Propositional Logic - Deductive Hypotheses

A hypothesis is a conjunctive set of premises that imply a conclusion. In symbolic form, a hypothesis can be stated as:

$ P_1 ∧ P_2 ∧ ... P_n → Q$

Figure 7

However, when we set up a hypothesis, we often attempt to place it in a different form which is given as following:

Premise 1

Premise 2

∴ Conclusion

Figure 8

Here, the three dots ∴ represent the term ‘therefore’ in natural language. The hypothesis must still follow the rules as stated in figure 7, but from now on will be written as it is in figure 8.

Assume that we want to prove that $ P→Q $ and $ ¬P∨Q $ are really the same. We would be trying to prove equivalence, or the fact that the truth values in all possible worlds per each proposition match one to one. We can denote this as $ P→Q ≡¬P∨Q $. We can prove the equivalence between these propositions using truth tables as shown in table 11.

P	Q	P→Q	¬P	¬P∨Q
T	T	T	F	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

Table 11

Syntax	Semantics
Wffs, formulation rules	Valuation and Interpretation functions

Table 12

In order to talk about soundness and completeness, it is important to be reminded what consists of syntax and what consists of semantics (see table 12). Here we can define:

1. Syntactic consequence, Γ⊢φ, which says that conclusion φ can be derived from set of premises Γ without deriving any truth values from the premises. 
2. Semantic consequence, Γ⊨φ, which says that conclusion φ must be true given that the set of premises Γ are true. 

From these two concepts, we can define soundness and completeness:

Soundness: [Γ⊢φ] → [Γ⊨φ], If the premises syntactically imply a conclusion, then the premises semantically imply a conclusion.

In this case, the fact is that syntax has associated formation rules that means that these rules are truth preserving. The calculus is not nonsense.

Completeness: [Γ⊨φ] → [Γ⊢φ], If the premises semantically imply a conclusion, then the premises syntactically imply a conclusion.

In this case, there is no semantically valid conclusions which are also syntactically valid. The language can capture all inferences.