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In this You Tube video, at 3:39 : https://www.youtube.com/watch?v=enZpq8jvFEs

Monsieur Phi ( understand " Monsieur Philosophie") a philosophy teacher ( and also a logician) gives a visual presentation of Euclid's deductive system of geometry as an ordered set that looks like a lattice.

Is actually a deductive theory a lattice? ( It seems doubtful however, since the axioms of a theory are all minimal elements).

If not, how to caracterize formally the order of a deductive theory?

Are there different possible types of order for deductive theories?

Is it possible for two deductive theories to instantiate two different kinds of order?

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    $\begingroup$ You might be interested in Lindenbaum–Tarski algebras. It is indeed the case that the Lindenbaum–Tarski algebras for different deductive systems have different structures. Conjunction and disjunction give the meet and join operations of a lattice; this lattice is a Heyting algebra if the system is intuitionistic, and is a Boolean algebra if the system is classical. This isn't exactly what you're asking, though, which is why I haven't posted this as an answer. $\endgroup$ – Clive Newstead May 4 at 13:50
  • $\begingroup$ @CliveNewstead. Thanks for this link. $\endgroup$ – Ray LittleRock May 4 at 13:54

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