The following truth table shows that (p v q) is logically equivalent to (a)-(c) and (p + q) is logically equivalent to (d): they have the same truth conditions, respectively.

p | q | p & q | p v q | p + q | (p v q) v (p & q) | (p + q) v (p & q) | (p + q) + (p & q) | (p v q) + (p & q) | |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | |

1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | |

0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

This result puts (yet) another nail in the coffin of the
`ambiguous *or*' hypothesis. The truth table shows that
if *or* were ambiguous between an inclusive and an exclusive
interpretation, then a sentence like `Bill smokes or drinks, or
both' should be two-ways ambiguous: it should have both an
inclusive and an exclusive interpretation. That is, even
though we could assign it four different representations in
propositional logic (the ones in the last four columns in
the truth table), these only represent *two* different
interpretations, because three are logically equivalent.
The problem is that *this sentence is not ambiguous!*
`Bill smokes or drinks, or both' has only an inclusive reading---
the exclusive intepretation is completely unavailable. Thus
the `ambiguous *or*' hypothesis makes exactly the wrong
prediction.

- Kim is taller than either Lee or Chris is.

**l**= Kim is taller than Lee

**c**= Kim is taller than Chris

**l & c**

Arguably, this sentence can also have a true "or" reading ("or" = v), but unlike e.g. (2), "or" can quite clearly also mean &.

- Either Kim or Lee is taller than Chris is.

**k**= Kim is taller than Chris is

**l**= Lee is taller than Chris is

**k v l** - Kim is exactly as tall as either Lee or Chris is.

l = Kim is exactly as tall as Lee

c = Kim is exactly as tall as Chris

**l v c**

While this sentence seems at first glance to have an "and-like" interpretation, it just doesn't seem to be equivalent to "Kim is exactly as tall as Lee and Kim is exactly as tall as Chris". If we eliminate the "exactly", however, it's quite clear that "or" can mean "and":

*Kim is as tall as either Lee or Chris is*. - Few students smoke or drink.

**s**= few students smoke

**d**= few students drink

**s & d** - Few students or professors smoke.

**s**= few students smoke

**p**= few professors smoke

**s & p** - Most students from Ohio or Indiana smoke.

**o**= most students from Ohio smoke

**i**= most students from Indiana smoke

**o & i** - Every student from Ohio or Indiana smokes.

**o**= every student from Ohio smokes

**i**= every student from Indiana smokes

**o & i** - Every student smokes or drinks.

**s**= every student smokes

**d**= every student drinks

**s v d** - Students rarely smoke or drink.

**s**= students rarely smoke

**d**= students rarely drink

**s & d** - Students always smoke or drink.

**s**= students always smoke

**d**= students always drink

**???**While

**s & d**is clearly wrong here, it's just as clear that**s v d**doesn't seem to capture the truth conditions of this sentence: it doesn't say that students always smoke or students always drink, rather it's saying something like "it's always the case that students are enganged in smoking or they're engaged in drinking". (We could make a similar case about (8): the propoisitional expression above is defective in exactly the same way, and the actual interpretation of the sentence is something like "all of the students are such that they engage in smoking or they engage in drinking".)

The descriptive generalizations that we can draw from
these sentences center around the interpretation of "or".
In some contexts --- in the "than-clause" of a comparative,
in the subject or VP of sentences with *few*, in
the subject (but not the VP) of sentences with *every*
in the VP of sentences with *rarely*, but not *always* ---
"or" seems to mean "and", as far as the truth conditions
of the entire expression go.

Unfortunately, propositional logic doesn't provide a particularly useful tool for figuring out why this should be so. The important conclusion to draw from these facts is that different words or constructions and different syntactic environments can give rise to (what appear to be) different interpretations of the same expression. In order to figure out why this is so, we need to be able to talk in more detail about the semantic contributions of these words and constructions, as well as the influence of sentence position on interpretation. Since propositional logic doesn't represent meaning below the sentence level, it doesn't help us too much in this regard.

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Chris Kennedy Last modified: Mon Feb 1 13:37:33 CST