whenever both of its conditions have the same truth value—either both true or both false.  Or to put it symbolically, p ↔ q implies either p * q or ~p * ~q.  So, as we might suppose, the negation of a bi-condition implies that its two component statements have opposite truth values—one true and the other false; that is, ~(p ↔ q) implies either ~p * q or p * ~q.

We've saved the negation of a disjunction for last, because it's the only sure bet in the lot.  We express the negation of the disjunction p ˅ q using the same procedure for negating other operations: ~(p ˅ q).  Recall that the condition for a disjunction to evaluate as true is for at least one of its disjuncts to be true.  Another way of expressing this is that the only condition under which a disjunction evaluates as false is if both of its disjuncts are false.  Thus, ~(p ˅ q) implies both that p is false and that q is false, or simply ~p * ~q, with no other alternatives.

 CHARACTERISTICS OF OPERATIONS | ORDERING MULTIPLE OPERATIONS | COMMON COMBINATIONS | OTHER COMBINATIONS

 

OTHER COMBINATIONS

There are other standard ways to combine logical operations, of course, ranging from intuitively simple to mind-bogglingly complex, from general-purpose to highly specialized.  We'll briefly encounter a few of these in the next lesson.  However, it isn't necessary to commit them to memory, since some are so seldom used they'd likely be forgotten long before we came across any real-life applications for them.

But by far the greatest number of combinations we make up ourselves, as we apply reasoning to specific situations.
"We can have a picnic if the rain stops, and the forecast is that the rain should end before noon; so we should be able enjoy an outing afterward."
"If the sink drain is backed up but the shower drain runs freely, then the blockage must be above the junction between the sink and shower drain lines." 
"If we find that the square of the length of the longest side of a triangle doesn't equal the sum of the squares of the lengths of the other two sides, then either the triangle isn't a right triangle or there's something wrong with our measurements or calculations."

We make up such constructions "on the fly" to custom-fit situations as we encounter them.  Being aware of how various operations can and cannot work together helps us to make a better job of it, to make our points succinctly, to avoid generating gibberish too often, and to resist being duped by the rhetoric and bafflegab of advertisers, ideologues, politicians, and clueless innocents.

 

CHARACTERISTICS OF OPERATIONS | ORDERING MULTIPLE OPERATIONS | COMMON COMBINATIONS | OTHER COMBINATIONS


TERMS

In this lesson, we've introduced and analyzed some common and useful ways to combine various logical operations.  In the next lesson, we'll examine a few more handy combinations of logical operations, expanding our concepts of how some combinations logically imply others, and how some combinations are actually equivalent to each other.  In this way, we'll acquire an appreciation of the power, flexibility, and utility of logic.

Next: Implication and Equivalence

 

 
 

 

 

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