## Saturday, October 07, 2006

### Sommers-Englebretsen Term Logic, Part VIII

A quick wrap-up post. In previous posts I've covered some basics of SETL:

In Part I, I noted the basics of SETL, in a rough way.
In Part II and Part III, I discussed briefly some special cases and how SETL handles them.
In Part IV, I discussed some basics of argument using SETL.
Part V looked at some simple arguments for which SETL gives us a better sense of what's going on than ordinary predicate logic does.
In Part VI I looked briefly at Englebretsen's discussion of how to extend SETL to modality.
In Part VII I looked at Murphree's union of SETL with numerical syllogistic.

Here are a few on-line references for further reading. All of them except Purdy's review of An Invitation to Formal Reasoning are found at the Notre Dame Journal of Formal Logic at Project Euclid.

Englebretsen, George
A Note on Contrariety
Do We Need Relative Identity?
Opposition
Preliminary Notes on a New Modal Syllogistic
Sommers on Empty Domains and Existence
The Square of Opposition

Murphree, Wallace
Numerical Term Logic

Purdy, William
On the Question, "Do We Need Identity?"
Review of Sommers & Englebretsen, An Invitation to Formal Reasoning

Sommers, Fred
Predication in the Logic of Terms
The World, the Facts, and Primary Logic

1. Hi. I am going through Sommers's and Englebretsen's textbook "Invitation to Formal Logic." I was hoping you could check my work on one exercise they provide. One is to transcribe the following:

If some farmer is a noncitizen and some farmer is a citizen then no scholar is a thief

-[+[+F-C]+[+F+C]]+[-(+S+T)]

Seem right?

Thanks for these great posts. You clarify their text nicely.

2. Hi again. I was hoping you could clarify something related to transcribing natural English to TFL.

In TFL, "if" = - and "then" = +

But what about a statement like "x only if y"?

'x only if y' is the same as 'if x then y', but the transcription heuristic seems to go out the window, unless I am neglecting something.

3. Hi, Matias,

I am currently away from home, so I am short on time and lack access to my books. But I should have answers for you later this week.

However, in this case, despite the way it's sometimes put in the book, you should not think in temrs of 'if' being equal to - and 'then' being equal to +. Rather, if...then.... is equal to -...+... -- that is, you should not think of the quantity sign and the quality sign as entirely separate from each other, since they only get their meaning as a pair, and likewise, the individual words 'if' and 'then' don't map onto signs, but the pair 'if...then...' maps on to the quantity-quality pairof signs.

Sommers and Engelbretsen likewise are assuming (I imagine -- I don't have the book in front of me, so have to answer from memory) that you are first regimenting the natural English into a form where the logical function of the English is easy to see. So you'd turn '...only if...' expressions into 'if...then...' expressions and only then use the transcription heuristic.

4. Hey, thanks so much for getting back to me. I see your point on not treating if and then separately. I think you can preserve the English and the heuristic somewhat if you say that x only if y is -(+x-y).

I wanted to ask something else once you have the time. I'm not sure how to transcribe x even if y. I take it to mean something like if (either y or not y) then x. The transcription seems to be ---y---y+x, but it doesn't work out with a modus ponens, for example.

Again, thanks for the help, and get home safe,
MF

5. A strict translation of 'even if' would, in my opinion, require a modal extension of the logic; 'even if' is a modal connective. Likewise, one wouldn't expect 'even if' to follow ordinary modus ponens rules: 'Even if' is not a conditional but a deliberate anti-conditional: it says that a proposed condition, a possible antecedent, is irrelevant and can be ignored (for truth-value purposes, anyway).

If we stay with the standard notation, then I suspect that the closest we could get would be something like:

--(+x+y)--(+x-y)

This actually is equivalent to

--x--x

which is essentially right. If I say 'x even if y', I mean something along the lines of 'x is true (and y's being true doesn't change that)'. Thus 'x even if y' is a more complicated way of saying 'x' -- to that extent, anyway.

6. I see what you mean. Still, my formation of a conditional, if (either y or not y), then x, is tautalogous, so its as good as x. Yet, when I try to transcribe that, it seems wrong. Why, specifically, does if (-) either (--) y or (--) not (-) y, then (+) x [---y---y+x] seem to fail?

7. OK, I see what you mean. It looks like your minuses aren't being distributed properly. I.e., the formulation should not be ---y---y+x but instead:

-(--y---y)+x

The quantity-minus distributes over the entire disjunction; it applies to the whole subject term, which in this cause is 'either y or not y'.  When distributed we get ---y----y+x, which is equivalent to -y+y+x, and thus to x. Hence the tautology. And it allows modus ponens: since either y or not y is a necessary truth, we can plug it in:

-(--y---y)+x
(--y---y)
Therefore +x

Here again, I think it's worth remembering that the -...+... is a package deal, not two completely separate signs. When we put the antecedent in after the minus, the minus from the if...then... has to distribute over everything in the antecedent (because the antecedent is the subject term, and quantity has to apply to the subject term as a whole).

8. Now I understand. Thanks a lot for the clarification.

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