Logical Implication
Logical implication is a fundamental building block of logical arguments, where 'if A then B' means that if A is true, B must be true by logical necessity. This contrasts with everyday uses of 'if...then' that express causation, threats, promises, or personal taste.
THE IDEA
DOES CHOCOLATE MAKE YOU HAPPY?
IF I EAT CHOCOLATE then I am happy. Is that logical?
If I touch wood after referring to something ominous, I feel better. If you fly from Chicago to Manchester via London, it can be cheaper than just flying to London on the very same flight and not going any further. If you drop some money in the street, someone will probably take it.
If you are white, then you have white privilege.
Are these things logical?
The innocuous little word "if" has a whole range of slightly different meanings. Some of them, but not all of them, encapsulate the most important building block of logical arguments: logical implication.
A logical argument is a way to demonstrate or verify that you are right. In life there are many ways to demonstrate that you are right. One is to shout loudly. Another is to tell anyone who disagrees with you that they are stupid. These are not good ways to persuade people that you are right, but unfortunately these ways are quite common.
The scientific method for demonstrating what is true involves carefully gathering evidence, analysing it, and then documenting the whole process in a way that could be reproduced by someone else if they followed the same steps. Importantly, it also comes with a way of discovering you are wrong.
Mathematics is at the heart of science but is a bit different from it. Mathematics uses logic, rather than evidence. It uses logic to decide when something is true, and it also uses logic to detect when something is wrong. We can sum this up by saying:
Logic is to mathematics as evidence is to science.
That is to say that the role of logic in mathematics is analogous to the role of evidence in science, but logic and evidence are fundamentally different. Unlike evidence, logic tells us when something has to be true, not by cause and effect, not by probability, not by observation, but by something inherent that will never ever change.
Logical arguments are built using logical implication. Logical implication is how you move from one true statement to another. It doesn’t make more things true, it just uncovers more true things than we saw before. Logical implication says that “if” one thing is true “then” another must be true, using logic.
It becomes complicated because in normal life we say “if ... then ...” in situations that are not logical. It might be personal taste, like “if I eat chocolate then I am happy”. It might be a threat, like “if you say that one more time then I will scream”, or a bribe, like “if you eat your broccoli then you can have ice cream”, or a promise, like “if you confide in me I won’t tell anyone”. It might be an agreement, like “if you walk my dog for me then I’ll pay you twenty quid”. It might be causation rather than logic, like “if you drop that glass then it will break”. It might be rules, such as “if you are over 75 then you don’t have to take your shoes off when going through airport security”. It might be a personal opinion about standards of behaviour: “If you loved me you wouldn’t say that!” really means “I personally don’t think that is a loving thing to say.” Another problem is that we use “imply” to mean “insinuate” in normal language, as in “Are you implying I’m stupid?” The difference is a bit blurry in normal life, but we can try and find the distinction by thinking about examples.
When to use it
In mathematics, proving theorems by deducing consequences from axioms using logical implication.
In programming, using conditional statements (if-then) to control program flow based on logical conditions.
In scientific reasoning, distinguishing between logical deductions and causal inferences when interpreting experimental results.
In everyday debate, clarifying whether a statement is meant as a logical necessity or as a probabilistic or emotional claim.
FROM THEORY TO EVERYDAY LIFE
In practice
Chocolate and Happiness
Eating chocolate may make a particular person happy, but happiness does not follow from eating chocolate by logic alone. The claim depends on that person’s response and circumstances. Logical implication concerns what must follow from the premises; it is not another name for a causal effect.
Broken Glass
Whether a dropped glass breaks depends on physical conditions, including the surface it lands on. The statement is an empirical causal claim. It does not follow from the meaning of dropping alone that the glass must break.
A MOMENT TO REFLECT
Put it to work.
Think of an 'if...then' statement you used recently. Was it a logical implication, or did it express a causal relationship, a threat, or a promise?
How can distinguishing logical implication from other uses of 'if' improve critical thinking and argumentation?
Identify a real-world argument where someone confused a correlation or personal opinion with a logical implication. How would you explain the difference?
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)