Notes on truth-functional paraphrase
EMR 17, Fall 2015
First, make sure that you have read part 1A of Goldfarb’s Deductive Logic, paying special
attention to §8. You should consider what follows to be a kind of augmentation of the
method laid out in that section.
The aim, in producing a logical paraphrase, is to illuminate as much of the logical structure
as possible of the statement that is being expressed by the sentence you are considering. For
now, that means the following:
• We want to identify the simplest statements out of which the target statement (i.e., the
statement for which we are producing a logical paraphrase) is built.
• We want to determine whether the mode of construction (by which the target statement is
built out of these simpler ingredients) is truth-functional.
• If it is not truth-functional, there’s not much we can do at this stage (though wait until later
in the course, when our logical tools will become more powerful).
• If it is truth-functional, we want to make sure that our paraphrase exhibits the right truthfunction.
Let’s illustrate these points, by means of some examples.
Example 1: “Randolph went shopping in the morning, and went to the movies in the
afternoon.”
There are two basic statements at work here:
Randolph went shopping in the morning.
and
Randolph went to the movies in the afternoon.
Now we have to rely on our understanding of the target statement to see whether its truthvalue is determined entirely by the truth-values of these two basic statements. And surely it
is: the target statement is true if both of the basic statements are true, and false otherwise.
We now pick – arbitrarily – the letter p to stand for the first statement, and the letter q for
the second, and arrive at this logical paraphrase:
p.q
Example 2: “Despite the fact that Randolph went shopping in the morning, he still
went to the movies in the afternoon.”
This is a little trickier. We can see an obvious commonality: the very same basic statements
go into the construction. But is the mode of construction truth-functional? My (Ned’s) own
judgment is “yes”: in fact, the statement expressed here has (I think) exactly the same logical
paraphrase as in Example 1. As far as truth is concerned, it is true if both basic statements are
true, and false otherwise. In addition, it insinuates that Randolph’s shopping trip somehow
made it less likely that he would go to the movies; but that claim is merely insinuated, and not
actually asserted. Still, we should be honest, and recognize that we have nothing to draw upon
other than our own shared sense of the ‘insinuated’/‘asserted’ distinction. One way to bring
that shared sense into focus is via a thought-experiment: Suppose you learn that, far from
making a visit to the movies less likely, Randolph went to the movies precisely because he’d
already gone shopping (as a sort of reward for completing his tiresome shopping chores,
say). Would that lead you to conclude that the statement asserted here is false? Or would it
rather lead you to conclude that it is misleading? If the former, you’ll disagree that the
statement in this example has the same logical paraphrase as the statement in Example 1; if
the latter, you’ll agree.
Example 3: “Because Randolph went shopping in the morning, he went to the movies
in the afternoon.”
This one is much clearer. Again, we have the same two basic statements as constituents. But
the way they are combined is not truth-functional. And that is because there is a logically
possible combination of truth-values – namely, true, for both basic statements – that fails to
settle a truth-value for the whole. To see this, imagine two different scenarios: In the first,
Randolph rewards himself for shopping by going to the movies. In the second, Randolph’s
movie-going plans were set long ago, and did not hinge at all on whether he went shopping
(though he did). In each scenario, the two basic statements are both true. But in the first
scenario, our target statement is true; whereas in the second, it is false.
So how do we logically paraphrase this statement? Alas, given the non-truth-functional
character, the best we can do is this
p
That is because our truth-functional tools cannot discern the richer inner structure this
statement contains.
Example 4: “Exactly one of Billy, Suzy, and Ahmed will enroll in logic.”
First, satisfy yourself that the basic statements are these:
Billy will enroll in logic.
Suzy will enroll in logic.
Ahmed will enroll in logic.
Second, convince yourself that any assignment of truth-values to these three statements will,
in fact, unambiguously determine a truth-value for the target statement. More specifically,
the target statement will be true if one of the three basic statements is true, and the other
two false; in all other cases, the target statement will be false.
Third, we need to find a suitable combination of connectives that will yield this truthfunction. There are many. (As you will see soon enough, there are always many.) But
probably the simplest, and most perspicuous, is this:
(p . –q . –r) ∨ (–p . q . –r) ∨ (–p . –q . r)
Example 5: “Provided they have adequate preparation, Suzy and Billy will enroll in advanced
Urdu.”
This is a little tricky, albeit for a different reason than example 2. Start by identifying the
basic statements (with sentence-letter labels added):
p
q
r
s
Suzy has adequate preparation.
Billy has adequate preparation.
Suzy will enroll in advanced Urdu.
Billy will enroll in advanced Urdu.
It seems pretty clear that the truth-value for the target statement is determined by whatever
the truth-values are for these constituents. But you have to be careful, in identifying the
exact way in which it is determined. If you’ve looked at simpler examples, you’ll be primed to
think that a statement of the form
Provided that A, B
should be paraphrased
A⊃B
(“Provided that Billy gets enough sleep, he’ll do well on the quiz” – that gets paraphrased as
just such a conditional.) Try that here, and you get
(p . q) ⊃ (r . s)
That yields a reading where what is asserted is that if both Billy and Suzy have adequate
preparation, then both will enroll. Put another way: all that is ruled out by the statement,
given this paraphrase, is a situation in which both Billy and Suzy have adequate preparation,
and yet at least one fails to enroll. But is that really what is intended? Consider a situation in
which (i) Billy lacks adequate preparation; (ii) Suzy has adequate preparation; (iii) neither
enrolls. According to the paraphrase we are considering, this is a situation in which the target
statement is true. But that seems wrong: it seems that since Suzy has adequate preparation, it
should follow from our statement that she enrolls. In other words, what the statement rules
out is really this: any situation in which Billy has adequate preparation but doesn’t enroll, and
likewise any situation in which Suzy has adequate preparation but doesn’t enroll. So the
correct paraphrase is as follows:
(p ⊃ r) . (q ⊃ s)
There’s an important lesson here. Once you’ve convinced yourself that an interesting truthfunctional paraphrase is possible (i.e., that we’re not dealing with something like Example 3),
don’t just mechanically translate English into symbols, in order to arrive at that paraphrase.
Think hard about exactly how the truth of the whole statement is determined by the truth of
its basic constituents.