While it’s perhaps not immediately evident, the statement It is not Tuesday or it is raining does at least imply the statement If it is Tuesday, then it is raining even in everyday language: if the first statement is true, and if today really is Tuesday, then it must in fact be raining. The second probably does not imply the first in most people’s everyday language, because in everyday usage if ... then is normally taken to imply some connection $-$ perhaps not a truly causal connection, but something along those lines. Material implication ($\to$) is definitely not the same as everyday if ... then, so a formal equivalence between statements involving material implication may not translate into an equivalence between the apparent everyday counterparts.
The same problem arises to some degree with disjunction ($\lor$). In everyday usage the word or is often closer to exclusive or ($\veebar$) than to ordinary inclusive $\lor$, especially when preceded by either: It was either John or Charles; Apologize to your sister, or leave the table!
More generally, there’s a problem translating between formal and everyday language. Consider the statement Touch me, and you’ll lose some teeth! On the face of it that is $p\land q$, where $p$ is you touch me, and $q$ is you will lose some teeth, but the purely formal version closest to the real sense is $p\to q$.
The inference is valid and derivable.
In general, a counter example to an invalid inference consists of a structure in which all the premises are true but the conclusion is false.
Your argument is incorrect because it does not precisely show (by the means of providing a concrete counter model) that this interpretation of the predicates invalidates the sequent, and your English example is not an appropriate deformalization of the statement: You can't just drop one of the disjuncts in each of the sentences. If $R(x)$ is to stand for a rough object and $S(x)$ smooth, then the argument is "If exists an object which is rough or smooth, then either there exists a rough object or there exists a smooth one".
Normally it is advisable to start a natural deduction proof from the bottom to the top, performing backwards introduction rules on the main operator until you can get no futher, then switching to the top and working your way down from the premises by successive applications of elimination rules until you hopefully meet in the middle.
In this case, the conclusion is a disjunction, but attempting disjunction introduction as the last step won't lead to success, because this would require a proof of one of the disjunctions, which is obviously not possible, because you can not infer either of the two sides for sure, only the disjunction of them. So go to the top immediately and start disassembling the premises.
The only premise you are given is an existential statement, so the first thing to do is an existential elimination. Existential elimination means that you assume the quantified statement for some arbitrary object (say $a$), derive some conclusion from this assumption, then, since you know that at least one such object exists, you can infer the conclusion for sure. The conclusion you want to arrive at is $\exists x R(x) \lor \exists x S(x)$, so try to derive that under the assumption $R(a) \lor S(a)$:
| exists x(R(x) v S(x))
| | R(a) v S(a)
| | -----------
| | ...
| | exists x R(x) v exists x S(x)
| exists x R(x) v exists x S(x)
The next thing from the top is the disjnction $R(a) \lor S(a)$. So you do disjunction elimination, which means that you assume each of the disjuncts, derive the same conclusion from both, then conclude that since at least one of the two sides holds, the conclusion follows for sure. The conclusion to infer at the end, and in the two subproofs, is again $\exists x R(x) \lor \exists x S(x)$:
| exists x(R(x) v S(x))
| | R(a) v S(a)
| | -----------
| | | R(a)
| | |-----
| | | ...
| | | exists x R(x) v exists x S(x)
| | | S(a)
| | | ----
| | | ...
| | | exists x R(x) v exists x S(x)
| | exists x R(x) v exists x S(x)
| exists x R(x) v exists x S(x)
That's the scaffold of the proof. Can you fill in the ...'s on your own?
Best Answer
I can see two interpretations of the question " is propositional logic language dependant"?
First : does the meaning of the connectives depend on the meaning of natural language connectives? I think the correct answer is that they do not. Connectives do not aim at mirroring natural language connectives. Connectives are defined independently as truth functions, that is functions ( in the set theoretic sense) from the set {T, F} to the set {T,F} or from the set {T,F}$\times${T,F} to the set {T,F}. These functions are represented by truth tables. So it is rather the other way round: when we read propositional calculus formulas, we use ordinary language connectives to " mirror" as much as we can the meaning of logical connectives.
Does propositional logic depend on the evolution of vocabulary, on changes in the meaning of words? The answer is, according to me, that the evolution of natural language causes no change in logic as such; it can only cause changes in the truth values of elementary sentences, because this value depends on the meaning of proposition, and therefore, it induces a change in the truth value of compound sentences in which the elementary ones are involved.
For example, in 1955, the proposition " Elvis Presley is an indecent artist" might have been considered as true, so, the following conditional was true in 1950:
"if Elvis Presley sings rock'n roll then Elvis Presley is an indecent artist".
( True antecedent, true consequent ; hence true conditional). But nowadays, due to changes in moral standards, and subsequent changes in the meaning of the adjective "indecent" the consequent is false. And since the antecedent is still true, the whole conditional is false.
" If Pablo lives in Texas, then Pablo lives in the USA"
was false. ( True antecedent, false consequent). Some years later, it was true. It became false again for a while , and then true after the Civil War.
We could say that logic by itself is a ( truth functional) machine that never changes, that always does the same work ( according to the same mechanism). But since the inputs change ( as to their truth value) the machine gives changing outputs. ( With one exception : when the " machine" produces a tautological formula, the output never changes, whatever change as to truth values the inputs might undergo).
Last thing. Logic by itself is not concerned with the actual propositions of natural language. In logic, propositions are simply the abstract objects $P_1$, $P_2$, $P_3$, etc and all the formulas that can be built with them according to the laws of logical syntax. The question of knowing whether this or that elementary proposition, say $P_2$, is actually true or false is meaningless: the only thing that counts is that each of them can be either true or false ( at least in bivalent logic).