Hi, I need some help with Truth-Functional Logic please. Which of the following statements are true and which are false? Please explain if possible. Thanks.
1. If A ⊨ B, then B ⊨ A.
2. If ⊨ C and C ⊨ A, then ⊨ A.
3. If A ⊨ C, then A ∧ ¬C ⊨ C.
4. If A ∨ B ⊨ C ∧ D, then A ⊨ D.
First, clearly define your symbols in plain English. Then re-write your questions in English. Maybe then you will see the answers to your questions.
I thought the double turnstile means implication, but in that case what does
If ⊨ C and C ⊨ A, then ⊨ A
mean? I agree with Avi, what do these symbols mean in plain English?
Logic should be more than shuffling symbols around.
Hi, the double turnstile in this instance defines tautological entailment. So A ⊨ B would mean that A tautologically entails B. Assuming this, If you could please help me with these questions I would be eternally grateful. Thanks
Hi, the double turnstile means tautological entailment. When there is no letter before the double turnstile it means that there is nothing that can make this sentence false, so it is always true. The letters are symbolising any potential sentence in TFL. They are part of our metalanguage for talking about any arbitrary TFL sentence. In this case, could you please help me with answering these questions? Thank you!
Just to clarify: If there isn't a letter before the double turnstile, then this means that this is a tautology (it is true on every valuation).
If the double turnstile is between some sentences and a final sentence (conclusion), then it means that the sentences before the double turnstile tautological entail the sentences after it (tautological validity).
If there is just one letter before, and one after, this is tautological equivalence (they have the same truth value on every valuation)
You can use truth-tables to help demonstrate your answer if that helps. Thank you.
Ok Fred, Here's what I think :
1. This is false. Example : Let A = "The president was assassinated" and B = "The president is dead". A entails B but B doesn't entail A (because the president may be dead for other reasons).
2. This is an example of Modus Ponens, so it's true. Here's the truth table.
c a | c∧(c→a)→a
1] 1 1 | 1 ✓
2] 1 0 | 1 ✓
3] 0 1 | 1 ✓
4] 0 0 | 1 ✓
3. This is true although at first sight it may appear to be a contradiction and therefore false. Truth table :
a c | (a→c)→(a∧¬c→c)
1] 1 1 | 1 ✓
2] 1 0 | 1 ✓
3] 0 1 | 1 ✓
4] 0 0 | 1 ✓
4. This is True.
a b c d | ((a∨b)→(c∧d))→(a→d)
1] 1 1 1 1 | 1 ✓
2] 1 1 1 0 | 1 ✓
3] 1 1 0 1 | 1 ✓
4] 1 1 0 0 | 1 ✓
5] 1 0 1 1 | 1 ✓
6] 1 0 1 0 | 1 ✓
7] 1 0 0 1 | 1 ✓
8] 1 0 0 0 | 1 ✓
9] 0 1 1 1 | 1 ✓
10] 0 1 1 0 | 1 ✓
11] 0 1 0 1 | 1 ✓
12] 0 1 0 0 | 1 ✓
13] 0 0 1 1 | 1 ✓
14] 0 0 1 0 | 1 ✓
15] 0 0 0 1 | 1 ✓
16] 0 0 0 0 | 1 ✓
Disclaimer : I'm not a professional logician like Avi, so if I were you I'd wait for his reply. :wink:
Thanks for the reply - this is truly helpful. I'm sure Avi will agree!
Thanks for your intervention, Jack. I have not checked your work, but you seem to know what you are doing.
Hi Fred. Sorry, I'm writing at this time and too busy to concentrate on anything else. I'm glad Jack responded to your queries.
Still, I reiterate what I said. I personally avoid getting entangled in symbols. My way is to re-write such sentences in plain English, and then look at them.
Symbols do not function directly in logic (as they do in maths, to a greater extent). They require the mind to first translate into natural language and then understand and evaluate them. That is why it is best to do the translation on paper first, rather than let the mind have to do it mentally. Saves time and improves understanding and accuracy.
Are you working on a new book? I'm currently studying "Future Logic" and have a query, but it shouldn't take much of your time to answer. I see there's a thread already in progress for this so I'll ask it there.
Hi Jack, you're welcome. Yes, I'm writing again on various topics. Do I have your e-mail for announcements or documents? If not, go to TheLogician.net homepage and give it to me there.
Hi Jack. I've put you on my mailing list. I rarely communicate, but will keep you posted.
Thanks for your message. I just had one more question that I hoped you guys might be able to answer. I'm trying to symbolise the following sentence in TFL, using the symbolisation key below:
James is outraged unless Daisy doesn’t prefer salad to dessert, Ed doesn’t
hate Yorkshire puddings, and Ed doesn’t laugh.
I: James loves ice cream
L: Ed laughs
O: James is outraged
S: Daisy prefers salad to dessert
Y : Ed hates Yorkshire puddings
So far, I've got this:
(O ↔ ¬(S ∧ Y ∧ L).
If anyone could help me with this I'd be truly grateful. Thanks again.
@ Avi, thanks!
You have 3 propositions here. Let's take them one at a time.
1. James is outraged unless Daisy doesn’t prefer salad to dessert
"unless" is a conditional, and I find the easiest way to translate it is:
A unless B = if not-B, then A
Convince yourself that this is correct with some examples. e.g. "I'll go out unless it's raining" means "if it's not raining then I'll go out".
So using your key, proposition 1 is:
O unless not-S = if not-(not-S), then O.
ie, if S, then O (S => O)
The other two propositions are straightforward.
The 3 propositions are combined conjunctively, so the final combined proposition is
(S => O) & ¬Y & ¬L
Hello Jack. Thanks for your participation again.
