Assingment 1 Question 2

Hello there

Can any one help me understand Transparent propositional language, namely how to do an term interpretation...just don't get this SG.

Thanks

Difficult stuff - I also would like someone to explain how to do term interpretations.

Thanks

Let me have a bash at this. I write this in the hope
that someone would kindly offer to correct my
misconceptions. Please don't take this as the literal
truth; I'm not one of the lecturers.

Here we go ...

Firstly, since I cannot really represent the funny
squiggle symbols that logic needs in plain text,
note that I use the following notation:
1. The letters 'a', 'b' ... 'z' represent both
   the letters 'a', 'b', etc *AND* the greek
   equivalents. Please use context to decide
   which I meant. For example, using it as follows
   -- 'Ind = {a, b}' means the normal characters
   but using it like this -- 'a is a sentence in L'
   means that 'a' represents the greek equivalent.
2. I do something similar with super/subscripted
   elements. Read with context in mind; for example
   when you see '(P,1)' then you know '1' is not
   super-or-sub-scripted. When you read 'f1(a)=a'
   then you should know '1' is subscripted. For
   superscripted I would use the '^' character;
   for example 'D^n' means 'D to the n'.
   

The transparent propositional language has these
elements:
   1. A set of individual constants called 'Ind'
      e.g. Ind = {a, b}.
   2. A set of predicate /functions/[1] called 'Pred'
      e.g. Pred = {(P,1), (Q,1)}.
   3. A set of /operators/[1], normally assumed to be
      {negation, conjunction, disjunction, conditional, biconditional}
   4. A set of rules that determine what is a legal sentence
      in a language. This set is given in the SG[2] on page 14,
      Definition 1.2.1
   5. A set of atoms that is induced by the set of 'Ind'
      and 'Pred'. For example, with Ind = {a, b}, and
      Pred = {(P,1), (Q,1)}, we simply take every possible
      combination of (P,1) with elements from Ind and
      every possible combination of (Q,1) with elements
      from Ind. This gives us (in this example):
         set of atoms A = {P(a), P(b), Q(a), Q(b)}.
      If Pred was instead 'Pred = {(P,1), (Q,2)}', then
      our exhaustive set of atoms will instead be:
         set of atoms A = {P(a), P(b),
                           Q(a,a), Q(a,b),
                           Q(b,b), Q(b,a)}.

The important thing about the transparent propositional
language is this: it has an interpretation to allow us
to attach meaning to the symbols.


A set of interpretations, according to the prescribed text,
is a set of functions that each assign constants such that
every constant in 'Ind' is mapped onto itself and every
predicate function from 'Pred' is mapped onto every element
of the cartesian product of 'Ind'.

Lets look at that a little closer with an example to boot.
We have 
   Ind = {a, b}
   Pred = {(P,1), (Q,1)}
   A = {P(a), P(b), Q(a), Q(b)}

Now, the above definitions say that the function in the
interpretation maps each constant onto itself, so for
all the interpretations, 'f(a)=a, f(b)=b'.

The next step is to map each function from Pred onto
every element from the cartesian product of 'Ind', which
gives us D^n = { {}, {a}, {b}, {a,b}}. IOW, we get a set
containing an empty set and a set of only 'a', and a set of
only 'b', and a set of 'a' and 'b'. Lets write it out 
on a line by itself so we can see it clearly.

      D^n = { {}, {a}, {b}, {a,b}}
   
Now, we map each function from Pred onto D^n to give us
a set of interpretations. Note that we will exhaust the
elements from both sets long before we reach the end.

I1  =  f1(P,1)={},       f1(Q,1)={},      f1(a) = a,    f1(b) = b
I2  =  f2(P,1)={},       f2(Q,1)={a},     f2(a) = a,    f2(b) = b
I3  =  f3(P,1)={},       f3(Q,1)={b},     f3(a) = a,    f3(b) = b
I4  =  f4(P,1)={},       f4(Q,1)={a,b},   f4(a) = a,    f4(b) = b
I5  =  f5(P,1)={a},      f5(Q,1)={},      f5(a) = a,    f5(b) = b
I6  =  f6(P,1)={a},      f6(Q,1)={a},     f6(a) = a,    f6(b) = b
I7  =  f7(P,1)={a},      f7(Q,1)={b},     f7(a) = a,    f7(b) = b
I8  =  f8(P,1)={a},      f8(Q,1)={a,b},   f8(a) = a,    f8(b) = b
I9  =  f9(P,1)={b},      f9(Q,1)={},      f9(a) = a,    f9(b) = b
I10 =  f10(P,1)={b},     f10(Q,1)={a},    f10(a) = a,   f10(b) = b
I11 =  f11(P,1)={b},     f11(Q,1)={b},    f11(a) = a,   f11(b) = b
I12 =  f12(P,1)={b},     f12(Q,1)={a,b},  f12(a) = a,   f12(b) = b
I13 =  f13(P,1)={a,b},   f13(Q,1)={},     f13(a) = a,   f13(b) = b
I14 =  f14(P,1)={a,b},   f14(Q,1)={a},    f14(a) = a,   f14(b) = b
I15 =  f15(P,1)={a,b},   f15(Q,1)={b},    f15(a) = a,   f15(b) = b
I16 =  f16(P,1)={a,b},   f16(Q,1)={a,b},  f16(a) = a,   f16(b) = b

