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Chapter Four — Symbolization in SL

 

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Contents: Section 1: Expressive Completeness; Section 2: Symbolizing Complex English Sentences; Section 3: Controversial Symbolizations

 

In this chapter we further the work on symbolizations started in chapter two. So, before reading on in this reference, you may wish to look back at the introductory material on symbolization. Chapter two has a chart describing the basics, a table giving a good overview, a reference for symbolization, and the full tutorial.

1. Expressive Completeness

The connectives of SL are all truth functional. Remember, this means that they may be used to form compound sentences whose truth value depends only on the truth values of the immediate components. Thus one can give a truth table to define the truth conditions of all SL connectives.

Now, one might wonder: why not define more connectives? For instance one could have a single connective to express "neither P nor Q". We might write this as 'P$Q' and could give it the following truth table.

  P Q P$Q
row one: T T F
row two: T F F
row three: F T F
row four: F F T

But there is no reason to bother to define '$' and make it a part of our language: we may simply symbolize "neither P nor Q" with our standard SL connectives as '~P&~Q'. In fact it doesn't take much thought to see why any truth function (like '$' but perhaps having more than two component parts) can be represented with only the connectives of SL. (Hint: one way to represent the truth function will be to look at each row making the sentence true -- like the one row which does this for '$' -- and construct a sentence true in just that row. Then put the sentences from each row together. How? See the tutorial.)

2. Symbolizing Complex English Sentences

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.

3. Controversial Symbolizations

We are now able to understand symbolization a bit better than in chapter two because we have the semantical foundation of chapter three's truth tables. Thus T4.3 reconsiders the controversial symbolizations first described when SL was introduced. Below you will find a synopsis of some but not all of this tutorial's considerations.

The first controversy over symbolization is that of disjunction. The question is

Should "or" in English be interpreted in an "inclusive" way (true when both disjuncts are true) or an "exclusive" way (false when both disjuncts are true)?

The following example illustrates the question and the controversy.

(*) Either Bill or Hillary Clinton is Human.

This sentence just doesn't sound right. Clearly there is something wrong with (*). One diagnosis...

The Advocates of Exclusive "or" say: There is nothing complicated going on in this example. (*) is wrong simply because it's false. Thus English "or" is -- at least in this case -- exclusive.

On this point of view, 'v' is not a very good translation of English "or" because the former is (by definition) inclusive.

But there is an alternative philosophy on this matter. The other side has an account of (*) and why it seems so bad:

The Advocates of Inclusive "or" say: The problem with (*) is not that it says something false; rather than misdescribing the Clintons, (*) doesn't say enough about them. Its problem is being uninformative. Anyone who says (*) leaves open the possibility that one or the other isn't human. Leaving this possibility open, when it's obvious that both former denizens of the White House are indeed human, is insulting and misleadingly uninformative. Still to say something insulting or misleading is not necessarily to say something false. In this case what is wrong about (*) is not falsehood but it's misleading, insulting character. Thus English "or" -- at least in this case -- is inclusive.

Explaining why (*) seems so wrong in this way usually seems very odd in itself to most students...at least at first. Perhaps it seems a bit defensive. But the advocates of inclusive "or" have a good offense too!

Against the Advocates of Exclusive "or": If English "or" were exclusive then (taking 'B' and 'H' to stand for the two obvious truths that Bill and Hillary are human) we need only consider the first row of a table:

B H B or H
T T T F T

"B or H" is false assuming that "or" is exclusive. But, and now we note a simple consequence, if "B or H" is false, then "neither B nor H" is true:

B H ~ (B or H)
T T T T F T

But this means that someone committed to saying that "or" is exclusive for the Bill and Hillary example, is also committed to the conclusion that neither of the two denizens of the US White House are human! So, the advocate of exclusive "or" is clearly wrong.

Of course, this is not the end of the argument between the sides. But it indicates a good reason to proceed as we have been all along: When we see "or" in English, we just symbolize it with the 'v'.


The second major symbolization controversy described in chapter four has to do with the Material Conditional. A material conditional is any conditional of a language having the same truth conditions as we gave to define SL's horseshoe. (False only if its antecedent is true and consequent false; true otherwise.)

One point is uncontroversial. There are many conditionals of English which are not material conditionals. Examples of English conditionals which are not material are those which have to do with what would happen in counterfactual conditions. These are not truth functional, so clearly not material like the horseshoe.

This important, controversial question is this.

Are the conditionals of English which are not counterfactuals typically material conditionals?

We may hope so as we have long been translating conditional statements that way! But, this is a difficult question. And the mere beginnings of a case for a positive answer are given in the second half of T4.3. The following is a mere synopsis of a summary. Make sure you've read the tutorial in order to begin to get the ideas involved.

When we have an example like...

(**) If Bob Dole is US president, then he can jump over the moon.

...we are tempted to think: This conditional is clearly false even though antecedent and consequent are both false. Consequently, it cannot be a material conditional (which is true when both antecedent and consequent are false).

But let us not be too hasty. The example of "Bill or Hillary" should make us think twice about trusting our first intuition. Here's an alternative conclusion to reach about the Bob Dole example:

First, the English statement (**) should not be confused with "If Bob Dole were US president, then he could jump over the moon". This were-could conditional is surely not truth functional so surely not a material conditional. But (**) is different so may still be a truth functional material conditional.

Second, given that we all know that Bob Dole was defeated and is not US president, it's very funny to say "what if he is". (It's OK to say, "what if he were", but then we are thinking counterfactually.) The conclusion to draw here is much like that for (*): (**) is an odd thing to say, but not necessarily false.

Third, we have seen examples in English which do seem to have the truth conditions of a material conditional. So, perhaps (**) is like these.

The bottom line? Given the controversy, what should we conclude and what should we do on homework?

  1. We have given some reason to think that SL's disjunction ('v') and conditional ('>') are may be more like English disjunction and conditional than one might first suppose.
  2. No matter what you think of the case made for 1, we will just agree to use 'v' and '>' to translate the English disjunction and conditional because they are the best we have in SL.

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