CAS LX 502—Semantics 1
Fall 2009
Assignment 4 (due Thursday, Oct. 8 at the beginning of class)
I. Entailment, Equivalence, and (In)compatibility
For each of the following pairs of propositional logic formulas, determine:
(i) whether the (a)-formula entails the (b)-formula,
(ii) whether the (b)-formula entails the (a)-formula,
(iii) whether the (a)-formula and the (b)-formula are logically equivalent, and
(iv) whether the (a)-formula and the (b)-formula are logically compatible.
Justify your responses by providing the necessary truth tables.
(1)
a. ~p
b. p → q
(2)
a. ~q & p
b. p → q
(3)
a. ~p & ~q
b. ~(p v q)
(4)
a. ~p v q
b. ~p → ~q
II. Entailment and Material Implication
A. In each of the following pairs of propositional logic formulas, the (a)-formula
entails the (b)-formula. Demonstrate this by providing the necessary truth
tables.
(5)
a. ~p & q
b. q
(6)
a. p
b. p v ~q
(7)
a. p
b. q → p
B. Next, construct truth tables for the following propositional logic formulas:
(8)
(~p & q) → q
(9)
p → (p v ~q)
(10)
p →(q → p)
C. Now, compare (5) to (8), (6) to (9), and (7) to (10). What conclusion can you
draw about the relationship between entailment and material implication?
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CAS LX 502—Semantics 1
Fall 2009
III. &, v, and Informativity
In lecture, we saw that the notion of relative informativity amongst sentences
can be defined precisely in terms of entailment:
If a sentence A entails another sentence B, but B does not entail A
(i.e., if A ‘asymmetrically entails’ B), then A is more informative than B.
The notion of relative informativity amongst propositional logic formulas can be
defined analogously:
If a PL formula A entails another PL formula B, but B does not entail A
(i.e., if A ‘asymmetrically entails’ B), then A is more informative than B.
A. Which of the following two propositional logic formulas is more
informative?
(11)
p&q
(12)
pvq
Justify your response by providing the necessary truth tables. In your
discussion, be sure to relate these truth tables to the definition of ‘more
informative than’ given above.
B. Assume the following basic translations of English sentences into
propositional variables:
p =
q =
To complete a Linguistics degree, you must take Syntax I.
To complete a Linguistics degree, you must take Semantics I.
Give the propositional logic formulas that correspond to the following
English sentences:
(13)
To complete a Linguistics degree, you must take Syntax I and Semantics I.
(14)
To complete a Linguistics degree, you must take Syntax I or Semantics I.
C. What do your results from Part A lead you to expect about the relative
informativity of the English sentences in (13) and (14)? Do your intuitions
about (13) and (14) support this expectation? Discuss.
(The following observations might be useful to you when formulating your
discussion: in the context of degree requirements, more information
correlates with fewer options in how to satisfy the degree. In other words, a
list of degree requirements that fills up an entire page will typically leave
you with fewer options in how to satisfy the degree than a list of degree
requirements that is only a few lines long. So, it might be helpful to consider
how many options for completing a Linguistics degree are allowed in (13),
as opposed to (14).)
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CAS LX 502—Semantics 1
Fall 2009
IV. A Puzzle About or and v
(Note: there are two pages to Part IV. Be sure to carefully read through both
pages before formulating your response.)
In class, we have translated the English word or with the propositional logic
connective v (disjunction). Such a translation works well for sentences like (15):
(15)
(Uttered by the coach of BU’s hockey team)
This year, our team will beat Boston College or Northeastern.
This sentence is judged to be true if it turns out that BU’s hockey team beats
Boston College, but not Northeastern. The sentence is also judged to be true if
BU’s hockey team beats Northeastern, but not Boston College. Finally, the
sentence is still judged to be true if BU’s hockey team beats both Boston College
and Northeastern. These judgements about (15) are in accordance with the truth
table for v:
A
T
T
F
F
B AvB
T
T
F
T
T
T
F
F
However, logicians and linguists have long recognized that not all uses of
English or correspond so tidily to logical disjunction. To see this, consider (16):
(16)
(Uttered by a mother to her children)
You can have hamburgers, or you can have pizza.
If this is all that the mother says, her children will infer that they have to choose
between hamburgers and pizza—eating both is not an option for them. In other
words, from the mother’s utterance of (16), her children will conclude (17):
(17)
It is not the case that you can have (both) hamburgers and pizza.
Here are some other examples illustrating the same phenomenon:
(18)
When you buy a new car from Acme Motor Company, you get $2000 back
in cash or a 2% car loan.
Inference: It is not the case that when you buy a new car from Acme
Motor Company, you get (both) $2000 back in cash and a 2% car loan.
(19)
Your dinner comes with wine or beer.
Inference: It is not the case that your dinner comes with (both) wine and
beer.
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CAS LX 502—Semantics 1
Fall 2009
Schematically, in these examples we find an inference from sentences of the form
in (20) to sentences of the form in (21).
(20)
A or B.
(21)
not-(A and B).
Now come your tasks:
A. Use the entailment tests to show that sentences of the form in (20) do not
entail those of the form in (21). You may use the specific examples given in
(16) and (17) when applying the tests.
B. Next, show that (22), our (presumed) logical translation for (20), does not
entail (23), our logical translation for (21). Do this by providing the relevant
truth tables.
(22)
AvB
(23)
~(A & B)
C. Now, state as precisely as you can how an utterance of a sentence of the form
in (20) conversationally implicates a sentence of the form in (21). Again, you
may organize your discussion and step-by-step reasoning around the specific
examples in (16) and (17).
(Hint: Your results from Part III concerning the relative informativity of
formulas like A v B and A & B will be relevant here.)
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