"I'll go out unless it's raining" means "if it's not raining then I'll go out". I would add: If it is raining, I won't go out." Always remember 'the negative side' - it is often forgotten.
I find the statement "James is outraged unless Daisy doesn’t prefer salad to dessert" a bit obscure. Presumably it primarily means: James is outraged if it is not true that Daisy doesn’t prefer salad to dessert, i.e. James is outraged if it is true that Daisy prefers salad to dessert. But we should add: James is not outraged if it is true that Daisy doesn’t prefer salad to dessert.
I have not looked at the rest of it.
Fred, another word of advice if I may. It is better for you to present your proposed solution to the problem and then have that corrected by others, than to ask others to do the thinking for you. One learns best from one's errors (if any), than from ready-made work by others.
Thanks again for your message. I think we have been taught to use disjunction if a sentence is in the form of 'unless A, B'. But it is important to note that we can also offer alternative disambiguations if they exist. I've got this (O ∨ ¬(S ∧ Y) ∧ L) as what I think is right, because it can be paraphrased in English as 'James is outraged unless it is not the case that Daisy prefers salad to desert and Ed hates Yorkshire puddings, and that Ed laughs.
Does this make sense? Thanks.
Thank you Avi. I've noted my idea of the solution below, but I'm not 100% sure about it. If you can think of any disambiguations for this statement that would be a real help. No worries if you don't have time, I really appreciate it nonetheless!
Avi, you wrote:
Hi Jack and Avi,
Thanks for your messages. So far, I have come up with three possible disambiguations of this statement: James is outraged unless Daisy doesn’t prefer salad to dessert, Ed doesn’t hate Yorkshire puddings, and Ed doesn’t laugh.
The first one uses 'unless' to imply a disjunction, whereas the other two use a conditional. If you could comment on which ones you think most accurately represent the sentence in TFL I would be very thankful.
1. (O ∨ ¬(S ∧ Y ∧ L))
2. ((S → O) ∧ ¬Y ∧ ¬L)
3. ((S → O) ∧ (¬S → ¬O) ∧ ¬Y ∧ ¬L)
Jack, you're right, but not entirely. Inattention on my part, though.
"I'll go out unless it's raining" =
If it does not rain, then I will go out, AND
If it rains, not-then I will go out (i.e. in that event I may not go out).
"The plant will die unless you water it" =
If you don't water the plant, then it will die, AND
If you water the plant, not-then it will die (i.e. in that event it may not die)
A unless B =
If not B, then A, AND
If B, not-then A (i.e. if B, possibly not A)
The reason for this is to shut off the possibility of both "If not B, then A" and "if B, then A" being true, in which case A would be true categorically. The negative side ensures we have a conditional statement.
The negative side in both cases has a not-then (= a then-possibly-not) consequent.
I wrongly stated it as a then (or then-necessarily) statement.
My apology. Thanks for correcting me.
Thanks for the clarification. As far as I'm aware modern propositional logic doesn't have a way of stating that a proposition B does NOT follow from another proposition A, like your innovation of "if A, not-then B", and in my opinion is all the poorer for it!
Hi Jack, thanks for the reply. Whilst I do agree with you, do you think that the other representations are also acceptable? Could this sentence not be seen to use a disjunction rather than a conditional? And is it not also necessary to additionally note that (¬S → ¬O) is true in addition to what is presented in number 2? Thanks
Jack, logical conditioning is dealt in great detail in Future Logic,part III.
That said, the form If/not-then is nothing very new really. In ordinary discourse, we say commonly things like "if so and so, it does not follow that such and such". If X, not-then Y is stated in modern logic as simply the negation of if X then Y, i.e. as not(X->Y), i.e. X does not imply Y.
Keep in mind that for every form we need a counter form (though there are very rare exceptions, as I recall).
(Jack, please help Fred if you can. As I said, I am in the middle of a difficult bit of writing. It is difficult for me to turn my attention elsewhere. Many thanks.)
Hi Fred, I've taken the liberty of adding your e-mail to my mailing list.
Glad that Jack is answering your questions.
Statements 1 and 3 don't really work.
1. O or ¬(S & Y & L)
By De Morgan's rule, the right disjunct is ¬S or ¬Y or ¬L, so you have O or ¬S or ¬Y or ¬L.
Rearranging, this is (¬S or O) or ¬Y or ¬L. Now, (¬S or O) is equivalent to S => O, which is what we need, and the remaining propositions ¬Y and ¬L should be combined conjunctively with this conditional according to the original statement, but they are instead combined disjunctively.
I just wanted to remind you of the first few questions I asked if thats ok. I noticed you used conditionals to represent the double-turnstile symbol in the truth table. In my textbook, the double turnstile is used to symbolise tautological validity between arbitrary sentences of TFL. It is not a symbol of TFL, but rather a symbol of augmented English used to describe arbitrary sentences of TFL. The conditional is a symbol of TFL, unlike the double-turnstile. Also, the letters in the questions are used to symbolise any arbitrary sentence of TFL, not specific ones.
Based on this information, would this change any of your answers?
I have put the questions I am struggling with below. Sorry to bring this up again, I just wanted to check.
Question: Are the following claims about tautological validity true or false?*
*If you could explain your position on each question that would be really useful.
1. If A ⊨ B, then B ⊨ A.
2. If A ⊨ C, then A ∧ ¬C ⊨ C.
3. If A ∨ B ⊨ C ∧ D, then A ⊨ C.
I also want to say how grateful I am for your help. I am working through this textbook by myself and your explanations have really helped me over the last week.