Assignment 2

deSwart pp. 56-58
exercises 2, 4, 5

Problem 2

p q if p then q p only if q
1 1 1 1
1 0 0 0
0 1 1 1
0 0 1 1

As we discussed in class, on top of their basic truth-conditional meaning, conditionals often have associated with them a notion of causation. In particular, in conditionals involving propositions that express events, the proposition in the antecedent of the conditional (p in p --> q) is typically understood to introduces a cause, while the consequent clause introduces the resulting effect. This general mapping between cause/effect and antecedent/consequent can be viewed as an instance of Grice's maxim ``Be orderly'': causes precede effects in the real world, therefore we should associate the first clause in a conditional with the cause, and the second with the effect.

What also seems to be the case is that this general tendency to identify the first proposition in a conditional with the cause has been codified into a general rule about the morpheme if: treat whatever proposition is introduced by the morpheme if as a cause. Normally, this works fine, since the if-clause is usually the antecedent of a conditional. The one situation in which a problem arises is in the case of only if constructions, because here the if-clause is the consequent. We thus end up with a potential conflict: if tells us to identify the consequent as the cause, but the usual mapping from causes and effects to conditionals tells us to identify the antecedent as the cause. This conflict is the basis for the anomaly of examples like (ii)b and (iii)b.

Even though the logical relations between the two propositions in these examples remain the same as those in the (a) sentences, the causal relations are reversed. Thus (ii)b seems to be claiming that the `dying state of affairs' is the cause of the `boiling in oil state of affairs'. This does not accord with our knowledge of how such states of affairs typically interact causally, however, so the sentence is strange.

It is worth observing that the form of this explanation relies on the assumption that there general tendency for organizing information so that material that is somehow ``prior'' (causally or temporally) comes first in an utterance. The same effect can be seen in conjunctions. While the truth conditions for and tell us that propositions of the form p & q and q & p should be equivalent, actual sentences such as (1) seem to argue against this view:

(1) a. The Lone Ranger got on his horse and rode into the sunset.
b. The Lone Ranger rode into the sunset and got on his horse.

(1)b sounds quite strange, and certainly not equivalent to (1)a. We can explain this by assuming that in conjunctions, the maxim ``Be orderly'' tells us that the first sentence should be temporally prior to the second (when such issues are relevant --- i.e., in sentences involving states of affairs that can have time values, like events). On this view, the oddness of (1)b is quite similar to the oddness of the (b) sentences in exercise 2: the ``priority'' relations that are conveyed by the ordering of the clauses are the reverse of what we expect, given our knowledge of how the world works.

p q not-q not-p if not-q then not-p
1 1 0 0 1
1 0 1 0 0
0 1 0 1 1
0 0 1 1 1

The truth table shows that the truth conditions for `if not q then not p' are identical to those for `if p then q' and `p only if q'. In other words, all three expressions are logically equivalent.

We can extend our explanation of the oddness of (ii)b and (iii)b to the examples in (iv) and (v) as well, provided we add the extra assumption noted in the text: that only has a meaning that can be roughly paraphrased as `nothing except' (this assumption is very well-motivated, as you can convince yourself by constructing lots of sentences with only). In 2.2, we assumed that the clause marked by the lexical item if---more precisely, the proposition denoted by this clause---introduces a cause. (iv)a and (v)a fit in nicely with this picture, if we broaden our notion of `cause' a bit to include states of affairs that are `necessary requirements of' or `temporally prior to (in some connected way)' other states of affairs. So in (iv)a, `somebody taking my place' is claimed to be a necessary requirement for `my leaving', and in (v)a, `the police tapping your phone' is a cause (or an `indicator', via temporal priority) of `your being in danger'. However, even though the b sentences are logically equivalent to the a sentences, they reverse these causal/priority relations, triggering some kind of anomaly.

What's different about the c sentences, and what makes them natural paraphrases of the a sentences, is that they not only maintain logical equivalence (as shown by the truth table in 2.3), they also make claims that are quite compatible with the causal/priority relations associated with the a sentences. If only means something like `nothing except', then part of the claim made in, for example, (iv)a is that `nothing except somebody taking my place' will cause it to be possible for me to leave---the state of affairs corresponding to the if-clause is the only cause of the state of affairs corresponding to the other clause. If this is true, then we're certainly able to conclude that a failure to achieve a state of affairs in which `you have somebody to take my place' is true would lead to a failure to achieve a state of affairs in which `I leave' is true. This is exactly what (iv)c asserts.

Problem 4

The sentence `It is not the case that Jack is singing or Jill is dancing' is ambiguous depending on whether the negation attaches just to the first clause, or to the larger constituent consisting of the conjoined clauses. These two readings are symbolized in propositional logic in (1) and (2), respectively.

s = jack is singing
d = jill is dancing

1. (not-s v d)
2. not-(s v d)

The truth conditions of these sentences in the various situations can be shown by looking at a truth table that indicates just these situations:

situation s d not-s (not-s v d) (s v d) not-(s v d)
1 (jack is singing and jill is dancing) 1 1 0 1 1 0
2 (jack is not singing and jill is not dancing) 0 0 1 1 0 1
3 (jack is singing and jill is not dancing 1 0 1 0 1 0

As the truth table shows, the reading in (1) (in red) is true in situations 1 and 2, but false in situation 3, while the reading in (2) (in blue) is true in situation 2, but false in situations 1 and 3.

Problem 5

  1. Key: c = it was extremely cold; s = Sally stayed indoors
    Translation: (c & not-s)
  2. Key: j = Jim has been elected; s= Sally has been elected; n = a new era has begun
    Translation 1: ((j v s) & n)
    Translation 2: (j v (s & n))

  3. NOTE: This sentence is ambiguous!

  4. Key: s = Sally bought an umbrella; j = Jim bought an umbrella; d = they will stay dry
    Translation: ((s & not-j) --> d)
  5. Key: s = Sally goes to the party; j = Jim goes to the party
    Translation: s --> not-j
  6. Key: m = Mary goes to the party; b = Bill goes to the party
    Translation: m <--> b
  7. Key: m = you mean it; b = i don't believe you
    Translation: (not-m & (m --> not-b))
  8. It is important to point out here that this is the right interpretation only if (note the only!), as I suggested in class, we take this complex statement to be making the assertion that `you don't mean it' is true. If we want to allow for the possibility that `you don't mean it' is false, just in a situation when `I don't believe you', then we do indeed have to translate and as `v'. (Do the truth table that we did in class to convince yourself.) I will leave it to the class to decide what the facts are here....

  9. Key: g = we are going; r = it is raining
    Translation: g --> not-r
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Chris Kennedy
Last modified: Thu Jan 21 12:22:10 CST