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*Symbolic Logic: Syntax, Semantics, and Proof*

6.3 The Syntax of RL

*open sentence*or an

*open formula*is one with an

*n*-place predicate

**P**followed by

*n*terms, where at least one of those variables is free. However, a

*closed sentence*or a

*closed formula*is one with an

*n*-place predicate

**P**followed by

*n*terms, where none of those terms are free variables. The main operator in a well formed formula (wff) in RL is the one with the greatest scope, which means that the one that falls under no other operator’s scope is the main one. We consider the quantifiers as operators. Thus in (∃x)(Px∧Qx) the main operator is ∃x, because all the rest of the formula falls under the quantifier’s scope, and in ¬(∃x)(Px∧Qx) the main operator is the negation, because the quantifier falls under its scope, and the rest of the formula falls under the quantifier’s scope. And there are five rules that determine a wff in RL: (i) An

*n*-place predicate ‘

**P**’ followed by

*n*terms (names or variables) is a wff. (ii) If ‘

**P**’ is a wff in RL, then ‘¬

**P**’ is a wff. (iii) If ‘

**P**’ and ‘

**Q**’ are wffs in RL, then ‘

**P**∧

**Q**,’ ‘

**P**∨

**Q**,’ ‘

**P**→

**Q**,’ and ‘

**P**↔

**Q**’

**are wffs. (iv) If ‘**

**P**’ is a wff in RL containing a name ‘a,’ and if ‘

**P**(

*x*/a)’ is what results from substituting the variable

*x*for every occurrence of ‘a’ in ‘

**P**,’ then ‘(∀x)

**P**(

*x*/a)’ and ‘(∃x)

**P**(x/a)’ are wffs, provided ‘

**P**(

*x*/a)’ is not a wff. (v) Nothing else is a wff in RL except that which can be formed by repeated applications of (i) to (iv).

(1) (∀x)(Fx→Bx)∨Wx(Agler 256)

(2) (∃x)Mx∧(∃y)Ry(Agler 256)

(3) (∀y)(Mx∧By)(Agler 256)

*bound*and

*free variables*. Variables are bound if they fall under the scope of a quantifier that is quantifying specifically for that particular variable, and it us a free variable otherwise.

Bound variable: When a variable is within the scope of a quantifier that quantifies that specific variable, then the variable is a bound variable.Free variable: A free variable is a variable that is not a bound variable.(Agler 256)

(3) (∀y)(Mx∧By)

(1) when it is not contained in the scope of any quantifier,or(2) when it is in the scope of a quantifier, but the quantifier does not specifically quantify for that specific variable.(Agler 257)

(6) [(∀x)(Px→Gy)∧(∃z)(Pxy∧Wz)]∨Rz(Agler 257)

(∀x)(Px→Gy)

(∃z)(Pxy∧Wz)

(1) (∃x)(Px∧Qx)

(2) (∃x)(Px)∧(∃x)(Qx)

(3) ¬(∃x)(Px∧Qx)(Agler 257)

(4) (∀y)(∃x)(Rx→Py)(Agler 257)

(∀x)Px ..................................... ∀xWff...........................................Its Main Operator

(∀x)(Px) ................................... ∀x

¬(∀x)(Px) ................................. ¬

¬(∃y)(Py) .................................. ¬

(∀x)(Px→Qx) ............................ ∀x

¬(∀x)(Px→Qx) .......................... ¬

(∀x)(Px→¬Qx) .......................... ∀x

(∀x)(Px)→(∃x)(Qx) ................... →

¬(∀x)(Px)→(∃x)(Qx) ................. →

¬[(∀x)(Px)→(∃x)(Qx)] ............... ¬

(∃x)(Px∧Qx)∧¬(∀y)(Py→Qy) .... ∧

(∃x)¬(Px∧Qx) ............................ ∃x

(∃x)(∀y)[(Px∧Qx)→Dy] ............. ∃x

(∀x)¬(∀y)(Px→Qy) ................... ∀x

¬(∀x)(∀y)(Px→Qy) .................... ¬

(Agler 285)

