Logical Arguments. Syllogisms, and Logical Connectives.

As in the previous post, this will once again be an overview. There are many different methodologies and factors to keep in mind and I cannot be conclusive here. I suggest looking into all of these matters further should you be interested in strengthening your skills at argumentation.

There is my process in which a logical argument can be formed. Some are better then others, and some can only be used in specific circumstances. I will state it again: I won’t be covering all of them, instead I’ll be focusing on a few important logical processes: the Syllogism, and logical connectives.

A Syllogism formally is three lines where first you make a universal claim followed by a particular claim which is predicated (based on, directly related too) on the first universal claim. The third sentence is then composed from those first two sentence. As an example, I will use the most famous form of Syllogism posed by Aristotle:

1. All men are mortal.

2. Socrates is a man.

3. Therefore, Socrates is mortal.

I hope everyone can see how the third sentence here follows logically from the first two. We know from the first line (for the sake of this argument) that all men are mortal, so when we are also told that Socrates is a man, we know that Socrates must then be mortal.

Going back to my previous post it would be easy to rewrite the format of this argument in premises and conclusions, which I will do below:

P1. All men are mortal.

P2. Socrates is a man.

C. Therefore, Socrates is mortal.

This is one of the most basic forms of a logical argument and is based around the definitions of those terms it uses. It’s useful because, when we try to misuse Syllogism, it tends to be quite obvious. This is because the concluding line will not be predicated from the first two lines. For example:

P1. Some Greeks are mortal.

P2. Socrates is a Greek.

C. Therefore, Socrates is immortal.

Again I hope it’s clear why this doesn’t work. In the first premise we see there is room for some Greeks to to be not moral, so for the sake of this argument we could say that it is the case that any given Greek could be mortal or not moral (perhaps immortal perhaps something else, since it is not specified). So when we are told Socrates is a Greek we know there is some possibility he is not mortal, but that’s all we know. We cannot say he is moral or otherwise based on this argument. All we could say is C. Socrates is possibly moral. Nothing more.

These simple syllogisms can be extended into more complex forms, but the take away here is that you should be making sure that your conclusions are predicated on your premises. Otherwise you’ll at best end up making mistakes and at worst end up speaking nothing but gibberish as your conclusions end up lack any cohesion with your premises. It’s best to avoid that if you can.

Next are logical connectives which do not serve a propose in this post more than to lay the ground work for other posts.

I’ll briefly list them going into a bit more detail below. If you want to know a bit more about how they work I’d either Google logical connectives, or go play with red stone logic circus in Minecraft (make a locking door but make sure you look up the wiki: you need at least an and, and or gate, but I like to use xor gate for mine 😉 ).

As to what logical connectives are, they function basically the same way we use them in language: by connecting different statements together, and trying the truth of both statements in a particular way. Technically you can create a system which contains all of the following connectives with only “and” and “or” connectors, but it’s far easier to talk about these logical relationships without trying to tie them altogether:

… and… (&)

The whole statement is only true if both sides of the and connective are true.

… or…

The whole statement is true when at least  one side of the statement is true.

if… then…

“If…then” statements works such that if the “if” statement is true, then the “then” statement must be true for the whole connected statement. If the “if” is false, then the “then” can be true or false to no effect. If x happens, then y happens. The statement remains true even if y happens with out x. The statement is only falsified when x is true, but y doesn’t occur as well.

… if and only if…(iff)

This is like the “If…then” statement, but instead x can only occur if y occurs and vise versa. The statement is false only if one occurs without the other. Iff can also, in some cases, indicated equivalency, but this is not necessarily the case.

… Elusive or… (xor, either)

Opposite to iff, this statement is only true when only one side of the statement is true. You can either have pudding or cake, but not both.

negation… (-, not)

Negation is reversing the meaning of the statement. Where (n) is a cat (-n) is not a cat.

… Equivalency… (=)

When two or more things are the same. They are equivalent. 2+3 = 5 = 1+ 1 + 1 + 1 + 1

I’ve included formal logic terms, short hand, and math symbols above many of which double as grammar. Each of the above can and are regularly used in English. I’m certain if you’re unsure of how to figure any of this out, you can manage it with a Google search or two. The biggest reason to include this early on is to clarify some of the common terminology and expose those reading this to some common ways people talk about these connectives. Besides, all of these connective are used in language and argument, so it is important to understand how we ought to use them within our arguments so that others will understand what we mean.

Hopeful I haven’t bored you all out of your minds. Next time I’ll get to induction and deduction. Which I feel is far more interesting.

Withteeth