# The logic of analytic inferences

Consider the following argument:

• P1: A is a father.
• Therefore: A is a parent.

The above inference is analytic and valid: it is impossible that someone is a father without also being a parent. However, formalizing the above and similar inference as a valid argument seems beyond the power of either proportional or predicate logic.

For example, in predicate logic, we would get something like:

• P1: Fa
• Therefore: Pa

Which is not valid.

My question is, if proportional and predicate logic lack the power to express analytic inferences like the above, are there logics that can, and if so, which logic and how?

• The key is analytic=true by definition: father= parent and male. If you add it, the corresponding predicate logic argument is valid. Oct 18, 2022 at 13:04
• So you are saying the argument needs a further premise: (for all)x(if Fx then Px)? Oct 18, 2022 at 13:10
• Exactly........ Oct 18, 2022 at 14:42
• @Maverick "it is impossible that someone is a father without also being a parent" A priest is a father yet not a parent, so you really need to add a premise "x is a father implies x is a parent". Oct 18, 2022 at 16:42
• "Analytic" doesn't mean provable from logic alone; it means that it falls out of analyzing your concepts. If you want to represent analytical reasoning in predicate logic, you need to start with a theory that spells out the relationships that would be discovered by analysis of the concepts. Oct 19, 2022 at 0:19

There are a few options available here.

1. If you wish to represent the inference as a valid argument, you could treat it as enthymematic, i.e. as having a hidden premise. You could add the premise (∀x)(Father(x) ⊃ Parent(x)) where ⊃ is the material conditional. Then you can deduce Parent(a) from Father(a).

2. Some logicians hold that there is a separate kind of validity called material validity. This is a concept going back to Abelard. The idea behind it is that some arguments have the property that the conclusion follows necessarily from the premises, but not in virtue of form alone. Advocates of this position distinguish formal validity from material validity but regard both as part of logic. There is a defence of this position by Stephen Read in "Formal and Material Consequence", Journal of Philosophical Logic, 23(3), 1994, pp. 247-265. It didn't convince me, but it is a good summary of the position.

3. You might follow Rudolf Carnap and treat (∀x)(Father(x) ⊃ Parent(x)) not so much as a premise but as a meaning postulate. This is a way of expressing how certain terms within a language are related in such a way that relationships between them hold analytically. On this position, the meaning postulates can be used within an argument without having to state them explicitly. It is open to the usual objections about whether the concept of analyticity really applies within natural languages.

• Thank you, that is very helpful! Oct 19, 2022 at 11:44

Some things are true by definition, and thus do not require logic.

Since the definition of 'A father' is 'a male parent', a father must be a parent.

(The above obviously deals with applicable definitions: you could equally say 'a father is a priest'.)

• "and thus do not require logic" The question is about logic, not definitions. Your answer as it stands is beside the point. Oct 18, 2022 at 16:37
• But the issue is if you do what you are thinking here you are using the same word in a different context. The same word same pronunciation but different contextual meaning. This is also known as the fallacy of four terms or you can also say equivocation is another fallacy that applies to what you are doing. Oct 18, 2022 at 16:44
• " The question is about logic" If <a> is defined as a subset of <b> then logic has no relevance to a particular case. Oct 18, 2022 at 17:32
• "But the issue is if you do what you are thinking here". I wasn't doing anything. Merely pointing out that you need context. Oct 18, 2022 at 17:33