We begin with a prepositional language Lp containing conjunction (Λ), a class of sentence names {Sα}αϵA, and a falsity predicate F. We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names {Sβ}βϵB,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200007611/resource/name/S0022481200007611_Uequ1.gif?pub-status=live)
is a well-formed formula (we will use WFF to denote the set of well-formed formulae).
The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation function:
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200007611/resource/name/S0022481200007611_Uequ2.gif?pub-status=live)
For example, the LP sentence “F(S1)” (i.e., Λ{F(S1)}), combined with a denotation function δ such that δ(S1)“F(S1)”, provides the (or, in this context, a) Liar Paradox.
To give a more interesting example, Yablo's Paradox [4] can be reconstructed within this framework. Yablo's Paradox consists of an ω-sequence of sentences {Sk}kϵω where, for each n ϵ ω:
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200007611/resource/name/S0022481200007611_Uequ3.gif?pub-status=live)
Within LP an equivalent construction can be obtained using infinite conjunction in place of universal quantification - the sentence names are {Si}iϵω and the denotation function is given by:
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200007611/resource/name/S0022481200007611_Uequ4.gif?pub-status=live)
We can express this in more familiar terms as:
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200007611/resource/name/S0022481200007611_Uequ5.gif?pub-status=live)
etc.