A set Γ entails the statement P if and only if there is no truth-value assignment in which every member of Γ is true and P false.
This is a standard def of entailment, check any textbook on formal logic. I would include a reference but that might be a product endorsement?
A simple truth table will show a row where α is true and β is false so ~( α|= β) and ...
The question is how to understand why (A → B) v (B → A) is always true even when A and B are sentences that have nothing to do with each other. The reason is that the truth values for the conditional (→) and the disjunction (v) are defined to be true for three out of the four possible sets of valuations of A and B in such a way that their combination always ...
I'm not sure if I completely understand your question, but it sounds to me like you've divided the argument into two positions:
Vaccines are good.
Vaccines are evil.
Have you considered the possibility that the truth could be somewhere in between? We might argue that the first vaccines were developed by people who wanted to help other people, supporting ...
This might help on some questions I read inbetween the lines; Is everything just an opinion? What I learned (and is key) here is "probably". The more sources you have, the more likely it is that something might be true. That might not help in the conversation but it could help to mention that you can disprove almost anything if you are very sceptic and ...
I am trying to get this proof to work out and so far I feel like I have the first part right but I'm stuck on how to get the A→B part.
Begin by assuming A → B, then state LEM as a Tautological Consequence to use disjunction elimination to derive ¬A ˅ B. Also, your attempt needs to be tidied up, you have a few unecsessary steps and you seem to forget that ...
Here is a very similar question with three answers: Prove (¬P ∨ Q) ↔ (P → Q) Your question seems to be focused on the Fitch system and your approach to the problem seems to be different. I will only address how you might proceed without considering how you might do this in the Fitch system.
You are attempting to use disjunction elimination with the ...
1) The burden of proof is borne by whoever makes the claim that is easiest to prove.
Absolute non existence of something is impossible to prove, but please notice how your rephrasing into :
"There exists at least one universe in which Object A does not exist."
also narrowed the conditions. If you were to narrow it further into:
"A glass of water does not ...
Since one has to prove ¬E one place to start might be E as an assumption. The strategy is to derive a contradiction somewhere after that assumption and then derive ¬E which is the goal.
The rest of the steps I think you are aware of. I am including the results of the proof checker. The proof checker you are using is likely different and you will need to ...
It's invalid. When there can be a world in which the premises are true and the conclusion is false, the argument is invalid. So if you wanted to use Tarsky's World as tool, you would need to use block language. So lets make:
P = Tet(a)
Q = Small(a)
R = Cube(b)
S = Large(b)
(these are just random predicates that don't change the meaning of the argument)
By using a truth table generator one can show that one of the set of valuations for the sentence letters leads to the result being false.
To see this, use this input ((P=>Q)&&(R&&S))=>(~P=>(Q&&R)) in the Stanford Truth Table Tool. When P and Q are false and R and S are true then the conditional with the conjunction of the premises ...
P → Q, R ˄ S |- ¬P → (Q ˄ R) is not syntactically valid.
An assumption of ¬P will not allow you to eliminate the conditional in P → Q to derive Q .
Is there a typo?
P → Q, R ˄ S |- P → (Q ˄ R) is valid.
1| P → Q
2|_ R ˄ S
3| R ˄E, 2
4| |_ P
5| | Q →E, 1,4
6| | Q ˄ R ˄I, 3,5
7| P → (Q ˄ R) →I, 4-6
A burden of proof falls under the auspices of speech acts. Anyone disrupting a conversation by bringing disagreement to it has an obligation to make things right, as it were, and explain one's self. It is completely a social construct. This is the principle of "onus probandi", be the application in general conversation or in a legal setting.
A burden of ...