Remember that an argument is valid if the premises cannot be true and the conclusion cannot be false. So we look for a row on the truth table that contains all true premises but a false conclusion. If there is, we can conclude that the argument is flawed. As a result, this argument is sound.
The direct argument from this statement of the principle of contradiction is not valid because there are rows on the truth table where all the premises are true but the conclusion is false. For example, suppose that we wanted to argue that Hillary Clinton is a liar. One could formulate an argument like this: "Hillary Clinton is a politician. Lies are a part of politics. Therefore, Hillary Clinton lies." This argument is not valid because it has a flaw in logic's Achilles' heel: the use of arguments from ignorance. In this case, the argument claims that because something is a part of politics, then it must be true of Hillary Clinton. But one could also claim that since Hillary Clinton is a woman, she must therefore lie about being in love every time she opens up to the press about her personal life. This argument is invalid as well.
Arguments from ignorance are ones that claim that because two things are connected, then they are equal. For example, one could say that since Hillary Clinton is a politician and lies are a part of politics, then she must lie about everything.
In general, to determine validity, go through each row of the truth table until you locate a row in which ALL of the premises are true but the conclusion is false. Can you locate such a row? Otherwise, the reasoning is sound. The argument is invalid if there are one or more rows.
For example, suppose we want to show that this argument is invalid: "All animals on all islands eat fish so all birds on all islands eat fish." Even though it looks like it might be correct, it's not because some birds don't eat fish. They just may not be on any island. If we try to use this argument to prove that all birds eat fish, we won't be able to since not all birds are on all islands. Only those on some islands eat fish while those on others don't so the argument isn't valid.
Here's another example: "All dogs love cats; therefore, all cats taste good." Even though both dogs and cats belong to one single species (canis lupus familiaris), not every dog loves every cat and not every cat tastes good. Some dogs may hate cats and some cats may scare away even the best-loved dog. So, the argument isn't valid.
To show that an argument is valid, you need to only check each row of the truth table against the entire argument rather than against each premise in isolation.
To begin, examine the shape of the argument to see if the premises give support for the conclusion. If they do, the argument is sound. The question then becomes whether the premises are true or untrue in reality. If an argument satisfies both of these conditions, it is sound. A sound argument leads to a valid conclusion, or in other words, a conclusion that follows from the starting points.
An argument is called "logical" if it follows a certain order of reasoning, using only two types of propositions (or statements): 1 categorical propositions, which say that a given thing is true or false; and 2 hypothetical propositions, which say that something might be true or false. For example, let's take the argument: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. This argument is logical because it follows a precise order of reasoning and uses only two types of propositions. The first proposition, all men are mortal, is a categorical proposition because it tells us that either this statement is true or it is not. The second proposition, Socrates is a man, is a hypothetical proposition because it tells us that if this statement is true, then something else must also be true. In this case, the something else is that Socrates is mortal.
Now, what if we wanted to argue that Socrates is mortal? We could not use the same argument, because the first proposition did not work with this new argument.