wondering about that. if a set of linear properties is defined as linear if it follows for every $a,b \in \sum$ $a \Rightarrow b$ or $b \Rightarrow a$
if i wanted to prove that a set of propositions I is defined such that: I only consists of a single elementary proposition p, and the only logical property is $\rightarrow$, how can i prove that I is linear according to the given definition(for every $a,b \in \sum$ $a \Rightarrow b$ or $b \Rightarrow a$)
and to help understand this concept better, if i wanted to show that if $\sum$ is a set of linear propositions and each proposition in $\sum$ has a model in which the proposition holds true, how can i show that there exists a model for $\sum$?
my attempt:
for the first: we'll check for a single proposition: if I consists only oneproposition p such that the only logical operator is $\rightarrow$ that means that for every $a,b \in \sum$ $a \Rightarrow b$ or $b \Rightarrow a$. we'll assume it holds true for every n amount of elements in $\sum$ ad we'll check for validity for n+1 number of elements. here i have my problem - if we have more than two elements(n+1 in this case), how can we deduct that I will be linear, at about infinite elements, i think that there is a case where it might be problematic.
for the second: as far as i understand, if we can show that if $\sum$ is linear aמd each poperty in $\sum$ has a model in which the proposition is true, we can deduct that $\sum$ is not bounded, and then using the induction i couldn't yet prove, we can deduct that if all of the above holds true, then there exists a model for $\sum$.
please help me fix and improve that, using the correct mathmatical writing or sohw me what i've done wrong.
thank you very much!