Salle 5, Site Marcelin Berthelot
En libre accès, dans la limite des places disponibles
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The concept of proof may be given a first approximate explanation by saying that a proof is a chain of valid inferences from known truths such that at each inference step the conclusion is seen to follow from the premisses. A natural reaction to this explanation is to say that whether something "is seen to follow" may depend on a subject, and to ask whether this does not make proofs subjective in character.

We cannot simply skip the phrase "is seen to" in this explanation, if the validity of an inference is understood in the customary way in terms of necessary truth preservation or consequence. Doing so would give an explanation according to which every theorem of an axiomatic theory has a one-step proof, consisting of an inference that simply takes sufficiently many of the axioms as premisses and the theorem as conclusion. Unless the theorem is a trivial consequence of the axioms, this is not what we mean by a proof.

The concept of proof is epistemic and can obviously not be reduced to a non-epistemic concept of valid inference. The question I am raising in this lecture is if we can account for this epistemic nature of the concept in a less metaphoric way than saying "is seen to" and if proofs can then come out as something objective. The distinction between an inference being evidently valid and being merely valid was made already at the birth of logic when Aristotle distinguished between what he called perfect and imperfect syllogisms, but he did not explain the distinction any further. Nor does our modern idea of formal proof contribute to such an explanation.

The more general concept of ground as used in epistemology and philosophy of language is closely connected with the concept of proof. A speaker is expected to have some grounds for what she asserts, and an assertion is evaluated as justified or warranted when the speaker has a sufficiently strong ground for the assertion. Here I restrict myself to the strongest possible grounds, what we call conclusive grounds, which should not be confused with infallible procedures for arriving at knowledge. We conceive of deductive proofs as one way in which we can obtain conclusive grounds; in the sequel I drop the attributes "deductive" and "conclusive", but always mean deductive proof and conclusive ground when I say just proof and ground.

What constitutes a ground for an assertion obviously depends on the meaning of the assertion. Here I shall take the view that the meaning of a proposition is given in terms of what is considered to be a ground for the assertion that the proposition is true.