w4

Assume for contradiction that Halpin is wrong and English 'or' is normally exclusive.

Consider this example: "Either George or Laura is human". Then, it follows from our assumption that ...

this disjunction is false. But if the disjunction is false, it's negation is...

true.

Is that bad? Well, if "Either George or Laura is human" is false then it's true negation is

Neither George nor Laura is human.

Oops. This isn't true. So, apparently we cannot so readily suppose that English "or" is exclusive.

In chapter two, we became used to symbolizing fairly simple English sentences involving only a few connectives. But there are lots of everyday examples of more complicated compounding in English. We should be able to translate these into SL. For example,

Private schools implementing a voucher program will fail to provide equal educational opportunities across a community if they either skim off the best students and leave the poorer ones behind or they charge parents fees beyond what is paid for in private funds and so exclude children of poorer families.

Is this a conditional sentence? Or a disjunction? There are no parentheses to help us. But there are "grouping" words in the English. For instance, what goes between "either" and "or" will be a first disjunction and the word either works like a left hand parenthesis. Here's a reformulation.

Private schools implementing a voucher program will fail to provide equal educational opportunities across a community if [either(they skim off the best students and leave the problem students behind) or they exclude children of poorer families by charging parents fees beyond what is paid for in private funds].

This sentence is now more pretty clearly a conditional with consequent "Private schools implementing a voucher program will fail to provide equal educational opportunities". The antecedent is more complicated.

A hybrid formulation may help:

~P if [(S&L) or E]

(Here we've used P: "Private schools implementing a voucher program provide equal educational opportunities", S: "Private schools implementing a voucher program skim off the best students", L: "Private schools implementing a voucher program leave the problem students behind", E: "Private schools implementing a voucher program exclude...") But we recall that 'P if Q' is symbolized as 'Q>P', so '~P' is the consequent and we should symbolize the whole English sentence as:

[(S&L)vE]>~P

The following table summarizes some of the mechanisms English uses to group.

When one has a sentence of form:	Then:	Example (Hybrid):	Symbolization:
If P, then Q	Antecedent is P	If A and B, then C.	(A&B)>C
Either P or Q	First disjunct is P	Either both A and B, or C.	(A&B)vC
Both P and Q	First conjunct is P	Both if A then B, and C.	(A>B)&C

One other good indicator of grouping in sentences is a comma (or semicolon). These often mark a major structural division in the sentence corresponding to the main connective. So...

If Private schools implementing a voucher program succeed, then their students will benefit directly and traditional public schools will benefit from the competition.

Which might have the hybrid form...

If S, then D and C.

...and, because of the comma marking the main connective, would have symbolization as follows:

S>(D&C)

Now, make sure you have carefully read T4.2 and do lots of exercises. Experience is the key to symbolization.