*Propositional* logic concerns the relations of entailment between 'sentences', as in the tautology ¬(p ? ¬q) → (¬p ? q). The 'sentence variables' p and q stand for any grammatically well-formed sentence; however, the theory itself doesn't specify what a "grammatically well-formed sentence" might be. We could say that propositional logic 'takes sentences for granted' and concentrates on the various relations on and between sentences (negation, conjunction, disjunction, coordination, implication...).

 

*First order logic* or *predicate logic* gives a theory of sentence structure: a well-formed sentence has an object-predicate structure, as in Fa where the predicate "F" is satisfied by some object a. In the simplest (set-theoretical) understanding, "F" can be seen as some set, and "a" names or designates particular, individual member of that set (so, "Fa" is a *propositional function*). Thus, if Fa is true, we're saying that the set F is non-empty.

 

If we can give 'objects' by names (constants such as a, b, c...), then the veracity of our theory is just a matter of the object named by "a" having "F": imagine that "a" 'names' Barack Obama, "b" names Nicolas Sarkozy and that "F" ranges over "tall people". Fa is thus true insofar as Barack Obama satisfies the predicate "is tall".

 

We can also give 'objects' not on a one-to-one naming, but as sequences of objects x1,x2...xn such that, for any xk, xk satisfies "F": this is an *abstraction* of the propositional function "F" in "Fa". "Fx" is thus "thing that is F" or, in set theoretic terms, "member of F".

 

Here, I used the expression "for any xk, xk satisfies F" - the entities over which F ranges are given as a conjunctive sequence "Fx1 ? Fx2 ? Fx3 ?...? Fx-n ? Fxn". This is usually given in symbolic logic by an upside-down capital "A"; as RG no longer allows html posting, I'll give this by "#": thus (#x) Fx is read as "all x such that x satisfies F". This is UNIVERSAL QUANTIFICATION

 

Thus, "(#x) Fx → Gx" means "for any object x, if x is F, then x is G". This could be "all unicorns are one-horned horses" or "Sherlock Holmes lives at 221b Baker Street" - a universally quantified expression can be 'true' even though there are no 'real objects' that satisfy it.

 

If, however, we say that SOME thing satisfies F, we're saying that at least one xk in the sequence<x1,x2...xn> satisfies F. Here, "F" ranges over a *disjunctive* sequence "Fx1 ? Fx2 ? Fx3 ?...? Fx-n ? Fxn". This is usually given in symbolic logic by an backwards capital "A"; I'll give this by "$": thus ($x) Fx is read as "some x such that x satisfies F". This is EXISTENTIAL QUANTIFICATION.

 

The distinction between universal and existential quantification is that existential quantification entails *commitment*: "($x) Fx" is equivalent to claiming that the 'set' "F" is non-empty. In philosophical logic, what we hold as "existing" are those variables that are bound under existential quantification.

 

Thus, our ontological domain is given by the range of the universal quantifier, and our ontology is an interpretation on this domain given by existential quantification.

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