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Logic: Logical Status of Statement Forms 
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#1
Sep1711, 04:59 PM

PF Gold
P: 641

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 nitpick 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 truthvalues 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 truthvalues 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 truthvalues to be true, which it obtains from the tautology. Therefore, all of the truthvalues 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’ truthvalues must be equivalent. Since all of the truthvalues 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 truthvalues in the contingency will cause the truthvalues for the conditional to be false. Since some of the other instances of the conditional form will yield truthvalues 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. 


#2
Sep1711, 05:17 PM

Mentor
P: 18,346

Before I say anything, could you please state the definition of a major operator and of a contingency. I never saw logic in english...
But you want me to nitpick: what do you mean with a "column of T's"?? Please specify this. 


#3
Sep1711, 05:20 PM

PF Gold
P: 641

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



#4
Sep1711, 05:54 PM

PF Gold
P: 641

Logic: Logical Status of Statement Forms
Major Operator: is a truthfunctional, sentential operator that decides the truthvalue 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. 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 


#5
Sep1711, 05:58 PM

Mentor
P: 18,346

You do seem to know your stuff though 


#6
Sep1711, 06:40 PM

PF Gold
P: 641

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 truthvalues 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 truthvalues 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 truthvalues are assigned to p and q. 


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