23 terms

PL definitions

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quantificationally true
A sentence P of PL is quantificationally true if and only if P is true on every interpretation
quantificationally false
A sentence P of PL is quantificationally false if and only if P is false on every interpretation
quantificationally indeterminate
A sentence P if PL is quantificationally indeterminate if and only if P is neither quantificationally true nor quantificationally false
quantificationally equivalent
Sentences P and Q of are quantificationally equivalent if and only if there is no interpretation on which P and Q have different truth values
quantificationally consistent
A set of sentences of PL is quantificationally consistent if and only if there is at least one interpretation on which all the members of the set are true
quantificationally valid
An argument of PL is quantificationally valid if and only if there is no interpretation on which every premise is true and the conclusion is false
function
an operation that takes one or more elements of a set as arguments and returns a single value
A predicate is...
a function that takes one or more objects as the values of its variable and returns a truth-value
A functor is...
a function that takes one or more objects as the values of its variables and returns an object
quantificationally entails
a set gamma of sentences of PL quantificationally entails a sentence P of PL if and only if there is no interpretation on which all the members of the set are true and P is false
Sentence of PL
A formula of PL is a sentence of PL if and only if no occurrence of a variable is P is free
Open Sentence of PL
A formula of PL is an open sentence of PL if and only if there is at least one occurrence of a variable in P that is free
Free Variable
An occurence of a variable x in a formula P of PL that is no bound
Bound Variable
An occurrence of a variable x in a formula P of PL that is within the scope of the x-quantifier
Scope of a quantifier of PL
The scope of a quantifier in a formula P of PL is the subformula Q of P of which that quantifier is the main logical operator
Expression of PL
A sequence of not necessarily distinct elements of the vocabulary of PL
Atomic Formulas of PL
Every expression of PL that is either a sentence letter of PL or an n-place predicate of PL followed by n individual terms of PL
A
All As are Bs. For every x (Ax then Bx)
E
No As are Bs. For every x (Ax then ~Bx)
I
Some As are Bs. There exists an x (Ax & Bx)
O
Some As are not Bs. There exists an x (Ax & ~Bx)
An interpretation assigns...
a universe of discourse, truth-values to the sentence letters of PL, elements of the universe of discourse to the constants of PL, a set of ordered n-tuples
Substitution Instance of a sentence of PL
IF P is a sentence of PL of the form (for every x)X and (there exists an x such that)Q, and a is an individual constant, then Q(a/x) is a substitution instance of P. The constant represented by a is the instantiating constant