# Help needed with predicate interpretations- Wilfrid Hodges logic

I was going through Hodges' Introduction to Elementary Logic and was rather puzzled by a section where one is asked to translate an argument into a predicate interpretation.

No man may be beaten with above forty stripes; nor may any true gentlemen be punished with shipping, unless his crime be very shameful, and his course of life vicious and profligate. John Doe Jr is a true gentleman, and, though his crime is in truth very shameful, his course of life is in no manner vicious and profligate. Therefore, John Doe Jr may not be punished with shipping.

However, it is only the section

nor may any true gentlemen be punished with shipping, unless his crime be very shameful, and his course of life vicious and profligate

that I am concerned with- with the given interpretation:

M: No man may be beaten with above forty stripes

Gx: x is a true gentleman

Cx: x's crime is very shameful

Lx: x's course of life is vicious and profligate

Sx: x may be punished with shipping

The translation of that excerpt is given as ∀x[Gx->[¬SxV[Cx∧Lx]]], though, considering that the disjunct truth functor V allows the sentence functor [φVψ] to be true when both sentence variables are also true, doesn't this mean there is a situation where there is a true gentleman who may not be punished with shipping and who has both committed a very shameful crime and who has a vicious and profligate course of life? Is a weak reading being applied? Apologies if this is rather a trivial question- I am going through the book by myself and have no tutors to consult.

Your problem concerns the precise meaning of the word 'unless' in this context. At its most general, you are right to say that 'unless' has a weak reading under which it has the same truth conditions as inclusive or.

For example, a sign that says, "Swimming is not permitted here unless a lifeguard is present" does not mean that if a lifeguard is present swimming is definitely permitted. A lifeguard may be present and tell people not to swim because she has seen a shark. So in this context it is correct to understand the sign to say, "Swimming is not permitted or a lifeguard is present (or both)". Similarly, "You won't pass the exam unless you study hard" doesn't imply that if you study hard you definitely will pass the exam. Your example states that a true gentleman may not be punished with shipping unless his crime is very shameful and his course of life vicious and profligate. I would understand this as expressing a restriction on who may be shipped, but not a prescription that anyone satisfying these conditions must be shipped. Maybe someone meets these conditions but there are extenuating circumstances that make a more lenient punishment appropriate.

The reason examples involving 'unless' appear problematic is that sometimes when we say 'unless' the meaning we intend to convey is actually 'unless and only unless'. 'Unless' has the same truth conditions as inclusive or, while 'unless and only unless' has the same truth conditions as exclusive or. But saying 'unless and only unless' is a bit of a mouthful, so we tend to just say 'unless' and allow the context to do the work of conveying the intended meaning.

I find `unless` confusing to unravel when used in natural language. One nice trick if you're ever confronted with an `unless` is to paraphrase it into an `if`.

Here's how to do it.

First, let's note that the following sentences are all equivalent ... and we can convince ourselves of this intuitively without appealing to any specific formalism. This gives us a recipe we can use later for de-unlessing the sentence.

``````1) I may go outside unless it rains.
2) If it rains, I will not go outside.
3) If I go outside, then it isn't raining.
``````

So, we want to encode the following sentence fragment in first-order logic:

nor may any true gentlemen be punished with shipping, unless his crime be very shameful, and his course of life vicious and profligate.

We can paraphrase this into mathy language as follows and convince ourselves that the meaning is the same.

For any true gentleman x, if it is permissible to punish x with shipping, then x's crime was very shameful and x's course of life was vicious and profligate.

Let's define the following abbreviations for ourselves to make this easier.

`G(x)` holds if and only if `x` is a true gentleman.

`S(x)` holds if and only if `x` may be punished with shipping.

`B(x)` holds if and only if `x`'s crime is very shameful and `x`'s life is vicious and profligate.

``````∀x.(G(x) → S(x) → B(x))
``````

From here, the equivalence with the following should be clear

``````∀x.(G(x) → (¬S(x) ∨ B(x)))
``````