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Propositional logic - Help translating statements into math

In philosophy, we've just started talking about propositional logic, and I was wondering how it would be possible to rewrite the 8 laws of immediate deduction (in the attached file) into a math equation.
When reading those statements in french, there are statements that sound ambiguous, which is why I'm searching for a mathematical interpretation.
(S'il est vrai que certains dentistes sont menteurs, alors l'affirmation "Tous les dentistes sont menteurs" est [Indéterminée], {mais la façon que c'est écrite me fait croire que c'est [Faux]})
(If it's true that some dentists are liars, then the affirmation that "All dentists are liars" is [Indeterminate], {but the way it's said makes me think it's [False]})
Otherwise, I'll have to learn those 24 phrases by heart for an exam.
Perhaps those statements could help summarize the laws?
(¬∀⋅A)⇔∃x⋅¬A\left( \neg \forall \cdot A \right) \Leftrightarrow \exists x\cdot \neg A
(¬∃⋅A)⇔∀x⋅¬A\left( \neg \exists\cdot A \right) \Leftrightarrow \forall x\cdot \neg A
Logic.png94.5KB
Original post by MartyO
In philosophy, we've just started talking about propositional logic, and I was wondering how it would be possible to rewrite the 8 laws of immediate deduction (in the attached file) into a math equation.
When reading those statements in french, there are statements that sound ambiguous, which is why I'm searching for a mathematical interpretation.
(S'il est vrai que certains dentistes sont menteurs, alors l'affirmation "Tous les dentistes sont menteurs" est [Indéterminée], {mais la façon que c'est écrite me fait croire que c'est [Faux]})
(If it's true that some dentists are liars, then the affirmation that "All dentists are liars" is [Indeterminate], {but the way it's said makes me think it's [False]})
Otherwise, I'll have to learn those 24 phrases by heart for an exam.
Perhaps those statements could help summarize the laws?
(¬∀⋅A)⇔∃x⋅¬A\left( \neg \forall \cdot A \right) \Leftrightarrow \exists x\cdot \neg A
(¬∃⋅A)⇔∀x⋅¬A\left( \neg \exists\cdot A \right) \Leftrightarrow \forall x\cdot \neg A


I'm not sure exactly what your question is. However, you almost certainly don't want to learn 24 such statements - your approach using quantifier symbols looks sensible, but you also need to develop some intuition about why those statements are true.

As for the French sentence, it looks to me to be precisely equivalent to the English translation, and I see no ambiguity there. And who wrote the final part?:

"{but the way it's said makes me think it's [False]})"

You? Or was this part of what your lecturer told you? I'm slightly confused here.
Reply 2
Avatar for MartyO
MartyO
OP
7
The way the lecturer puts it, if it's true that some dentists are liars, then the statement that all dentists are liars is indeterminate.

That seems to me like the keyword "some" is inclusive of all dentists, otherwise the second statement would be false, as in: if some dentists are liars, some dentists are not liars. But, what makes it indeterminate is that we don't know if there exists dentists that are not liars.

I'm trying to write the expressions mathematically, like the 5.b) might become:
∃D,L(D)≡True⇔∀D,L(D)≡Indeterminate\exists D,\quad L(D)\equiv True\Leftrightarrow \forall D,\quad L(D)\equiv Indeterminate
There exists some Dentists for which Lying(Dentist) is true, therefore, the statement that for all dentists, Lying(Dentist) can be true or false,
(it would yield false if there exists a Dentist for which Lying(Dentist) is false,
and it would yield true if there are no Dentists for which Lying(Dentist) is false).

If I am to write the statements as mathematical expressions, I want to do it correctly from the start, since I'll be confronted again with propositional logic in discrete mathematics in two semesters.
Being able to do these kinds of things would also help me in computer science (and my programming teacher already said that conditional "else if" branches are inclusive disjunctions, instead of exclusive disjunctions).

So, how would I be able to translate word for word the expressions in math?

5. c) is easy to translate, because it's the direct opposite of the statement:
∃D,L(D)≡True⇔∄D,¬L(D)≡False\exists D,\quad L(D)\equiv True\Leftrightarrow \nexists D,\quad \neg L(D)\equiv False

But then, how can De Morgan's laws be applied?

And the 5.a) gets complicated:
∃D,L(D)≡True⇔∀D,L(D)≡Indeterminate∵∃D,¬L(D)≡True∨False\exists D,\quad L(D)\equiv True\Leftrightarrow \forall D,\quad L(D)\equiv Indeterminate \\ \because \exists D,\quad \neg L(D)\equiv True\vee False
De Morgan.PNG22.5KB
Reply 3
Avatar for InOrbit
InOrbit
9
You can think of the universal quantifier as a 'big conjunction' and the existential quantifier as a 'big disjunction'. Not only does it make sense to think of them this way, but it makes De Morgan's rules work for the quantifiers too.

I haven't done this in a while, but if I was to translate 5a:
∃D(L(D)=True)⇒∀D(¬L(D)=Indeterminate)\exists D(L(D) = {True}) \Rightarrow \forall D(\lnot L(D) = {Indeterminate}).
Three valued logic is weird.
(edited 7 years ago)

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