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Let's Be Logical


     Logic is something that is practiced everyday in a civilized society. When most people think about logic they most often think of axiomatic principals that allow us to live in relative harmony with the society, the universe, that humanity has created. Something wholly logical, something beneficial to all of mankind, is ensuring that trash and recycling goes into a trash bin or a proper receptacle so that our individual footprint does not negatively impact the image and cleanliness of our neighborhoods which helps ensure, as much as possible with unceasing human creation of waste, that we can maintain at least some semblance of a habitable planet for future generations of humans. If placing trash and recycling in the proper receptacle is logical then littering, not ensuring that trash and recycling is placed in the proper place of disposal is not only the activity of a misanthrope sociopath, it is categorically illogical.
     Even in something as beneficial to all of humanity as taking out the trash and not leaving it all over the street to rot and fester disease and pollute the air and all of society with the foul stench of human squander, there is an exception to the rule that everyone must put their trash and recycling into the proper bins. If all the bins in the all the world are full and there is an inevitable overflow of human filth all over the streets that is not just a consequence of humanity’s rampant materialist culture, it is a logical exception to the rule. This is also an example of informal logic.
     According to Canadian philosopher Leo Groarke, “informal logic is the attempt to build and to understand and improve thinking, reasoning, and argument as they occur in real life contexts: in public discussion and debate; in education and intellectual exchange; in interpersonal relations; and in law, medicine and other professions. It combines the study of argument, evidence, proof and justification with an instrumental outlook which emphasizes its usefulness in the analysis of real life arguing.” Informal logic is used by community leaders, politicians, business owners, aristocrats, educators and others who have influence over society to help govern, to help do what is best (hopefully) for the whole of society, or at least their constituents. Most would agree that in informal logic, despite the exception of a trash-burdened planet, polluted by the scourge of humanity’s inability to obtain satisfaction, littering is a trespass that society has to be against for the sustainability of our planet and the ability to continue the practice of argumentation.
             
    Example 1:

“If everyone littered like Jonny litters the world would become uninhabitable. Everyone in the world including Jonny litters. Therefore the world becomes uninhabitable.”

• Argument (or reasoning or proof that an idea is true most often governed by empiricism): If Jonny litters and all seven billion people on the planet litter the planet will become unlivable. (We have seen the effects of littering and pollution on the environment, especially since industrialization)

• Conclusion (or what the argument is trying to prove): Littering will make the world uninhabitable

• Premise (or what the argument assumes to be true): humanity cannot live in a world filled with garbage and dross

• Inference (the connection between the premise and the conclusion; the truth of the premise establishes the truth of the conclusion): Humanity cannot live in a world filled with garbage so we should not litter.
    Despite the perceived logic of being more conscious of how the environment is treated some misanthrope can argue that death is coming to us all and that the planet will eventually die just like of all of us, the planet has had many world ending events and maybe humanity is supposed to drown in disgusting heaps of our own vain materialism.
    Natural language can be used, debates of informal logic can become heated shouting matches. The arguments can be propaganda laden and only show a corrupt agenda to poison not only the environment but also the discourse of human progress. Informal logic is essentially critical thinking and is not bound by any true rules of conduct, its focus is on the premise and the inference and both can be evaluated using all manners of epistemology.'
    There is a way that logic is formalized, a manner in which you can deal with such statements that does not involve any ambiguous banter about the benefits or negative conclusions associated with these philosophical debates, moral imperatives; there is no empirical data to assess, a logic that does not appraise the premise of an argument, just the inference. Each inference is then evaluated as “True” or  “False” or  “Valid” or “Invalid.” We have to first regiment the language according to specific details that allow us to focus more on the structure of the language rather than the existential content. Symbolic logic, according to 19th Century Scottish mathematician, “may be defined as the science of reasoning by the aid of representative symbols; these symbols being employed to be synonymous substitutes for longer expressions. […] When any expression verbal or symbolic, of inconvenient length has to be written frequently in the course of an argument or investigation, we naturally cast about for some short and simple symbol to represent and replace it” (McColl, 493).
      We can prove “If everyone littered like Jonny litters the world would become uninhabitable. Jonny and the everyone else in the world litters, therefore the world becomes uninhabitable,” is valid, by using a highly regimented formal logical language. Rules and symbols must be established to prove that the proposition is valid. Since I am writing in English we will use the range of the English alphabet, capitol letters, to represent the main clauses in the propositions.
 We use the rules of inference specifically modus ponendo ponens or modus ponens which is Latin for “the way that affirms by affirming” and is denoted by the formula P → Q, P |- Q.
 This rule ensures that the condition is replaced by the consequence.
     Example 1.1:
Jonny = the letter “J”
Everyone in the world = the letter “E”
Uninhabitable = the letter “U”
 “If J and E. U. Therefore U”
     So if J and E imply U and J and E are asserted to be “True” or “Valid” then U must be true, or If J and E. U. Therefore U. We can also simplify the expression further by using more symbols completing the transformation to a complete logical expression formatted into the logical formal language.

