Is the argument A implies B, therefore A implies B or C valid?

  • MHB
  • Thread starter Ackbach
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    2016
In summary, the validity of an argument that states A implies B, therefore A implies B or C depends on the logical relationship between A, B, and C, as well as the truth values of these statements. This relationship can be determined by examining the premises and conclusion of the argument, as well as the use of logical operators. Implication and equivalence are two different logical relationships, with implication guaranteeing the truth of one statement based on the truth of another, while equivalence has the same truth value for both statements. An argument can be valid but not sound, meaning that the conclusion follows logically from the premises but the premises themselves may not be true. To test the validity of an argument, one can use techniques such as logical deduction or truth
  • #1
Ackbach
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MHB
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Here is this week's POTW:

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Construct a formal proof of validity for the following argument:
\begin{align*}
A &\to B \\
\therefore A &\to (B \lor C).
\end{align*}

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
Re: Problem Of The Week # 200 - January 26, 2016

This is Exercise 8.21 on page 212 of Language, Proof and Logic, by Barwise and Etchemendy.

No one answered this week's POTW. Here is my solution:

View attachment 5217
 

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FAQ: Is the argument A implies B, therefore A implies B or C valid?

Is the argument A implies B, therefore A implies B or C valid?

The validity of this argument depends on the logical relationship between A, B, and C. If A implies B, and B implies C, then A implies both B and C. However, if A implies B and B does not imply C, then A only implies B and not C. The validity of the argument also depends on the truth values of A, B, and C.

How do you determine the logical relationship between A, B, and C?

The logical relationship between A, B, and C can be determined by examining the premises and conclusion of the argument. If the premises support the conclusion, then there is a logical relationship between them. Additionally, the use of logical operators such as "implies" or "therefore" can also indicate the relationship between the statements.

What is the difference between implication and equivalence?

Implication is a logical relationship between two statements where the truth of one statement (A) guarantees the truth of another statement (B). On the other hand, equivalence is a logical relationship between two statements where they both have the same truth value. In other words, if one statement is true, then the other statement is also true, and if one is false, then the other is also false.

Can an argument be valid but not sound?

Yes, an argument can be valid but not sound. Validity refers to the logical structure of the argument, while soundness refers to both the logical structure and the truth of the premises. An argument can be valid if the conclusion logically follows from the premises, but if the premises are not true, then the argument is not sound.

How can you test the validity of an argument?

To test the validity of an argument, you can use techniques such as logical deduction or truth tables. Logical deduction involves examining the logical structure of the argument and determining if the conclusion logically follows from the premises. Truth tables involve assigning truth values to the statements in the argument and determining if the conclusion is always true based on those truth values.

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