Reams Of Further Logic

Logic is the study of argument. It is a tool, and can be used to analyse the structure of arguments, and does not necessarily deal in truth, but also falsity. It should sink in how logic is an inescapable condition of reason.

The most basic argument is of the form: P therefore P. It has a premise and a conclusion (both ‘P’). It is a tautology, since the conclusion is equivalent to the premise. Its simplicity makes it easy to see why it is valid, whilst this also renders it the least meaningful or informative type of argument (you learn very little from it.) Both the premises and the conclusion are propositions (and, incidentally, they are the same proposition). “It is wet. Therefore it is wet.” This is a valid argument (the premise leads to the conclusion). It is deductive (known with certainty), as against inductive (merely probable).

“She is smiling. Therefore she is happy” is an inductive argument, and fairly weak, because we cannot be certain that the premise leads to the conclusion (she might be faking the smile). It would have the form P therefore Q. Although we can see a connection intuitively between the propositions, it does not have a strong logical shape because there is no formal connection between the conclusion and the premise (P and Q are different propositions which do not recur with a logical operation to connect them.) We would have to make a more complicated argument involving propositions which say it is impossible to smile without being happy in order to logically verify it. It is weaker than if it were deductive. To be deductively valid an argument must convince us that, were the premises true, the conclusion would definitely be true also. Deductive logic is akin to certainty in mathematics. You cannot rationally argue with it. However, we cannot use deductive logic all of the time because we cannot always reason from solid foundations. Hence the existence of dialectical reasoning (the ascertaining of the truthfulness of opinion through discussion.)

There are two ways to criticise somebody’s argument. You can dispute the truth of their premises (say, for example, B could not have been caused by A because A never happened). Or you can find fault in the reasoning, which is where logic comes in.

Logic allows us to analyse complex arguments through taking a close look at their structure. Somebody could reason and be logical by utilizing connectives in their language:

“If, then”,

These words are in abundance in ordinary language and communication. Sometimes they almost seem superfluous to requirements, like saying “...er...”, something to fill a pause almost, however they perform a crucial function. They are pivotal with regard to validity, and affect propositions’ logical relationship to one other. This is perhaps why they occur so frequently – they allow us to change our minds about the relevance of the statement we are making, how it is angled, where it fits in our overall point. ‘And’ associates propositions, describing them as consistent. ‘If, then’ describes a causal connection. ‘Not’ contradicts a proposition. ‘Or’ says one or the other proposition may be the case (inclusive), but not both (exclusive).

'Therefore' is an argument indicator. It signals the conclusion, and no argument is complete without one. “Hence”, “since”, “thus” and “so” are also argument indicators. You might think you don’t make many arguments in your day-to-day life, but every ‘so’ indicates a conclusion, a statement of belief about a situation on the basis of a previously given or tacitly understood set of assumptions, which may or may not constitute a sound argument.

What do you think of this? Is it valid?:

“Either Scott is in the East Slope bar or at the racetrack. If Scott is in the East Slope bar then so are both Joe and Lisa. Joe and Lisa are not at the East Slope bar. Therefore Scott is at the racetrack.”

To demonstrate how logic can be used, this is what the argument might look like were it to be formalised into a (rough) logical language:

Sign. Use.  Name in Logic.
&    And (conjunctive)
-     Not (negation)
→   If, then (conditional)
V    Or (disjunctive)
├     Therefore (argument-indicator)


This is the argument:

1. SE V SR
2. SE → (JE & LE)
3. - (JE & LE)
├ SR

(Where S: Scott, J: Joe, L: Lisa; E: is the East Slope bar, and R: is at the racetrack.)

Understanding the parts in between propositions (the connectives and argument indicators) gives you insight into the significance of the arguments that people make. An argument may be valid and unsound - because it makes sense in theory but the propositions do not live up to the facts. The premises may be true but the conclusion does not follow from them, rendering the argument invalid. The conclusion may be true, but not follow from the premises, so the argument is invalid! How can the conclusion be true, but the argument invalid? To give an example of why this makes sense, if I spend 10 minutes making an argument, and we find after 10 minutes that you already agree with the conclusion, and my premises do not even lead to the conclusion, the argument is, for practical purposes at least, useless. We have simply expressed an agreement of opinion. On the other hand, you might agree with the conclusion for a different reason to me, in which case a valid argument coming from a different line of reasoning would not be a waste of time, it would strengthen the conclusion a fortiori. So validity is valuable to logic.

Logic is significant in everybody’s lives (sometimes we all genuinely want or need to get our point across clearly). A firm grasp of logic is important in management, for example, because managers have to generalise over large areas of a business (see the wood as well as the trees) which involve the consideration of large factors, and apply concepts like ‘economy’ or ‘efficiency’ to groupings of activity. The arguments made are of the essence. The figures which the accountant uses contribute to the arguments made, because they inform the rationale behind costing. Abstract notions of market, social and financial factors have to be comprehended in the manager’s head in the form of arguments, conclusions drawn and decisions based on those conclusions. This may sound like I’m complicating the process, but this is what a manager does. It is the job of the mathematician to number-crunch, yes, and it is the job of the logician to language-crunch.

Consider: There are flaws in your logic. If there are flaws in your logic, then there are flaws in your reasoning, which means there are flaws in your argument, in which case the conclusions you draw from those arguments could easily be false, therefore the decisions you make founded on those arguments may not be effective. Have I made a strong case? It has roughly the form:

Not L. If not L then not R. If not R then not A. If not A then not C. If not C then not D. Therefore not D.

Do you agree with all the premises? Does the conclusion follow from the premises in an argument which has that shape?

If you identify each capital letter in the above argument with a proposition, a statement of any kind. “John Major used to be Prime Minister” for ‘L’ for example. Or “John Major is the present Prime Minister.” – Anything you like. R might stand for: “Squirrels have furry tails.” Or you can play with sentences which you think make more sense together. Remember, you are looking for validity, not truth. You will find that every argument of the shape just shown is valid, because of its shape, whatever premises or conclusion you choose to slot in. Try to become accustomed to what validity really is, rather than what you would like it to be, or how it is popularly misused (e.g. as simply meaning ‘good’.)

Whether we are then able to judge that the argument is fundamentally a sound argument (the best compliment for an argument) depends on the truth or falsity of the individual propositions. That’s the bit where you say “that’s not true, because…” and I say “yes it is true, I read it in the news” and you say “ah, but I read a different article that had better sources, so it’s not true” and we debate the facts. This is separate from that which makes the arguments, in their skeleton, structurally sturdy. We understand how the elements of logic relate to each other, just as we understand how the materials of architectural design depend upon each other. The shape comes from a blueprint of the design of the building, which obeys certain rules of engineering and laws of physics. These instructions work on the page, on the scale model and on the real-life structure. Theory is not always true to form, however. So the built structure may have integrity but be unrepresentative of the original design. Or indeed it may be true to form, but in the event the building crumbles, revealing flaws in the design. One thing is for sure, logic itself is not at fault when the theory breaks down. Logic is the very foundation of reason, which is arguably the backbone of any discipline with a claim to knowledge. That doesn’t make the logician correct by default of course.

Finally, consider the nature of hypothesis. Hypotheses are useful, and yet a hypothesis needs no factual accuracy in order to be successful. In other words it might bear no relation to what is the case. It is by its very nature ‘imagined’, but it still aspires to be structurally sound. It is not even speculative. It does not predict what will happen, but says “if this were the case, so would that be.” Causal relations, real or conceived, are at the very heart of why logic works.




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