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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?

Maverick
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    The key is analytic=true by definition: father= parent and male. If you add it, the corresponding predicate logic argument is valid. – Mauro ALLEGRANZA Oct 18 '22 at 13:04
  • So you are saying the argument needs a further premise: (for all)x(if Fx then Px)? – Maverick Oct 18 '22 at 13:10
  • Exactly........ – Mauro ALLEGRANZA Oct 18 '22 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". – Speakpigeon Oct 18 '22 at 16:42
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    "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. – David Gudeman Oct 19 '22 at 00:19

2 Answers2

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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.

Bumble
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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'.)

PRL75
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