Logic: Logical Status of Statement Forms

Dembadon

The professor for my symbolic logic course requires us to be extremely precise with our explanations. Given the subject, I understand his reasoning and appreciate his rigor. I am studying for our first exam by doing some of the exercises at the end of the sections on which we're going to be tested. I'm hoping to get some feedback on the reasons I've provided for the following cases. Are they precise enough; verbose; false? In other words, please nit-pick the hell out of them!

I'd also like to deviate a bit from the standard H.W. submission form, if it's okay with the mentors. I would like to place the questions and answers in the same section to keep those who wish to help from having to scroll up and down to match the questions to their corresponding answers.

1. The problem statement, all variables and given/known data

2. Relevant equations



3. The attempt at a solution


What is the negation of a tautology? Why?

The negation of a tautology is a contradiction.

Reason: In a tautology, all of the truth-values under the major operator are true, and since all of the values under the major operator are false for a contradiction, it follows that the negation of a column of T’s will be a column of F’s.

What is the negation of a contradiction? Why?

The negation of a contradiction is a tautology.

Reason: In a contradiction, all of the truth-values under the major operator are false, and since all of the values under the major operator are true for a tautology, it follows that the negation of a column of F’s will be a column of T’s.

What is the disjunction of a contingent form and a tautology?

This will be a tautology, because a disjunction requires only one of the truth-values to be true, which it obtains from the tautology. Therefore, all of the truth-values for the disjunction will end up being true. Hence, the conclusion will be tautological.

What is the biconditional of two contradictions?

This will be a tautology, because for a biconditional to be true, both premises’ truth-values must be equivalent. Since all of the truth-values in a contradiction are false, then the biconditional will be true for every instance of the form.

What is a conditional with a tautology as an antecedent and a contingent form as a consequent? Why?

This will be another contingency.

Reason: The false truth-values in the contingency will cause the truth-values for the conditional to be false. Since some of the other instances of the conditional form will yield truth-values that are true, then the conclusion will be a mix of true and false instances of the form, thus making a contingency.

What is the biconditional of two contingent forms? Why?

This has the possibility to be anything: tautology, contradiction, contingency.

Reason: It’s possible to find instances of the contingent forms that yield all three logical statuses.

micromass

Before I say anything, could you please state the definition of a major operator and of a contingency. I never saw logic in english...



OK, this is good. Were you taught to write it in sentences like this?? If you would ask me that question, then I would write it out in symbols.

But you want me to nitpick: what do you mean with a "column of T's"?? Please specify this.

The same as above
OK

"both premises' truth values must be equivalent". I see what you mean, but what do you mean with "equivalent"?

Good, but could you specify exactly when the conditional will be true and when it will be false.

Please give three examples where those three things can happen.

Dembadon

Perfect, micro! This is just what I needed. I will work-up a reply and get back to you. Thank you for your help.

Dembadon

Sure!

Major Operator: is a truth-functional, sentential operator that decides the truth-value for the entire formula/compound.

Contingency: a single statement form that is false for some substitution instances and true for others; that is, it has both T’s and F’s in its truth table under the major operator.

Another note: I’m required to know both the symbols and English translations for this course. I’ve used sentences here because it is more difficult for me than using symbols.

By saying “column of T’s,” I’m referring to the truth table values under the major operator.

I mean the same thing here.

Good question. By “equivalence” I mean that the truth-values, T or F, in the truth table must be the same for the substitution instance (row in the truth table).

The conditional will be false if and only if the antecedent is true and the consequent is false. All other cases will make the conditional true.


Contingent forms → Tautology
T iff T → T
T iff T → T
F iff F → T
F iff F → T

Contingent forms → Contradiction
T iff F → F
F iff T → F
T iff F → F
F iff T → F

Contingent forms → Contingency
T iff F → F
T iff T → T
F iff F → T
F iff T → F

micromass

Yes, but the antecedent is always true. So...

To nitpick further: you've only shown me truth tables. How do I know that there are actual formulas that satisfy this truth table?? That is, can you give me actual formula's??

You do seem to know your stuff though

Dembadon

Thanks! Here are some examples I worked out earlier. They aren't very complicated, but they still illustrate my points.

But first, some definitions!

Let "≡" mean "if and only if".

~ ≡ not
• ≡ and
⊃ ≡ conditional operator

Contingent forms, joined by a biconditional, that make a tautology:

~(p ⊃ q) ≡ (p • ~q)

In the case above, it is not possible to obtain a false conclusion no matter what truth-values are assigned to p and q.

Contingent forms, joined by a biconditional, that make a contradiction:

~(p ≡ ~q) ≡ ~(p ≡ q)

In the case above, it is not possible to obtain a true conclusion no matter what truth-values are assigned to p and q.

Contingent forms, joined by a biconditional, that make a contingency:

(p ≡ q) ≡ (p ⊃ q)

In the case above, it is not possible to obtain a tautology or a contradiction no matter what truth-values are assigned to p and q.

micromass

That sounds ok!!!!

Dembadon

Excellent. Thank you for your help!