Editing Truth function (section)

== Principle of compositionality ==

Instead of using [[truth table]]s, logical connective symbols can be interpreted by means of an interpretation function and a functionally complete set of truth-functions (Gamut 1991), as detailed by the [[principle of compositionality]] of meaning.
Let {{mvar|I}} be an interpretation function, let {{math|Φ, Ψ}} be any two sentences and let the truth function ''f''<sub>nand</sub> be defined as:
* ''f''<sub>nand</sub>(T,T) = F; ''f''<sub>nand</sub>(T,F) = ''f''<sub>nand</sub>(F,T) = ''f''<sub>nand</sub>(F,F) = T

Then, for convenience, ''f''<sub>not</sub>, ''f''<sub>or</sub> ''f''<sub>and</sub> and so on are defined by means of ''f''<sub>nand</sub>:

* ''f''<sub>not</sub>(''x'') = ''f''<sub>nand</sub>(''x'',''x'')
* ''f''<sub>or</sub>(''x'',''y'') = ''f''<sub>nand</sub>(''f''<sub>not</sub>(''x''), ''f''<sub>not</sub>(''y''))
* ''f''<sub>and</sub>(''x'',''y'') = ''f''<sub>not</sub>(''f''<sub>nand</sub>(''x'',''y''))

or, alternatively ''f''<sub>not</sub>, ''f''<sub>or</sub> ''f''<sub>and</sub> and so on are defined directly:

* ''f''<sub>not</sub>(T) = F; ''f''<sub>not</sub>(F) = T;
* ''f''<sub>or</sub>(T,T) = ''f''<sub>or</sub>(T,F) = ''f''<sub>or</sub>(F,T) = T; ''f''<sub>or</sub>(F,F) = F
* ''f''<sub>and</sub>(T,T) = T; ''f''<sub>and</sub>(T,F) = ''f''<sub>and</sub>(F,T) = ''f''<sub>and</sub>(F,F) = F

Then
{{Bulleted list|style=font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif;
| ''I''(~) {{=}} ''I''({{not}}) {{=}} ''f''<sub>not</sub>
| ''I''(&) {{=}} ''I''({{and}}) {{=}} ''f''<sub>and</sub>
| ''I''(''v'') {{=}} ''I''({{or-}}) {{=}} ''f''<sub>or</sub>
| ''I''(~Φ) {{=}} ''I''({{not}}Φ) {{=}} ''I''({{not}})(''I''(Φ)) {{=}} ''f''<sub>not</sub>(''I''(Φ))
| ''I''(Φ{{and}}Ψ) {{=}} ''I''({{and}})(''I''(Φ), ''I''(Ψ)) {{=}} ''f''<sub>and</sub>(''I''(Φ), ''I''(Ψ))
}}
etc.

Thus if ''S'' is a sentence that is a string of symbols consisting of logical symbols ''v''<sub>1</sub>...''v''<sub>''n''</sub> representing logical connectives, and non-logical symbols ''c''<sub>1</sub>...''c''<sub>''n''</sub>, then if and only if {{math|size=100%|''I''(''v''<sub>1</sub>)...''I''(''v''<sub>''n''</sub>)}} have been provided interpreting ''v''<sub>1</sub> to ''v''<sub>''n''</sub> by means of ''f''<sub>nand</sub> (or any other set of functional complete truth-functions) then the truth-value of {{tmath|I(s)}} is determined entirely by the truth-values of ''c''<sub>1</sub>...''c''<sub>''n''</sub>, i.e. of {{math|size=100%|''I''(''c''<sub>1</sub>)...''I''(''c''<sub>''n''</sub>)}}.  In other words, as expected and required, ''S'' is true or false only under an interpretation of all its non-logical symbols.