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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 where 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.

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  • $\begingroup$ It is .⊃. (⊃ is material implication) $\endgroup$ – wa7d Sep 25 '18 at 2:26
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The periods are a (somewhat archaic, I think) way to indicate how the sentence should be parsed, without adding more parentheses. So:

$$r .\supset. s \supset r$$

should be read the same as:

$$r \supset (s \supset r)$$

because connectives without periods take precedence over ones with periods. Some books will only put periods on one side or the other to control how they behave on the left vs. right as well. So:

$$r \supset s .\supset. r \supset s \equiv (r \supset s) \supset (r \supset s)$$

but

$$r \supset. s \supset. r \supset s \equiv r \supset (s \supset (r \supset s))$$

Some of this is still visible even in notation that doesn't use this convention in general, though. It is common to see something like:

$$\forall n. P(n) \supset P(s\ n) \equiv \forall n. (P(n) \supset P(s\ n))$$

where the period is simply considered part of the syntax of $\forall$, but the parsing rules are similar to the rule for . earlier, and in a syntax where the period has its own meaning we might be able to also write:

$$\forall n P(n) \supset \forall n P(s\ n) \equiv (\forall n P(n)) \supset (\forall n P(s\ n))$$

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  • $\begingroup$ I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks! $\endgroup$ – wa7d Sep 25 '18 at 2:49
  • $\begingroup$ The dot notation is used in Russell and Whitehead's Principia Mathematica written more than a century ago. It went out of style quite fast, being replaced by parentheses. You can look up page 10 of quod.lib.umich.edu/u/umhistmath/AAT3201.0001.001/… for an explanation of its use. $\endgroup$ – Bernard Massé Sep 25 '18 at 3:21

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