Hey yall! anyone know how to solve this proof only using replacement rules and valid argument forms? (no assumptions/RA)

I am trying to do truth tables and derivation but it doesn’t make sense could someone help me out?

Can one prove a deduction theorem for propositional or first-order logic using a metalogic that doesn't include induction?

Basically the idea is: The only reason people choose action A is because they think that everybody else in the sample will choose action A, and choosing anything besides A will put them at a disadvantage given that everyone else chooses A. Now everybody would prefer to not choose action A, but only do so because they believe that they’ll be the only ones that haven’t.

Real world example in case my wording sucks: Say you have an election and everyone hates the two major candidates. People would prefer to vote for NOT those two, but because they believe that everyone else is going to vote for one of those two, they believe they MUST vote for one of the two.

I think this is bad logic, but I see so many people utilizing it and it pisses me off… regardless, is there a name for this?

PLEASE don’t bring politics into this NOT a political post, just an example.

I will provide an example:

There are 3 parents, one continuously has still borns, one is infertile, one is extremely unattractive to where they cannot find a partner at all.

Example 2:

Person 1 fails their test because of procrastination, person 2 fails their test because of anxiety , person 3 fails their test because their car breaks down on the way to school.

It should be concluded that in either example, the severity is the exact same for all situations given that the outcome is the same, however this often does not happen.

K(A, B, C) = A - AB' + B'C'

Hello, I'm working through An Introduction to Formal Logic (Peter Smith), and, for some reason, the answer to one of the exercises isn't listed on the answer sheet. This might be because the exercise isn't the usual "is this argument valid?"-type question, but more of a "ponder this"-type question. Anyway, here is the question:

‘We can treat an argument like “Jill is a mother; so, Jill is a parent” as having a suppressed premiss: in fact, the underlying argument here is the logically valid “Jill is a mother; all mothers are parents; so, Jill is a parent”. Similarly for the other examples given of arguments that are supposedly deductively valid but not logically valid; they are all enthymemes, logically valid arguments with suppressed premisses. The notion of a logically valid argument is all we need.’ Is that right?

I can sort of see it both ways; clearly you can make a deductively valid argument logically valid by adding a premise. But, at the same time, it seems that "all mothers are parents" is tautological(?) and hence inferentially vacuous? Anyway, this is just a wild guess. Any elucidation would be appreciated!

https://preview.redd.it/bk8cotoeolxd1.png?width=1883&format=png&auto=webp&s=bcf2e46ffba967a72fcd4723c08ac5fc1041a67c

Consider a language L with only unary relation symbols, constant symbols, but no function symbols. Let M be a structure for L. If I have a sequence of subsets M_n of M with each M_n definable in an admissible fragment L_A of L_{omega_1,omega}, can I guarantee that the intersection of M_n’s is also definable in L_A?

I know the answer is positive if the set of formulas (call it Phi) defining the M_n’s is in L_A.

My doubt is, what if Phi has infinitely many free variables?

Edit: Just realized Phi can have at most one free variable as the language has only unary relation symbols. Am I correct? Does this mean that the intersection is definable in L_A?

So I've been learning logic online but I really didn't get the vacously true statement part, I didn't understand it at the moment so I moved on thinking "It wasn't that important as it's 'exceptional case'" and now it has snowballed into me struggling with truth tables so yeah... Any help would be appreciated.

Premise:

(1) Everyone must belief in god (2) Not following any religion is permitted

'Not following any religion' has 2 subsets: Subset (a), do not follow any religion but belief in god. Subset (b), do not belief in god.

Question: does (2) contradict (1)?

I've read several explanations of this logic puzzle but there's one part that confuses me still. I tried to find an explanation on the many posts about it but I'm still lost on it. What am I missing?

• Each person can conclude that everybody sees, at most, two people with blue eyes and everybody knows that everybody knows that.

This is because each person independently sees that at most one person has blue eyes and it's themselves. So they will be thinking that everyone else may see them with blue eyes and wonder if they're a second person with blue eyes, but then they'd know that at most two people have blue eyes, the person hypothesizing this, and themselves. However, this can't go any further because you know that under no curcumstances will anyone see two or more people with blue eyes.

So it seems to me that everyone can leave on the third night, not the 100th.

i’ve been stuck on this for an hour and a half and i still can’t figure it out. i’m only allowed to use rules for conjunction disjunction. i can’t figure out how to derive B

Prove that for all formulas A and B:

1. A ⊨ B and B ⊨ A if and only if A ↔ B;

2. A and B have all their logical consequences in common if and only if ⊨ A ↔ B.

I am a beginner in logic, but I can’t manage to do 2. In fact, let's imagine that A has as its only consequence "there are cherries." Let's imagine that B has as its only consequence "there are cherries." Let's imagine that A is "there are apples" and B is "there are pears." Suppose that if there are apples, there are cherries, and if there are pears, there are cherries. I don't see how this implies that if there are apples, there are pears.

Hello everyone,

I'm not very advanced in mathematics; I’m currently in my first year of university. I recently encountered the "drinker’s paradox," which asks if there is always someone PPP in a bar such that if they drink, then we know everyone else in the bar also drank. The question is : is there a guest P in every bar so that if P drinks -> we know for sure that everyone else drunk ?

My answer is: the statement is true in every case, simply due to the existence of someone in the bar.

• If everyone is drinking, then anyone can be PPP.
• Otherwise, PPP would be the "last" person who would drink if everyone else did. This means that if not everyone drinks, PPP also wouldn’t drink, as they would only drink if everyone else drank before them.

My answer was rated as incorrect without much explanation, and I’m not entirely convinced. I believe that PPP always exists, even if not everyone is drinking (in which case, PPP simply wouldn’t be drinking).

