The semantics of relations proceeds in much the way one would expect-- the new item that has to be taken account of is the order of the terms (because, for example, Tab is not at all the same thing as Tba -- Arthur being taller than Beryl is not the same as Beryl being taller than Arthur).
Let us start with an Interpretation
Interpretation 1
Universe= {a,b}
F={a}
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[The core of this is from Leblanc and Wisdom [1972] p.117 and f.]
‘Everything is lovely’ we understand as meaning ‘Whatever thing you may choose, it is lovely’ and this becomes(x)Lx
Thus far we have considered only 'monadic' predicates-- our atomic formulas consist of a predicate followed by only one term-- for example, Fx. But in English we regularly encounter dyadic predicates or relations. For example, 'Arthur is taller than Bert' cannot be symbolized with the tools we have used so far; what is needed is a relation to represent '...is taller than ...' Txy, say, and then the proposition would be symbolized Tab.
Existential Instantiation permits you to remove an existential quantifier from a formula which has an existential quantifier as its main connective.
The circumstance that Existential Instantiation gets invoked looks like this.
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There is a rule for adding a Existential Quantifier. This permits the step illustrated by the following proof fragments.
5 Fa 6 (∃x)Fx 5 EG
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There also is a rule for adding a Universal Quantifier. This permits the step illustrated by the following proof fragments.
5 Fx 6 (x)Fx 5 UG
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To understand the concepts of scope, free, bound. To meet substitution and a simplified version of the rule for removing a Universal Quantifier.
We will certainly wish to discuss the truth and falsity of formulas with quantifiers in them.
Let us start with an Interpretation
Interpretation 1
Universe= {a,b}
F={a}
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To learn how to use the Universal and Existential Quantifiers in symbolizing propositions.
In Predicate Logic there are two new logical connectives, the Universal Quantifier (x) and the Existential Quantifier (∃x). These are used for symbolizing certain English constructions (they also have their own rules of inference and their own semantics, which we will learn about later).
The propositional rules of derivation carry over unchanged into Predicate Logic