(i) Ann-place predicate ‘P’ followed bynterms (names or variables) is a wff.(ii) If ‘P’ is a wff in RL, then ‘¬P’ is a wff.(iii) If ‘P’ and ‘Q’ are wffs in RL, then ‘P∧Q,’ ‘P∨Q,’ ‘P→Q,’ and ‘P↔Q’are wffs.(iv) If ‘P’ is a wff in RL containing a name ‘a,’ and if ‘P(x/a)’ is what results from substituting the variablexfor every occurrence of ‘a’ in ‘P,’ then ‘(∀x)P(x/a)’ and ‘(∃x)P(x/a)’ are wffs, provided ‘P(x/a)’ is not a wff.(v) Nothing else is a wff in RL except that which can be formed by repeated applications of (i) to (iv).(Agler 258)

(i) Ann-place predicate ‘P’ followed bynterms (names or variables) is a wff.

*open sentence*or an

*open formula*is one with an

*n*-place predicate

**P**followed by

*n*terms, where at least one of those variables is free. However, a

*closed sentence*or a

*closed formula*is one with an

*n*-place predicate

**P**followed by

*n*terms, where none of those terms are free variables (Agler 259).

Open formula: An open formula is a wff consisting of ann-place predicate ‘P’ followed bynterms, where one of those terms is a free variable.Closed formula: A closed formula is a wff consisting of ann-place predicate ‘P’ followed bynterms, where every term is either a name or a bound variable.(259)

(ii) If ‘P’ is a wff in RL, then ‘¬P’ is a wff.

(iii) If ‘P’ and ‘Q’ are wffs in RL, then ‘P∧Q,’ ‘P∨Q,’ ‘P→Q,’ and ‘P↔Q’are wffs.

(iv.a) If ‘P’ is a wff in RL containing a name ‘a,’ and if ‘P(x/a)’ is what results from substituting the variablexfor every occurrence of ‘a’ in ‘P’ ...(Agler 259)

(1) Pb(Agler 259)

*x*/b)

(2) Pbb

*z*/b).

(iv.b) ... then ‘(∀x)P(x/a)’ and ‘(∃x)P(x/a)’ are wffs, provided ‘P(x/a)’ is not a wff.(Agler 260)

*n-*place predicates with

*n*variables coming after them. It seems that because a variable was substituted, that thereby necessitates a quantifier, but I am not sure why. I might guess that by substituting, we are somehow treating the variable that replaces the name as being bound and thus needing a quantifier that would bind it. I wonder if it makes sense to think about the situation as if we began from the other direction. So suppose we were to substitute a name in for a variable. Perhaps this can only be done if the variable is bound. But why might it be impossible to substitute a name in for an unbound variable? Recall the example from before: (∀y)(Mx∧By). Is there some reason why we cannot substitute a name in for the x of Mx? I do not know. Does it have something to do with the fact that by not quantifying the variable, we have not determined what can be substituted in for it? At any rate, the main idea is that (for some reason) we need to add a quantifier to the variable-substituted predicate to make it a wff. Let me quote.]

The consequent of (iv) says that by putting a universal quantifier or existential quantifier in front of the formula that is the result of substituting a variable x for every name ‘a,’ the resulting formula is a wff. For example, using (1) from above, note that ‘Pb’ is a wff and contains a name ‘b.’ Second, take ‘P(x/b),’ which is the formula that results from substituting the variable x for every occurrence of ‘b’ in ‘Pb.’ This gives us ‘Px.’ The consequent clause of (iv) says that both of the following will be wffs:(∀x)Px(∃x)Px(Agler 260)

(v) Nothing else is a wff in RL except that which can be formed by repeated applications of (i) to (iv).

(3) Pab∧Ra

(260)

(4) ¬Qa→(∀x)Rx

*x*/a) results when we substitute

*x*for any occurrence of a, then (∀x)Rx is a wff. We can establish that Ra is a wff on the basis of rule i. Thus we can infer that (∀x)Rx is a wff. We can then bring the two parts together using rule iii. Here is how Agler works it out:

Agler then uses a similar procedure to show that

(5) (∀x)Pxx→¬(∃y)Gyis a wff (for the details, see Agler, page 261).

*Symbolic Logic: Syntax, Semantics, and Proof*. New York: Rowman & Littlefield, 2013.

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