Here are some examples of valid arguments in English. (I’ve borrowed these from Pospesel, Arguments: Deductive Logic Exercises.) For each argument, first symbolize it, then construct a derivation to show that it is valid.
The purpose of these exercises is two-fold. First, they give you more practice symbolizing, and more practice constructing derivations. But, second, they help bring the two tasks together, showing you how we can use our tools to show that actual arguments presented in English are valid.
The Pragmatic Justification of Induction (see Salmon, The Foundations of Scientific Inference, pp. 52-54):
If there are uniformities in nature, then induction will work. If there is a method of inference about the future which works, then there are uniformities in nature. So, if induction does not work, then no method of inference about the future will work.
Do We All Descend from Adam and Eve?
Given that Cain married his sister, their marriage was incestuous. Adam and Eve were not the progenitors of the entire human race if Cain didn’t marry his sister. Hence Adam and Eve were the progenitors of the whole human race only if Cain’s marriage was incestuous.
Meaning is Conventional (see Locke, Essay Concerning Human Understanding, II 8):
The connection between words and the ideas for which they stand is arbitrary. Humans don’t share a common language. If there were a natural connection between words and the ideas they stand for, all humans would share a common language. And, further, the connection between words and the ideas is arbitrary if it is not natural.
‘Santa Claus does not exist’ is false (For discussion, see Quine, “On What There Is” Review of Metaphysics 1948; Richard Cartwright, “Negative Existentials,” Journal of Philosophy 1960, 629-630)
The sentence ‘Santa Claus does not exist’ is about Santa Claus. If it is about Santa Claus, then there is such a thing as Santa Claus. But if there is such a thing as Santa Claus, it is false. So the sentence ‘Santa Claus does not exist’ is false.
There is design in the world. Provided that there is design, there must be a designer. So there must be a designer of the world.
Consider the sentence, L: ‘L is not true’. Here are two valid arguments about L:
If L is true, then L is not true. So L is not true.
If L is not true, then L is true, so L is true.
Symbolize both arguments and show that they are valid. For your scheme of abbreviation, let T represent the sentence, ‘L is true’.