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Logical equivalence: wedge–arrow

In this video, York student Joseph proves the logical equivalence of '(P arrow Q)' and '(tilde P vel Q)' using a truth-table.
Here’s another example of logical equivalence. It’s called the Wedge-Arrow Law and it’s about equivalences between disjunctions and material implications. The Wedge-Arrow Law says that ‘(P arrow Q)’is logically equivalent to ‘(not P or Q)’. So let’s construct the truth table. We’ll need two basic sentences, ‘P’ and ‘Q’, and lay out how things could be with their-truth values. Then the truth table for ‘(P arrow Q)’ is the same as for any material implication which is true, false, true, true. Now, let’s fill in the truth table for ‘(tilde P, wedge Q). ‘Q is going to be easy ‘ 00:46.400 –> 00:51.200 to fill in: we just copy it over - true, false, true, false.
‘Tilde P’ is false, false; true, true.
Now, we do vel: false,
true is true: false, false is false; True, true is true and true, false is true. Now, we can see the truth tables for ‘(P arrow Q)’ and the truth table for ‘(tilde P vel Q)’ are both the
same: true, false, true, true. So they are logically equivalent. Now, there are similar forms like this. Let’s see if we can find an arrow sentence equivalent to ‘(P vel Q)’. The truth table for ‘(P vel Q’) is true, true, true, false. Now, let’s think about arrows. This [‘(P arrow Q)’] is false when the antecedent ‘P’ is true and the consequent ‘Q’ is false, so going back to ‘(P vel Q)’ on this row, where it’s false, if we wanted a conditional which is false on that row, what could we put? Well, ‘Q’ on that line is false so we could try ‘….arrow Q’. But ‘P’ is false and we’re looking to have ‘true arrow false’, so we’d use ‘tilde P’.
OK, let’s fill in the truth table. ‘Q’ is true, false, true, false. ‘Tilde P’ is false, false, true, true. For
arrow: false, true is true; false, false is true; true, true is true; and true false is false. And again, we have equivalence. So, as with De Morgan’s Law, there is a pattern for converting arrows into wedges and back again.

This video shows a truth-table proof of the Wedge–Arrow Law. This law says that ‘(P (rightarrow) Q)’ is logically equivalent to ‘(~P (vee) Q)’.

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Logic: The Language of Truth

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