Difference between revisions of "Semantics of logical connectives (FOL)"

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Definition

Let [ilmath]\mathscr{L} [/ilmath] be a given first order language and let [ilmath]C[/ilmath] denote the collection of all logical connective symbols[Note 1], the semantics of a logical connective symbol[1] is a function from the Cartesian product of one or more set of truth values to the truth values, written [ilmath]\mathbf{B}_*[/ilmath] where [ilmath]*[/ilmath] is a logical connective symbol, [ilmath]*\in\{\neg,\vee,\wedge,\rightarrow,\leftrightarrow\} [/ilmath], this mapping is commonly called the truth table[1] of the connective.

Suppose [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] are variables which may only take on truth values (eg, are the semantics of a formula) then we define [ilmath]\mathbf{B}_*[/ilmath] as follows:

Uniary connectives

Truth table for logical negation

[ilmath]\mathbf{X} [/ilmath] [ilmath]\mathbf{B}_\neg(X)[/ilmath]
[ilmath]T[/ilmath] [ilmath]F[/ilmath]
[ilmath]F[/ilmath] [ilmath]T[/ilmath]

Binary connectives

Truth table for [ilmath]\vee[/ilmath], [ilmath]\wedge[/ilmath], [ilmath]\rightarrow[/ilmath], [ilmath]\leftrightarrow[/ilmath], that is logical or, logical and, logical implication and logical equivalence respectively:

[ilmath]\mathbf{X} [/ilmath] [ilmath]\mathbf{Y} [/ilmath] [ilmath]\mathbf{B}_\vee(X,Y)[/ilmath] [ilmath]\mathbf{B}_\wedge(X,Y)[/ilmath] [ilmath]\mathbf{B}_\rightarrow(X,Y)[/ilmath] [ilmath]\mathbf{B}_\leftrightarrow(X,Y)[/ilmath]
[ilmath]F[/ilmath] [ilmath]F[/ilmath] [ilmath]F[/ilmath] [ilmath]F[/ilmath] [ilmath]T[/ilmath] [ilmath]T[/ilmath]
[ilmath]F[/ilmath] [ilmath]T[/ilmath] [ilmath]T[/ilmath] [ilmath]F[/ilmath] [ilmath]T[/ilmath] [ilmath]F[/ilmath]
[ilmath]T[/ilmath] [ilmath]F[/ilmath] [ilmath]T[/ilmath] [ilmath]F[/ilmath] [ilmath]F[/ilmath] [ilmath]F[/ilmath]
[ilmath]T[/ilmath] [ilmath]T[/ilmath] [ilmath]T[/ilmath] [ilmath]T[/ilmath] [ilmath]T[/ilmath] [ilmath]T[/ilmath]

See next

Notes

  1. ↑ Usually this means:
    • [ilmath]\neg[/ilmath], [ilmath]\vee[/ilmath], [ilmath]\wedge[/ilmath], [ilmath]\rightarrow[/ilmath] and [ilmath]\leftrightarrow[/ilmath]
    However we can define almost all of these symbols in terms of the others, keep this in mind.

References

  1. ↑ 1.0 1.1 Mathematical Logic - Foundations for Information Science - Wei Li

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