Introduction to Logic
Harry J. Gensler
Chapter 5: Basic Quantificational Logic
Quantificational Logic concerned with arguments whose valid depends on "all", "no", "some", and similar notions.
5.1 Easier Translations
Ir - Romeo is Italian
Ix - x is Italian
(x)Ix - For all x, x is Italian
(∃x)Ix - For some x, x is Italian
Capital letters are used for general terms or categories.
Small letters are used for singular terms or specific things or particulars.
- Single capital letters denote statements = S
- Capital letters followed by a small letter denote a general term = Ir
- Capital letters followed by two or more small letters denote a relation = Lrj
- a small letter can stand as either a constant or a variable
A quantifier is a sequence in the form of either:
1) (x)
2) (∃x)
(x) is a universal quantifier. It claims that the formula that follows is true for all values of x. i.e.
(x)Ix - for all x, x is Italian (All are Italian)
(∃x) is an existential quantifier. It claims that the formula that follows is true for at least one value of x. i.e.
(∃x)Ix - for some x, x is Italian. (Some are Italian)
English - Quantificational Language
all (every) - (x)
not all (not every) - ~(x)
some - (∃x)
no - ~(∃x)
All A is B - (x)(Ax⊃Bx)
Some A is B - (∃x)(Ax·Bx)
No A is B - ~(∃x)(Ax·Bx)
5.2 Easier Proofs
Reverse squiggle rule
~(x)Fx - (∃x)~Fx
~(∃x)Fx - (x)~Fx
Existential Instantiation
(∃x)Fx - Fa
The variable 'x' is substituted as the constant 'a'. 'a' is a hypothetical reality.
When there is more than one existential quantifier: i.e.
(∃x)Mx
(∃x)Fx
More than one hypothetical must be used: i.e.
(∃x)Mx - Ma
(∃x)Fx - Fb
Universal Instantiation
Since (x)Fx states all x is Fx any constant can be instantiated. Thus if a problem looks like this:
(x)Fx - Fa
(x)Rx - Ra
(x)Gx - Ga
(x)Lx - La
Since it is universal it is true (or false) for all.
In doing a proof the order now looks like this:
1. asm: the opposite of the conclusion
2. reverse squiggle
3. existential instantiation
4. universal instantiation
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