As you can see, all we do is make sure that we exhaust every
possible combination of combining Pred with D^n; it's really very
simple, much like generating a truth table was in cos161.

Of course, those are all possible interpretations, so there are
quite a few. It's not usually feasible to generate all possible
interpretations; for example the one from assign1,Q2 would have
had I1 to I32 possible interpretations (which is why they only
asked for the one that fits the situation).


To generate valuations from the above interpretations, we simply
examine the interpretation and assign the atom a T or a F, depending
on what the interpretation says. Lets examine I8 from above:

I8  =  f8(P,1)={a},      f8(Q,1)={a,b},   f1(a) = a,    f1(b) = b
          |                   |
          |                   |
          |                   |  +------------------------------+
          |                   +--+ This means that Q(a) and Q(b)|
+---------+----------+           | are both true.               |
|This means that P(a)|           +------------------------------+
|is true.            |
+--------------------+

So, for valuation W8, you set the atoms P(a), Q(a) and Q(b) to
true and all the other atoms will be false; our valuation is
therefore:
W8 = {(P(a),T) , (Q(a),T) , (Q(b),T) , (P(b),F)}

Remember, each valuation has to assign either T or F to
every single atom! So all the ones that a certain interpretation
contains will be T, the reminder atoms for that interpretation
will be F.

You can go ahead and use a string of '1's and '0's to represent a
valuation, but make sure that you write out the set of atoms so
that the reader is not confused.

The assignment question (Q2) merely asks you to work backwards
from the valuation to get the interpretation. So using this:
   Ind = {a, b}
   Pred = {(P,1), (Q,1)}
   A = {P(a), P(b), Q(a), Q(b)}

and given a valuation of (for example) this:
   W = {(P(a),F) , (P(b),F) , (Q(a),T) , (Q(b),F)}

we find that the only atom that is T is Q(a), therefore
our interpretation will have f(P,1) as an empty set and
f(Q,1)as the set containing only 'a', like so:

I  =  f(P,1)={},      f(Q,1)={a},      f(a) = a,    f(b) = b



---------------------------------------------------

[1] For want of a better word.
[2] Study Guide, 2007

Jeez

Must of taken u a day to type that...


But Thanks man, i think i understand it a bit better

NinjaMojo Wrote:
-------------------------------------------------------
> Jeez
>
> Must of taken u a day to type that...
>

From around 15:00 to around 15:45, I think
(went out at 16:15; took the wife on a date

>
> But Thanks man, i think i understand it a bit
> better

I'm still not very convinced that *I* understand it
thoroughly, though. I'm fairly certain that the
above is correct, but not 100% certain, so feedback
is essential.

Anyway, I'm wondering whether to typeset the
above into a pdf or similar, so that I can use
all the proper symbols, etc ... only thing is, I won't
be able to post it here again, only able to post a
link to it.

Regards
goose.

Ps. Don't forget the feedback; I need to correct it if
it is at all wrong.

Hie Goose

It is advisable when you want to help someone to give an example of a question similar to that of an assignment other than writing an answer to an assignment on Osprey. Now that assignment marks count towards your final year marks we do not expect such a thing to be done.

Lecturer COS 361-F

bester Wrote:
-------------------------------------------------------
> Hie Goose
>
> It is advisable when you want to help someone to
> give an example of a question similar to that of
> an assignment other than writing an answer to an
> assignment on Osprey.

I did try my best not to give any assignment answers
out. Rereading the above, I still find no assignment answer
(well, not directly anyway).

If the above is actually an answer to an assign question,
please accept my apologies; I don't intend to ever give out
the answer I wrote in an assign. question (as my answers
can frequently be wrong, and I'd hate for others to lose
marks as a result of an error on my part).

> Now that assignment marks
> count towards your final year marks we do not
> expect such a thing to be done.
>

Thank you for that; I did not realise that.

> Lecturer COS 261-C

Can you possibly comment on my explanation above?
I'd rather like to know if I got it correct.

Thanks

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Hie Goose

From your explanation above it shows you did understand term interpretation for transparent propositional logic.Thats Good. You can also study the solutions coming together with your assignment.

Lecturer