Introduction
While at University, it seems that Logic is a pet hate for those who have never studied it, I personally see use in logic – even if only to help me beat my brother at an argument while sat at the dinner table. While this article’s main aim is not to teach logic, it does intend to educate people who have never encountered it before of it’s purpose and value. I do so in the hopes that by the end, you will have a basic understanding of what logic and feel compelled to go and learn some yourself!
Logic, is found in analytic philosophy; it aims to take the structure, terminology and language of an argument – and determine whether such an argument could even be true.
There are 2 key forms of evaluating an argument : validity and soundness. Validity, is concerned with the logical structure of an argument – an argument is valid if and only if the premises are true, then the conclusion is true. Here we aren’t concerned with the truth of any premises, but whether if we take the premises as true is the conclusion true. Soundness, by comparison considers the truth of any premises used. An argument is sound if and only if : the argument is valid, and the conclusion is true. It is this evaluation that determines whether an argument is true.
Logic, or analytic Philosophy – predominantly focuses on validity. It does so because we can determine whether an argument is false (and discount it) before even discussing the truth of a premise. If the argument could be true, It can also tell you which circumstances the argument is (e.g. true when premise 1 is true and premise 2 is false.)
Consider this argument for how validity works.
Premise 1 :If I it is raining, I will take an Umbrella.
Premise 2 : I will take an Umbrella.
Conclusion : So it is raining.
This simple argument adopts the logical fallacy of confirming the antecedent, and so isn’t valid. This is because, premise 1 tells us that if it is raining, I will take an Umbrella – but just because I have an Umbrella does not mean it is raining. So in this case, even if both premises are true, the conclusion does not have to be true.
Through the use of this method, we can also determine that the argument cannot be true – or sound, without even discussing the truth of any of the premises – because we may not even know whether it is raining.
– So here, it seems that Logic can allow us to easily determine whether any argument is true.
– There are numerous logical methods used to determine whether an argument could be true. In logic, they do not stop at validity – instead, they also consider satisfiability.Meaning they consider the situations where the argument can be true – Generally, you will determine the truth value (instances where the conclusion is true) which will look something like this..
Conclusion is true where Premise 1 = True, and Premise 2 is False. And in no other circumstance.
You would then take this truth value, and discuss the real truth of the premises to see if they match with the truth value. In this case, if premise 1 is true and premise 2 is false : then the conclusion of the argument is true.
Propositional Logic and the tools used.
Propositional logic, aims to simplify and join together propositions through the use of placeholders (where you take a proposition and change it to a singular letter, for example : P.) It tries to remove ambiguity in an argument – which can sometimes be important when dealing with ambiguous terminology and the intended purpose of an argument.
It does so in the hopes of avoiding ambiguity, and laying out an argument in such a way so that you can determine whether the argument is satisfiable. It creates a new language which represent wording within the argument.
For example : Either it is raining, or it is sunny would translate as “P v Q” – let me explain why this is the case :
- Propositional logic is made up of connectives and propositions. So the P/Q represent the propositions “It is raining/sunny” and the v represents the Disjunction “either…or.”
The Connectives in Propositional Logic (These are the basics)
- Negation : “Not” – is true when the proposition is false. (e.g. it is NOT raining is true when “it is raining” is false.)
- Conjunction :”And” – is true when both propositions are true, otherwise it is false.
- Material Implication – “if… then” – is only false when the antecendent (if) is true and the antecendent (then) is false. Otherwise it is true.
- Biconditional : “If and only if” – is true when P and Q are both the same; true or false.
- Disjunction : “Either…or” – is true when either P or Q is true… or both.
How would you use this? – Truth Tables.
While there are numerous ways to use these tools, I wanted to focus on truth tables as one use.
Truth tables are used to determine 2 things – 1. The truth value of a proposition (e.g. under what circumstances a sentence can be true, and 2. The importance of the proposition.
The first use I have explained previously : we can determine whether the argument could be true, and under what conditions it is. But the second use, generally tells us the importance of the proposition : as it tells us whether a sentence is either : a) a contingency, b) a contradiction or c) a tautology.
Contingencies, are things that could be true or false, hence why we need to determine under what condition a sentence may be true. Alternatively, contradictions cannot be true – they are false under all scenarios (so we can discount them.) Finally, tautologies cannot be false : they are always true.
Contingent truths generally offer value in one form or another; but tautologies are usually known as truths of logic and so are often branded as useless. (By example, what is the value in arguing that : “Jon is Jon” when it cannot logically be false?)
Here is one of the many uses that propositional logic has.
Formal Logic.
Formal logic is very similar to propositional logic : it uses connectives in a similar way. The only difference is that it aims to reduce ambiguity even more within the propositions. It does so through the introduction of another new language.
It uses quantifiers, to categorise a subject. By example :
The existential quantifier (there exists an x) refers to one person, while the universal quantifier refers to many people.
To avoid confusion, I am not going to lay out any formal logic here – but instead recommend you reading an online guide on how to undergo it. – but it’s main purpose is similar to that of propositonal logic : to determine truth, the only difference is it does this in a more precise and accurate way.
How is this used? – Proof Trees.
Proof trees work by breaking down an argument into it’s simplest form and presenting each simple proposition (or primitive sentence) onto a branch. It does so – as it allows the Philosopher an easy way to determine whether there exist contradictions in the branches of the argument. Contradictions occur when there exists a sentence and it’s negation – e.g. “Jon is here” and “Jon is NOT here.” – this branch would then be closed as we know this cannot be true.
When the proof tree is finished, the Philosopher will be left with open or closed branches which tell us important things :
- Firstly, whether the argument is satsifiable, valid or a tautology (or a contradiction.)
- Secondly, if it is satisfiable – under what circumstance the argument is true.
While these uses are similar to those mentioned in tools previously, it is important to note that some arguments are so complex that other tools do not suffice : and proof trees are as such an easy way to come to these key conclusions.
Overall, I have illustrated that logic is an important tool when determining whether an argument could be true, before discussing the empirical truth of the premises.
This is a powerful tool that will allow you to closely evaluate arguments, whether that be in Philosophy – or something you encounter in every day life.
If you have studied Logic, I would recommend applying it to some of the arguments used by politicians – to determine whether their claims could even be true.
If you have’t studied it before, I hope I have inspired you to go and investigate – if for no other reason than to use it when debating with a sibling.
Other uses not mentioned : domains, negation/conjunction/disjunctive normal form, and much more!
Thank you for taking the time to engage with me,
Comment below, or tweet me @theapeironblog
Jon Hawkins.
