Monadic tautologies and contradictions?

[Wyman]

Owen:

I agree that we do include propositional variables as a primitive notion, which take on the values T and F. Perhaps we can consider the application of the truth functions without variables as the 'arithmetic' of logic and truth functions with variables as the 'algebra' of logic.

Now you're speaking my language (see my post analogizing sentence logic with algebra). Algebra would deal with variables(x, y z), the domain of which would be all real numbers. Arithmetic deals with Instantiations of these variables (represented as a,b,c in algebra and 1,2,3.5 in arithmetic I suppose). The domain of propositional variables in logic, rather than being the infinite set of real numbers, is the set of just two possible values (t,f).

That's what I have been thinking about lately, reading a book by Susan Haack on metalogical issues, when this thread came up. Your 'perhaps we can consider the application of the truth functions as...' comment goes to the heart of logical semantics - i.e. 'interpretation' of the primitive terms. I'm still working this out in my own mind, so I'm not saying I'm correct in my analogies and interpretations and my previous posts may not have been clear.

I think Arising is asking something like - Isn't the interpretation of propositional variables as true or false, but not both (since the domain consist entirely of two and only two choices) somewhat odd or circular? If the semantics allows only two values to variables, then is the principle of non-contradiction (or tautology) semantical (implicit the interpretation) or syntactic (stated as an axiom)?

[Arising_uk]

... rather than being the infinite set of real numbers, is the set of just two possible values (t,f).

Or as Boole had it, {0,1} which he said was the thing that made the algebra of logic different from the algebra of maths. Just my two cents.

I think Arising is asking something like - Isn't the interpretation of propositional variables as true or false, but not both (since the domain consist entirely of two and only two choices) somewhat odd or circular? If the semantics allows only two values to variables, then is the principle of non-contradiction (or tautology) semantical (implicit the interpretation) or syntactic (stated as an axiom)?

Nah! Not as bright as that, I go with Boole in that Logic is, or can be, concerned with two things, one, the relations between facts which boils down to the relations between propositions and, two, the relations between things, which for me boils down questions of 'is' or 'being' or 'existence'(or something like this as I'm not too clear what I think about this in the first place, which is why I'm asking such simple questions).
p.s.
ta for your replies as whilst I may not respond, either due to not having a response or not understanding what was said and having to think about it, I thank you for the food for thought.

[Owen]

Owen:

I agree that we do include propositional variables as a primitive notion, which take on the values T and F. Perhaps we can consider the application of the truth functions without variables as the 'arithmetic' of logic and truth functions with variables as the 'algebra' of logic.

Now you're speaking my language (see my post analogizing sentence logic with algebra). Algebra would deal with variables(x, y z), the domain of which would be all real numbers. Arithmetic deals with Instantiations of these variables (represented as a,b,c in algebra and 1,2,3.5 in arithmetic I suppose). The domain of propositional variables in logic, rather than being the infinite set of real numbers, is the set of just two possible values (t,f).

That's what I have been thinking about lately, reading a book by Susan Haack on metalogical issues, when this thread came up. Your 'perhaps we can consider the application of the truth functions as...' comment goes to the heart of logical semantics - i.e. 'interpretation' of the primitive terms. I'm still working this out in my own mind, so I'm not saying I'm correct in my analogies and interpretations and my previous posts may not have been clear.

I think Arising is asking something like - Isn't the interpretation of propositional variables as true or false, but not both (since the domain consist entirely of two and only two choices) somewhat odd or circular? If the semantics allows only two values to variables, then is the principle of non-contradiction (or tautology) semantical (implicit the interpretation) or syntactic (stated as an axiom)?

If the semantics allows only two values to variables, then is the principle of non-contradiction (or tautology) semantical (implicit the interpretation) or syntactic (stated as an axiom)?

I think there are two methods of proof for propositional logic, Syntactic (by deduction) and semantic (by calculation).
The theorem (p v ~p) is not an axiom (of any system that I have seen), rather it is deduced from (primitive terms, definitions,
axioms, and rules of inference). The theorem (p v ~p) is considered true for any p, |- (p v ~p).

In a semantic proof (p v ~p) is true for every value of p. |= (p v ~p). That is, (all p)(p v ~p). ie. (T v ~T ) & (F v ~F).
We can calculate the truth value of (p v ~p) by assumed tables of functions of p.
For example: given that T and F are primitive, we tabulate ~T=F, ~F=T,
(T v T)=T, (F v T)=T, (T v F)=T, (F v F)=F.
(T & T)=T, (F & T)=F, (T & F)=F, (F & F)=F.
From which we can calculate the truth value of (p v ~p). ((T v ~T) & (F v ~F)) <-> ((T v F) & (F v T)) <-> ((T) & (T)) <-> (T).

By the semantic method we can expand propositional logic to first order propositional logic involving quantifiers over p, q etc..

Even first order predicate logic in finite and determined domains can be decided (proven) within this semantic method of decision.

[Wyman]

That's a good summary - thanks.

Here's a question - is logic better seen as a formalization of the arguments and reasoning of ordinary language, or as a formalization/abstraction of mathematical reasoning? I come down firmly on the latter side.

[Owen]

That's a good summary - thanks.

Here's a question - is logic better seen as a formalization of the arguments and reasoning of ordinary language, or as a formalization/abstraction of mathematical reasoning? I come down firmly on the latter side.

I side with Russell here. Logic as the boy and mathematics as the man. Deduction seem to be a universal method of proof applied to both
formal logic and formal mathematics.
The truth value analysis that I have suggested seems to only have application to formal and informal logic.

That Russell and Whitehead demonstrated that 'mathematical analysis' is reducible to formal logic in 'Principia Mathematica', is convincing
evidence that mathematics is advanced logic, (higher order logic).

Wittgenstein's assertion that all propositions are truth functions of elementary propositions seems to support Russell's view.

Logic seen as "as a formalization/abstraction of mathematical reasoning" leads us to conclude that propositional logic is a mathematical
concern, which I find difficult to maintain.

But for the non-logical axioms of mathematics (e.g. the axiom of infinity) we could say logic is mathematics.
The deductive aspects of logic are the same as the deductive aspects of mathematics.

That's a good question - thanks.

[Wyman]

Logic seen as "as a formalization/abstraction of mathematical reasoning" leads us to conclude that propositional logic is a mathematical
concern, which I find difficult to maintain.

Not sure what you mean by 'concern' - i.e. as opposed to 'formalization.' I was really concentrating on the distinction between logic as a formalization of ordinary discourse v. mathematical discourse. For instance, the creation of modal or relevance logic to account for concepts not found so much in mathematics as in ordinary (or perhaps non-mathematical scientific) discourse.

On whether mathematics is advanced logic or logic is an abstraction of mathematical reasoning - your isolating 'deduction' as common between mathematics and logic is perhaps what I am getting at here. Deduction is an abstraction of a common pattern in mathematical reasoning. So I think that deductive reasoning came from mathematics in its several branches and that Russell and Whitehead created a deductive system that generalized mathematical reasoning - using propositions as variables and generalizing argument forms.

Maybe this is clearer: mathematics, over history, has created various abstract concepts that model certain aspects of the world - such as geometrical figures, numbers, functions. Physics utilizes other abstractions which also model the world - such as mass and force and velocity(along with functional analysis). In a similar way, logic abstracts certain concepts from mathematical reasoning and attempts to create a model of mathematical reasoning based on those abstractions.

The problem with saying that mathematics is advanced logic, I think, is that historically, mathematics came first; it is somewhat like saying that arithmetic is advanced algebra, or that basic geometry is advanced analytic geometry.