Logic is defined as (from Dictionary.com):
- the science that investigates the principles governing correct or reliable inference.
- a particular method of reasoning or argumentation: We were unable to follow his logic.
- the system or principles of reasoning applicable to any branch of knowledge or study.
- reason or sound judgment, as in utterances or actions: There wasn't much logic in her move.
- convincing forcefulness; inexorable truth or persuasiveness: the irresistible logic of the facts.
Logic is used primarily as an informational filter. We all apply logic to some extent to information that is introduced to us through external sources such as books, television, radio, the Internet, etc. in an attempt to filter out true information from false information. This isn't, however, to say that the logic applied by everyone is equally effective at filtering true information from false information. Most often the logic we use is a simple consistency check. we internally and often subconsciously check the information to determine if it is consistent with what we already know or believe about the universe. For instance if someone tells us a story about the time they jumped off of their roof and flew we will probably reject the accuracy of the story because it is inconsistent with what we know and/or believe about the universe. The story would directly contradict our knowledge of gravity and unless a good explanation for how the person flew was presented our logic would tell us that the story was almost certainly false. The conclusions that we draw from this most basic form of logical inference will always depend on what we already know or believe about the universe. Because of this it may lead different people to different conclusions depending on what an individual knows or believes about the universe. For this reason a simple consistency check alone is often insufficient to determine with certainty whether information is actually true or false, because if what the individual believes, or believes he knows, about the universe is false then the inference drawn from a simple consistency check may be logically valid, but unsound. This means that the conclusion may or may not be true because one or many of the premises may or may not be true. If that didn't make any sense, just keep it in mind. It will be explained more later.
Logic is also applied by everyone as a problem solving tool. For this paragraph I will be using a very broad definition of problem to include any instance in which an individual has to make a decision between two or more options. Here again experience plays a large role, but this is not necessarily the case. When faced with a problem we always apply some form of logic to the solution. Again this is not to say that all logical systems that we could apply are necessarily equal. Most often for this we reason cause and effect. We base our reasoning of cause and effect on experience. More specialized problems will often require different applications of logic. But for simple decision making we attempt to model in our minds as best we can the options and what would happen if we chose each option. We draw upon experience and other trails of reasoning depending on the problem to try to determine what the best option to chose might be and based on what we determine from this process we make our choice. This is not necessarily always the case. In many instances we have no experience of one or more of the options and we make our decision based on other factors. The more information we have, however, the better a decision we can make when presented with options, through the application of logic. The game of chess is an excellent logical game. It's a game of complete information, which means that both players each know all there is to know about the state of the game at any given point in time during the game. Making a decision about your next move will depend partially on cause and effect reasoning which is partially experience based and partly based on other factors, such as the rules of the game, the other player's strategy and the current state of the game.
A more powerful application of logic, though, is to infer, from what we can directly observe, information about things we cannot directly observe. This is how logic is used in philosophy, math, and science. In our day to day lives we may also apply logic for this purpose to infer, from what we have observed, what we have not observed. This application of logic is the basis for all of the knowledge we have gained in philosophy, mathematics and science. This fact brings us now very naturally to my next question.
Is Logic Valid?
This may, at first, seem like a difficult question to answer. There is, after all, no way to logically prove the validity of logic. All such attempts would naturally be begging the question. In order to prove the validity of logic logically one would have to already accept the validity of logic. Likewise, however, how do you disprove the validity of logic in such a way that the proof of the invalidity of logic would have to necessarily be true based on the premises? The only way to do this that anyone knows of is through logic. But if you need to apply the very thing you seek to disprove in order to disprove it you are again requiring us to accept the validity of logic in order for us to accept your proof that logic is invalid. Here we arrive at a contradiction. Thus the validity of logic is accepted as axiomatic as Logic is observed to be useful for understanding the universe, and logic itself is defended by retortion. Retortion is the act of turning something back against itself, as in the example I just gave of how one might go about proving the invalidity of logic using logic. It is also a fact in logic's favor that if the structure of an argument is valid and all of the premises of the argument are true then the argument's conclusion must necessarily be true. This fact has never been disproved. No valid and sound argument has ever been presented which supported a false conclusion. And here we have a falsifying point of logic. If someone could present a logical argument that was both valid and sound but supported a demonstrably false conclusion you would force us to completely rethink our understanding of the universe.
Understanding Logic
Now let's get into how logic works. Logic is applied to logical arguments. Arguments, in logic, are statements which are trying to convince you to do, buy or believe something. Logical arguments consist or a set of premises which are connected by a set of logical connectors, such as the conjunctions "and" and "but", the disjunction "or", and the conditional "if...then" statements. I'll give a more complete listing of these logical keywords a little later and what logical connective they correspond to. The purpose of a logical argument is to support a conclusion, which is usually separated from the argument by words such as "therefore" or "because". The validity of an argument is determined solely from the argument's structure. Not all argument structures are valid, and an invalid argument structure will allow for an argument in which all of its premises are true to support a conclusion which is false. However an argument with a valid form does not allow for such a thing. Logic is, in reality, only concerned with the validity of an argument. In logic a valid argument is one in which the argument's structure is valid and a sound argument is an argument whose structure is valid and in which every premise is true. The premises in a sound logical argument may be axioms, or they may be proven or provable assertions. Axioms are facts which cannot be logically proven but are observably or necessarily true. For clarity I will give an example of a necessarily true axiom.