^ is used for a conjunction, substitute for the word “and”

→ is used for conditional propositions, "if and then propositions.

 ⊢ is used for provability; Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates.

Now we can complete the logical expression using the formal logical language:

J^E→U, J^E ⊢U

The statement still holds true in the new logical language, but to ensure the truth of the proposition a proof must be offered.
1. J^E → U    P
2. J^E        P
3. U    1, 2 MP

The proof works out so this is a valid argument.

       There is another form called modus tollens which is Latin for “the way that denies by denying.” It is another rule of inference that makes a valid argument by denying the consequent of a particular proposition.
We use the symbol “~” for the word not or for negation.
         Example 2:
“If Jonny is in his apartment then he will answer the door. Jonny does not answer the door. Therefore he is not at home.”

J → D, ~D ⊢ ~J
 Proof:
If Jonny is in his apartment then he will answer the door. Jonny does not answer the door. Therefore he is not at home.”

J → D, ~D ⊢ ~J

1. J → D       P
2. ~D          P
3. ~J    1, 2 MT

This argument is valid.

Using this argumentation we can prove that the misanthrope may not be so wrong about their indifference about a trash filled planet.
Example 2.1:

“If everyone littered like Jonny litters the world would become uninhabitable. Esther wants to live in an uninhabitable world. Jonny and everyone litters therefore Esther is happy.”

Esther = S

J^E → U, J^E, S → U  ⊢  U

1. J^E → U         P
2. J^E             P
3. S → U           P
4. S         1, 2 MP
5. U         3, 4 MP

 This argument is valid.
     
     Who knows how Esther will survive in a world consumed by all of our humanly dander but she is going to try and be delighted in doing so. The application of the formal language of logic hopefully is not being used for such devious planning or argumentation as the want for our world to be swallowed by human filth but that is how you formalize logic. It limits the abstraction of ideas so that we do not lose time and focus on debate about the principalities involved in every detail of existence.
     There are still arguments that are truly invalid based upon the premise and structure of the proposition that the symbols just cannot follow. The following are classic example of using invalid reference forms and how we can prove the invalidity of an argument. There are two inference forms, the fallacy of assuming the consequent and the fallacy of negating the antecedent.
Here is an example of a fallacy assuming the consequent:
Example 3:

“If the course isn’t easy then we’ll need to study. We don’t need to study. If the course requires very little reading then it’s easy. Therefore the course requires very little reading.”

~E → S, ~S, L → E |- L

1. ~E → S          P
2. ~S              P
3. L → E           P
4. E         1, 2 MT
5. L        3, 4 FAC


 Here is an example of a fallacy negating the antecedent:

Example 3.1:

“If he’s the best candidate then he’ll get the job. He’s the best candidate. If he doesn’t get the job then he won’t be happy. Therefore he’ll be happy.”

B → J, B, ~J → ~H |- H

1. B → J            P
2. B                    P
3. ~J → ~H        P
4. J 1,                       2 MP
5. H                       3, 4 FNA

      That is how logic becomes a formal language. It removes all the ambiguity and all of the ancillary concepts and big ideas that could get us caught up in this long, drawn out, Aristotelian philosophical debates about proper etiquette and decorum and what is in the best interest of humankind or us individually. Formal logic or symbolic logic lets us with the near certainty of mathematics find a basis in argumentative structure that can help us to understand at least the modes that can help us argue better, which can help us make better decisions, which can help humanity not succumb to a planet plagued by our excess.

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