Inference operation

The inference operation is a function in logic that assigns the set of all formulas that logically follow to a (possibly empty) set of formulas ( assumptions or premises ) . ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$

Inference operation and derivability relation

If there is a derivability relation , the associated inference operation is to be defined as follows: = . Conversely, for a given inference operation, the derivability relation can be determined as follows: iff. ${\displaystyle \vdash }$ ${\displaystyle Cn(\Gamma )}$ ${\displaystyle \{\mathrm {A} \mid \Gamma \vdash \mathrm {A} \}}$ ${\displaystyle \Gamma \vdash \mathrm {A} }$ ${\displaystyle \mathrm {A} \in Cn(\Gamma )}$

Properties of an inference operation

Just as there are different derivability relations for different logics ( propositional logic , predicate logic , intuitionistic logic , modal logic , etc.), there are also different inference operations.

So although there are different inference operations, there are a number of properties that all (or most) inference operations have. These were first examined by the logician Alfred Tarski . Tarski names the following properties

• Extensivity : , states that assumptions are always consequences, alternatively, that any statement which is thought may also conclude. ${\displaystyle \Gamma \subseteq Cn(\Gamma )}$
• Idempotency : , states that consequences of consequences always are already consequences, alternatively, if one assumes a statement which follows from the assumptions, so you get so no additional conclusions. ${\displaystyle Cn(Cn(\Gamma ))\subseteq Cn(\Gamma )}$
• Monotony If , then , means, if two sets of assumptions are contained in each other, the corresponding sets of consequences are also contained in each other, alternatively: if a statement follows from a set of assumptions, then the same statement still follows if further assumptions are added. (If the monotonic property is abandoned, one speaks of non-monotonic logic .) ${\displaystyle \Gamma \subseteq \Delta }$ ${\displaystyle Cn(\Gamma )\subseteq Cn(\Delta )}$
• Compactness : is finite and means that conclusions from infinite sets of assumptions can always be deduced from a finite subset of the set of assumptions. ${\displaystyle Cn(\Gamma )=\cup \{Cn(\Delta )\mid \Delta }$ ${\displaystyle \Delta \subseteq \Gamma \}}$

The first three properties make the inference operation a hull operator .