7/27/26
Introductory video to the games
Main connective
The notion of the 'main connective' is important in many of logic. It is central to the evaluation of truth tables and it is central to many rules of inference. Make sure that you can get 10 out of 10 right in 30 seconds. [Sample formula M ⊃ (J ∨ ~J) ]
Truth tables
Satisfiability
Here the idea is to try to produce a valuation for the atomic propositions that will make the whole formula true (or 'satisfy' it). This cannot always be done (for example A∙ ~A is not satisfiable). And, if it can be done, sometimes there are different ways of doing it (for example, A∨B is satisfiable with A true and B false or with B true and A false or with A true and B true). Here you 'toggle' the atomic propositions (not the connectives) to set them true or false.
Make sure that you can get 10 out of 10 right in, say, 2 minutes.
[There is a mechanical, or semi-mechanical, way of finding assignments of truth values to satisfy a formula, if that is possible, using what is known as 'Trees'. But, whether or not you learn about trees, there is value in exploring the task by hand to get some sense of what 'satisfiable ' means and what has to be done to establish it.]
Consistency
If it is possible for all of a collection or list of formulas to be true at one and the same time (ie they are simultaneously satisfiable), those formulas are Consistent. Try the Consistent exercise. Make sure that you can get 10 out of 10 right in, say, 7 minutes.
[As above, there is a mechanical, or semi-mechanical, way of finding assignments of truth values to simultaneously satisfy several formulas, if that is possible, using what is known as 'Trees'. But, whether or not you learn about trees, there is value in exploring the task by hand to get some sense of what 'simultaneously satisfiable' means and what has to be done to establish it.]
Invalidity
If it is possible for all of the premises of an argument to be true, and the conclusion false, at one and the same time, that argument is Invalid (and the assignment that does this amounts to a Semantic Counter Example) . Try the Invalid exercise. Make sure that you can get 10 out of 10 right in, say, 7 minutes.
[Yet again, as above, there is a mechanical, or semi-mechanical, way of finding semantic counter examples to sentential arguments, if that is possible, using what is known as 'Trees'. But, whether or not you learn about trees, there is value in exploring the task by hand to get some sense of what 'semantic counter example' means and what has to be done to establish it.]