Specializations > Semantics and Pragmatics

Propositional Logic and "...I don't believe you"

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**capjac145**:

I am curious about propositional logic.

How would a sentence like "It isn't raining, and if it is, I don't believe you" be translated into propositional logic?

I was thinking something like: (¬p∨(p→q))

Or would it need the biconditional? Or would it be something else all together?

Any insight about this would be great :)

**Daniel**:

I think the notation you wrote reveals that you intended a different version of the original sentence:

It isn't raining, and if you say it is, I don't believe you.

(Alternatively you would be opening yourself up to false beliefs and false assertions, such that you're saying maybe it's raining but stating that it is not. That's anomalous and I'm not sure how to analyze it.)

There is also a complication of the phrase "believe you", which is I guess some sort of grammatical metonymy where "you" is substituted for "what you said".

Regardless, this is going to involve some sort of embedding of propositions beyond what can be expressed with basic predicate logic. You would need second-order logic of some sort including potentially dealing with issues of factivity (compare "I know P" versus "I believe P", where only in the first is "P" entailed as true).

As a general approach, you can try substituting complex expressions for simpler parts, then building up the complex expression. (That still won't account for more advanced issues like quotation, though.)

--- Quote ---Or would it need the biconditional?

--- End quote ---

Biconditionals don't generally correspond to natural language conditionals,* except in formal statements like "if and only if". Material implication refers essentially has a gap in the relevant truth conditions because it is unimportant what happens when "[if] P" is false, having no bearing on the outcome of Q one way or the other, so it is always true if P is false. That would seem to apply here as well. You are not making any sort of assertion about whether you would believe them if they said something other than that it is raining. (And again this would get into complex issues of embedded propositions if you wanted to consider the case of the other person saying it is not raining, but regardless that is not specifically accounted for in the original statement-- maybe they are a pathological liar so if they agreed with you, instead of believing them, you would question your own beliefs instead!)

[*In fact, no simple propositional logic corresponds to all general usage of conditionals. There has been a lot of research on this topic, and it's interesting but complex. Among other issues are conditionals of the type "There's food in the fridge if you get hungry."]

**capjac145**:

Thanks :D

You're right. I am curious about "believe" still, what about something like, You don't mean it, and if you do, I don't believe you.

Would that be like (¬p∨(p→¬q))

or it is it more like that if and only if that you mention?

**Daniel**:

Again, there's a literal contradiction there: first you assert not-P, then you conditionally allow for the possibility of P. (There's nothing technically wrong with writing out propositional logic like that, but it's basically meaningless as an English sentence.)

We're forced to repair it by adjusting the assertions, such as interpreting the first part to mean something like "I think you don't mean it, but..."

In short, basic propositional logic corresponds only to a narrow subset of natural language utterances (and sometimes only approximately). In order to tackle more complicated issues like belief, you will need to get a lot deeper into the research on these topics. They've all been researched extensively, but that will be a lot of reading. One problem is trying to jump ahead to analyze sentences beyond the scope of what your currently available tools allow. There's a lot more to be added to a complete theory than basic propositional logic, but it begins to break down when trying to stretch it to new sentence types without revising the foundation of the theory. While analogy and the sort of what-about-this approach you're taking can often be useful in exploring new aspects of scientific analysis, in the case of a formal system like propositional logic, it just doesn't stretch like that, because it's not meant to. It applies only within very strict parameters, which is why it's consistent and useful in the first place. It can be extended, but not just by looking at new sentences and trying to bend the expressions to fit. As somewhere to start (at least to understand why this must be the case), see: https://en.wikipedia.org/wiki/Second-order_logic

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