# Symbolic logic, need help with truth table definition

I don't really know if this is an acceptable topic to be placing in a "Sciences" homework help forum. It's a problem I'm having in my Symbolic Logic class.

## Homework Statement

Every sentence has a truth-table which can describe it, does every truth table have a sentence?

We're dealing with the SD and SD+ logic derivations.

N/A

## The Attempt at a Solution

It's an extra credit question, and isn't really based in any particular chapter, so I'm having trouble on where to look for the answer. My gut says yes, but I'm led to believe otherwise. For any two variables, by using the four main connectives with or without the negation modifier:

Code:
[U]A | B | A&B | AvB | A>B | A=B |[/U]
T | T |  T  |  T  |  T  |  T  |
T | F |  F  |  T  |  F  |  F  |
F | T |  F  |  T  |  T  |  F  |
F | F |  F  |  F  |  T  |  F  |

So with those 4 main connectives, each row has one possible outcome. Using those four connectives, each possible combonation for a row (TTTT, TTTF, TTFT, TFTT, ..., TFFF, FFFF) CANNOT be achieved (see FFFT, for example), even by using the negation modifier (~). Does this mean I have proved that every truth table does not have a sentence to go along with it?

But I have no clue what I'm doing, and I think I'm looking at this the wrong way. Any insight?

Thanks.

Borek
Mentor
I would say you should write all 16 tables and try to find a sentence for each. And you may try more complicated sentences like A&(Av~B) - not that this one is of any use, just an example.

AHA! I think I understand what I forgot when first trying to intuitively think about it.

The negation modifier can be moved around inside, while I was just thinking I could put it in front of the whole sentence. For example, to come up with the full 16 truth tables, I could have ~(A>B), A>~B, etc. This could form the full 16 truth tables. But I can't really know how to tell if I've PROVED anything. I can't figure out how to extrapolate this to include ALL sentences, rather than just the basic atomic sentences with the main 4 connectives +/- negation modifier.

Also, just in case it was confusing, the '>' is supposed to be a horseshoe (ex. P>Q = If P then Q).

Edit: Thanks Borek for your help. I got so excited that I thought I found something that I forgot to thank you. :D

So now I just need to figure out how to extrapolate my conclusions to include all sentences of SD.

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