The axiom of the existence of an objective reality in which we all exist is necessarily true. This might get confusing, please leave a comment letting me know what is specifically confusing if it is and I will do my best to clarify. This axiom is necessarily true for any logical system trying to understand the universe, however as the axiom is not logically provable it is not necessarily true in the universe. Yes I am aware that sounds contradictory but it isn't. The reason why the axiom is accepted as necessarily true follows from the only thing that we can logically prove to ourselves with certainty through deductive inference. This is Descartes's argument which proves the existence of our consciousness. "Dubito, ergo cogito, ergo sum." This means, "I doubt, therefore I think, therefore I am." This argument proves deductively only to a given individual that his consciousness exists. There is no deductive logical argument that can prove the existence of a reality that exists externally to any given individual's consciousness. Because of this we cannot conclude with certainty the reality of such an external reality. For all any given individual knows that individual is the only consciousness that exists and the reality that the individual perceives is merely a very elaborate dream like state generated by a compartmentalized subconscious, which would allow for the individual to think of ideas and concepts that he was not consciously aware of having thought up that could then be presented to the individual's conscious awareness through made up "person-like" constructs of the individual's subconscious. If this were the case, though, what would that change about that individual's experience of the universe? Most likely nothing, esspecially since the individual could never prove that what he percieved as reality was in fact not real. The individual would still have to experience every consequence of that individual's actions. The individual would most likely still not be able to exercise complete god-like control over his reality just as the individual has had no such god-like control up until the moment he decided or realized that the reality he experiences is not real. For this reason, among others, it is sufficient for us to treat an objective reality as necessarily true for all intents and purposes.
If that last paragraph was at all confusing, once again, please comment and I will attempt to clarify whatever was confusing about it. It probably wasn't worded as well as I would have liked. So most of the confusion is probably my fault.
Getting back to validity and soundness, though, it is important to know what it means for an argument to be invalid or unsound. If an argument presented is invalid or unsound it does not necessarily mean that the conclusion is false. An invalid or unsound argument may support a true conclusion, but the argument would not be convincing. The reason the argument fails to convince is because based on the structure of the argument or the fact that one or more of the argument's premises are false or unproveable, there is no way to know from that argument whether or not the conclusion is true, even if it is. Thus you cannot conclude that a person's conclusion is false based solely on the validity or soundness of the argument presented. You can explain why the argument is invalid and ask that the person try to formulate a sound argument to support their conclusion. Remember if an argument is valid and sound then it's conclusion is necessarily true, that's the power of logic.
The Logical Connectives
A logical argument will often contain within premises, many statements which are connected by logical connectives. These connectives determine how the statements relate to each other within the premises and also how the premises relate to each other to support the conclusion. In most formal logical systems the set of logical connectives are conjunctions, disjunctions, conditionals, negation and equivalence.
Conjunction
The conjunction connective is the logical and operator. The logical statement (A and B) is true only if A is true and B is true. If either A or B are false then (A and B) evaluates to false. In a truth table this looks like:
From this truth table we see that when both A and B are true then (A and B) is true. However if A is true and B is false then (A and B) is false, and if A is false but B is true then (A and B) is false, and if both A and B are false then (A and B) is false.
Disjunctions
The logical disjunction is the logical or operator. The logical statement (A or B) is true if either A is true or B is true. It can also be true if both A and B are true. The or operator evaluates to false if both A and B are false. It's truth table looks like:
From this truth table we see that (A or B) is true when both A and B are true. It is also true when A is true and B is false, and when B is true but A is false. (A or B) evaluates to false when both A and B are false.
Negation
The logical negation is the logical not operator. The logical statement (not A) just means that if A is true then (not A) evaluates to false, and if (not A) evaluates to true then A evaluates to false. On a truth table is looks like:
From this truth table we see a little better that when A is true (not A) is false, and when (not A) is true A is false.
Conditionals
The logical conditional is the implication operator. (A implies B) can also be read (if A then B) evaluates to true when A is true and B is true. It is also true when B is true but A is false and when both A and B are false. (A implies B) is false when A is true but B is false. It's truth table looks like:
Equivilence
The logical equivalence, also known as the biconditional is the "is equivilent to" or the "if and only if" operator. (A is equivilent to B) can also be written (A if and only if B), sometimes abbreviated (A iff B). (A iff B) evaluates to true when both A and B are either true or false. If A is true but B is false then (A iff B) is false, and likewise if A is false but B is true then (A iff B) evaluates to false. It's truth table looks like:
