### at least two formally valid arguments are not valid.

Here’s a formal proof of the liar:

1. (

*x*)(*p*)((C*x*• S*(*_{x}*p*)) ⊃ ~*p*)
2. Cc • S

_{c}((*x*)(C*x*• S*(*_{x}*p*) ⊃ ~*p*)) ∴ ~(*x*)(*p*)((C*x*• S*(*_{x}*p*)) ⊃ ~*p*)
3. Cc • S

_{c}((*x*)(*p*)((C*x*• S*(*_{x}*p*)) ⊃ ~*p*) ⊃ ~(*x*)(*p*)((C*x*• S*(*_{x}*p*)) ⊃ ~*p*1, UI
4.
~(

*x*)(*p*)((C*x*• S*(*_{x}*p*)) ⊃ ~*p*) 2, 3, MP
In English:

1. for any

*x*and any*p*if*x*is a Cretan and*x*says*p*then ~*p*.
2. Chris is a Cretan and Chris says for
any

*x*and any*p*if*x*is a Cretan and*x*says*p*then ~*p*.
3. If Chris is a Cretan and Chris says
for any

*x*and any*p*if*x*is a Cretan and*x*says*p*then ~*p*then it’s not the case that for any*x*and any*p*if*x*is a Cretan and*x*says*p*then ~*p*. (From 1 and universal instantiation)
4. it’s not the case that for any

*x*and any*p*if*x*is a Cretan and*x*says*p*then ~*p.*(from 2, 3, and*modus ponens*)
Here might be a way out:

The argument’s

*form*is valid, but there is no time in which the string of symbols which compose lines 1 and 2 express a proposition. Since strings of symbols which fail to express propositions*eo ipso*fail to express premises, any instance of lines 1 and 2 will fail to constitute an argument. Since only arguments may be valid and lines 1 and 2 fail to constitute an argument, nothing follows from lines 1 and 2. Ergo, there is no liar’s paradox.
Objection: Wow. Way to be

*ad hoc*.
Reply: False. I have independent
reasons for saying that some strings of symbols which appear to express
propositions don’t actually express propositions. For example, here’s a formal
proof of Zeus’ existence:

1. (

*x*)*x*=*x*∴ (∃*x*)*x*= z
2. z = z 1, UI

3. (∃

*x*)*x*= z 2, EG
In English:

1. Everything is identical to itself.

2. Zeus is Zeus. (from 1 and universal instantiation)

3. There is something such that it is
Zeus. (from 2, and existential
generalization)

I think that this argument’s form is
valid, and I also think that its one and only premise is true. However, I don’t
think the string of symbols in line 2 expresses a proposition because in order
for it to express a proposition, the letter ‘z’ must refer to Zeus. Since Zeus
doesn’t exist, ‘z’ doesn’t name
anything, and therefore line 2 is not even false. Since the proof only goes
through if each line expresses a proposition, lines 1, 2, and 3 fail to
constitute a proof.

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