# Natural language into propositional logic

Need some help putting these two examples of natural language into sentence logic. For reference, use the transcription guide below:

D = you think so; E = I think so; F = it is true

1. If you think so, I think so. And if I think so, you think so. (is it possible to express this using just one connective?)

2. Unless it isn’t true, you don’t think so.

• Seems like a homework question and you are not showing any effort. Commented Mar 29, 2019 at 14:28
• These may fly in Mathematical logic but these are not even meaningful propositions by Philosophy standards. They would need more details than provided. The point of deductive logic is to prevent or reduce deception or ambiguity traps. Mathematical logic doesn't always adhere to that purpose. They do their own thing. Commented Mar 29, 2019 at 22:48
• How about the fact that I’m trying to learn all of this out of a book and need a little help that I cannot get elsewhere? I put in plenty of effort, it’s just when I’m not totally sure of an answer, I like to receive confirmation before I move onto a different problem set or topic. I don’t understand why you have to pass judgement. Commented Mar 30, 2019 at 4:39
• You have not shown that effort -- that is, you have not posted what you have tried so we might offer advice on where you are having trouble. Commented Mar 30, 2019 at 5:07

1. This sentence is a conjunction of two conditionals:

(D→E) ∧ (E→D)

You can put it into a single connective by using CB: D↔E

1. I was taught that "unless" is a flag for the "or" connective, so I will write my answer like that. If you rewrite the sentence to "You don’t think so unless it isn’t true", then the logic you get is:

(~F)∨~D

• Thank you so much for your response. For #2, however, I was recently taught that “X, unless Y” is the same (usually) as ~Y > X. Would it be possible to write it out then as ~~D > ~F, which would just be D > ~F? Commented Mar 29, 2019 at 2:53
• Hello, if you were taught that "X unless Y" meant X ∨ ~Y, then "You don’t think so unless it isn’t true" would be "You don’t think so" ∨ ~"it isn’t true" = ~D ∨ ~~F = ~D ∨ F Commented Mar 29, 2019 at 4:29
• Unless is not always a contropositive. Unless expresses a negative term. For example, you will fail this class unless you score an 85 or above. This would be if you do not score 85 or above then you will fail this class. That is not a contrapositive. Another example, you are hell bound unless you accept Christ as a savior. This means if you do not accept Christ as a savior then you are hellhound. Notice whatever verbiage after UNLESS becomes the antecedent of the conditional. You can perform logical equivalence after to translate it correctly. Commented Mar 29, 2019 at 22:44