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I found this: English to Logic essay

It is very clear right up to the paragraph on material implication. In that paragraph it gives many examples of English language expressions that can be construed to be essentially an "if-then" sentence core. Then it gives about as many examples that seem to be the same, i.e. a possible "if-then"sentence core, but are not. Then the author states that "most speakers" are asserting the truth value of one of the propositions when they use one of these latter sentence cores and that is a source of multiple problems.

This leads to the question. Language is not static. What may have been a clear implication in the 1950 English that leads to a clear "if-then" statement might not do the same in 2020. Is propositional logic language dependent? Of course, the answer is yes on the surface. However, if you grew up and learned English in the corn belt in the 1950s, you do not speak the same English as a 20 year old native New Yorker. That the two of you can communicate with each other is attributable to the flexibility of the human brain. The symbolic language of propositional logic is not flexible in the same way.

kimchi lover
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  • I am not sure what you are after. The non-examples the text quotes seem to all express causality (a concept not captured by propositional logic) instead of implication. At any rate, I would call the abstract concept of propositional logic independent of the natural language used to sometimes express relations between propositions that are somewhat like some relation in propositional logic. The meaning of "$\to$", say, is perhaps much stricter and more constant than the meaning of "if" - but that's because we spend pages defining it for the sake of universality and unambiguous durability – Hagen von Eitzen Mar 14 '20 at 21:16
  • The issue for me is language. In a particular statement, I do not see any causality, for instance, but you do. I write p → q, you do not. Is that a language issue? If it is, how would I "correct" the language I speak so that it matches yours? DO I just learn a new language? If it is not a language issue, then there could be a list of expressions that meet the criteria for p → q and a list of those that do not. Probably not all inclusive lists, even if they existed. –  Mar 14 '20 at 21:55
  • The formal model of natural language defined by propositional logic is a very very simplified one: formal languages are used to formalize computer programs and mathematical theories. One of the implicit assumptions is that mathematical truths are not time-dependent. – Mauro ALLEGRANZA Mar 15 '20 at 09:46
  • @MauroALLEGRANZA, Your comment prompted me to think about my question again. It is not the language of propositional logic that I am asking about, nor is it the language spoken by the logician. If I say I speak English, it does not mean that I speak the same English that another speaks. Even if the difference is small, there is the possibility that my translation and the other's translation will not be the same. –  Mar 15 '20 at 13:15
  • An example of why I have this question: https://math.stackexchange.com/questions/3460037/logical-expression-that-someone-takes-a-class-and-talks-to-others-in-the-class –  Mar 16 '20 at 17:26
  • Great question! I wish I had seen it sooner. I have a related question: how much do mainstream logic theories owe to the syntax of (Indo-)European languages? – kjo Apr 18 '22 at 10:59

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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.

  • Note however that changes in the meaning of words is not the only factor that can cause a change in the truth values of sentences. Changes as to the facts of the world can also do this. Consider the case of Pablo who lives in Texas in 1844. At this time, the conditional

" 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).

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