Symbolic logic, with truth table definition

In summary, Borek is having trouble figuring out how to solve an extra credit question in his Symbolic Logic class. He is unsure if he has proven that every truth table does not have a sentence to go along with it. He is looking for help from the community to figure out a solution.
  • #1
DyslexicHobo
251
0
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.

Homework Equations


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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
Last edited:

1. What is symbolic logic?

Symbolic logic is a formal system of reasoning that uses symbols to represent logical statements and connect them using logical operators such as AND, OR, and NOT. It is used to analyze arguments and determine their validity.

2. How is symbolic logic different from traditional logic?

Symbolic logic differs from traditional logic in that it uses symbols to represent logical statements, while traditional logic uses natural language. This allows for clearer and more precise analysis of arguments and their logical structure.

3. What is a truth table in symbolic logic?

A truth table is a table that displays all possible combinations of truth values for a logical statement or argument. It is used to determine the truth value of a complex statement based on the truth values of its components.

4. What is the purpose of using truth tables in symbolic logic?

The purpose of using truth tables in symbolic logic is to evaluate the validity of logical statements or arguments. By listing all possible combinations of truth values, we can determine if a statement is true or false under different conditions.

5. What are some real-world applications of symbolic logic?

Symbolic logic has various applications in fields such as computer science, mathematics, philosophy, and linguistics. It is used to design computer algorithms, construct mathematical proofs, analyze arguments in philosophy, and study natural language syntax and semantics.

Similar threads

Replies
2
Views
929
  • Precalculus Mathematics Homework Help
Replies
6
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
7
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
15
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
25
Views
4K
  • Introductory Physics Homework Help
Replies
12
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
9
Views
1K
Back
Top