I'm trying to to determine whether the following is a tautology, contingency, or contradictory:

(p ⊃ q) ∨ (q ⊃ p) .⊃. (r .⊃. s ⊃ r)

This is school work. I'm getting that it's a tautology, but only through looking at patterns of solutions of textbook problems and exercises. I would like to know what exactly is meant by the dot prior and after the material implication (.⊃.)?

I understand that material implication - in a truth table - has false only when p is true and q is false. I also understand the or operator where it's true when either both or one of the two, namely: p or q is true. I just need to understand what the dots are supposed to mean. I couldn't find good explanations online.

  • 1
    You shouldn't be using more than one conventional notation. If the dots express what parenthesis does why are you using BOTH simultaneously. PICK only ONE -- not both. If you don't use dots at first then use brackets after parentheses and so on. ( ) then [ ] then { } and so on. There should be no dot notation if you did not start that way.
    – Logikal
    Sep 26, 2018 at 13:41

2 Answers 2


The dots function like parentheses: they disambiguate an otherwise ambiguous expression. In this case the expression is equivalent to:

((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r))

Note that without the dots the consequent would be:

(r ⊃ s ⊃ r)

Which is ambiguous between the following two:

((r ⊃ s) ⊃ r)

(r ⊃ (s ⊃ r))

The dots just tell you which connective is the main one.

  • I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
    – wa7d
    Sep 25, 2018 at 2:48
  • 1
    @wa7d If in doubt, parentheses are universal and unambiguous!
    – Cai
    Sep 25, 2018 at 10:20
  • The Principia Mathematica dot notation has been out of style for many decades and should really only be encountered in historical texts. Sep 29, 2018 at 15:11

The use of dots like that is an alternative to using nested parentheses. Putting dots around a connective indicates that its binding priority is lower than that of another connective in the sentence. Your sentence could also be written ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r)).

To give a simpler example:

A ∨ B ⊃ C

is syntactically ambiguous. It could mean

(A ∨ B) ⊃ C


A ∨ (B ⊃ C)

An alternative way to specify the first version would be to write:

A ∨ B .⊃. C

It is a matter of style as to which you prefer, but the dot notation is not as common as it used to be.

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