I’m feeling a bit confused and would appreciate any help in understanding this better.

Thank you, everyone!
P.S. I’m studying computer science, but I really enjoy Logik and am glad to have found this subreddit.

Hey all! Came across an interesting logical paradox the other day, so thought I'd share it here.

Imagine this: I offer you a game where I flip a coin until it lands heads, and the longer it takes, the more money you win. If it’s heads on the first flip, you get $2. Heads on the second? $4. Keep flipping and the payout doubles each time.

Ask yourself this: how much money would you pay to play this game?

Astoundingly, mathematically, you should be happy paying an arbitrarily high amount of money for the chance to play this game, as its expected value is infinite. You can show this by calculating 1/2 * 2 + 1/4 * 4 + ..., which, of course, is unbounded.

Of course, most of us wouldn't be happy paying an arbitrarily high amount of money to play this game. In fact, most people wouldn't even pay $20!

There's a very good reason for this intuition - despite the fact that the game's expected value is infinite, its variance is also very high - so high, in fact, that even for a relatively cheap price, most of us would go broke before earning our first million.

I first heard about this paradox the other day, when my mate brought it up on a podcast that we host named Recreational Overthinking. If you're keen on logic, rationality, or mathematics, then feel free to check us out. You can also follow us on Instagram at @ recreationaloverthinking.

Keen to hear people's thoughts on the St. Petersburg Paradox in the comments!

Why do we use conjunction rather than material implication when formalizing “Some S is P” . It would seem to me as though we should use material implication as with universal quantification no? I can talk about some unicorns being pink without there actually being any.

Construct a proof of the following fact: (Z ∨ T) ↔ P, Z, (P ∨ R) → ¬(Q ∨ T)   ⱶ  ¬(Q ∨ T).

Construct a proof of the following fact: ¬(P∨ Q)  ⱶ  A → ¬P. 

i need to proof these two examples and despite spending hours i cant figure it out

Hello,

I'm writing a brief newsletter for a nonprofit group and I noticed (or think I noticed) an issue with the statistics I'm quoting. I am not educated in philosophy and logic so I'm having a hard time wrapping my head around it.

The statements go like this:

"20% of persons with disabilities live in poverty."

"40% of people who live in poverty are disabled."

both statements refer to populations in the same country

Aren't these two statements referring to the same demographic, IE, people with disabilities who live in poverty? How can the percentages be different?

Not 100% which paper this is from but can anyone explain why the answer is B? And what is the difference between B and D. Most of the people I’ve asked reached the conclusion that the answer is C as well, however our current understanding after breaking down the question is that it all breaks down into B? (Implies lack of extinguisher is related to the occurrence of car fires, however this also assumes the fire extinguisher can put out the fires?)

Hey, can someone please recommend me any resources that go over truth trees? I understand the concept of truth tables relatively well but I'm having some issues understanding truth trees.

I’m a beginner in logic. Here is an exercise. I’m struggling with question 3. I answered that we cannot conclude A iff B from these tautologies because I made a truth tree with (A → B) ↔ (B → C) and ¬(A ↔ B), and I found that with A, ¬B, and C, both formulas (A → B) ↔ (B → C) and ¬(A ↔ B) are true. But ChatGPT o1 told me that I was wrong, and I’m having trouble understanding its explanation. Can you show me where I made a mistake?

Here is the exercise:

In each of the three questions, A and B are propositional logic formulas about which nothing is known in advance.

1. Demonstrate that if we have:

A ⊨ A → B and B ⊨ B → A then we have: A iff B.

2. If there exists a formula C such that A ⊨ C and B ⊨ C, can we conclude that A iff B ? (justify the answer).

3. If for any formula C we have:

A ⊨ C iff B ⊨ C,
can we conclude that A iff Bb? (justify the answer).

Hello everyone, I'm currently working on a problem in propositional logic and I'm having trouble verifying whether a set of premises logically entails a conclusion. The problem is about finding which values of  X  make the following implication true: 

Problem Statement:  

Given the premises:  A ∧ X  and  X → ¬ B ,   determine for which  X  it holds that  A ∧ X, X → ¬ B ⊧ C → (A → B) . 

I was given three options to consider as potential values for  X :  

1.  C → ¬ A   

2.  A ∧ C   

3.  ¬ B    

To tackle this, I’ve tried creating truth tables for each potential value of  X  and checking if the conclusion  C → (A → B)  holds whenever the premises are true. However, I’m having difficulty determining the correct logic behind this and interpreting the results from the truth tables correctly. 

Let us say a formula A is structurally consistent for a certain consequence relation iff, for any substitution s, there is a formula B such that s(A) doesn’t imply (with respect to the aforementioned relation) B.

Correct me if I’m wrong, but in classical logic the only structurally consistent formulae are tautologies, right? Contradictions are structurally inconsistent, and we can always find a substitution that maps a contingency onto a contradiction. (Or so I think. I have an inductive proof in mind.)

Are there logics/consequence relations without any structurally consistent formulae? Any other cool facts about this notion?

A fallacy wherein "understanding" something requires being within its own specific in-group.

For example (not a political statement just a demonstration) if someone says that "you have to be a Republican in order to understand Republican ideology" or similar?

Is there a name for this?

Hoping someone here has experience teaching logic at the high school level! I need some advice…

I teach an elective philosophy/critical thinking class to high school juniors and seniors. I just introduced the basics of inductive reasoning and how it contrasts with deductive.

My question is what kinds of inductive arguments should I teach? They already know how to identify strong vs weak / cogent / uncogent, but I don’t want to get too far into the weeds with a dozen types of inductive argument forms.

Can anyone recommend where to go from here?

Thanks!

Would be very grateful 🙏