Material Validity
- the validity of the original argument depends on the meaning of the constituent propositions that make up the premise and conclusion.
Formal Validity
- the validity of the argument does not depend on the specific meaning of its constituent propositions.
–Instead, the validity depends on the ‘logical structure’ or ‘form’ of the argument.
–Specifically, the validity of the argument depends on the meaning of the “if…then…” phrase in the argument.
propositional operator
- “if…then…”
- logical expressions or connectives that transform propositions into more complex propositions.
- Propositional variables [variables such as A; B; etc. that stand in for propositions] are combined with an argument’s logical expressions/propositional operators in order to isolate the argument’s logical structure or form.
- An argument has a particular argument form if substituting its basic propositions for propositional variables results in that form.
Formal invalidity
1) If P then Q,
2)Not P
∴Not Q
1. Any argument with this form will be invalid even if it has true premises and a true conclusion.
2. but, counterexamples cannot establish validity
Formal Symbols:
~ (tilde)
the negation of a proposition.
~P = “It is not the case that P”
Formal Symbols:
→ (arrow)
a conditional relation between two propositions
P → Q = “If P then Q
Formal Symbols:
appropriate capital letter
a particular proposition
- So, we might symbolise “My ex-girlfriend is a liar” by the letter L.
Conditionals
- A conditional states that if one proposition is true, then another proposition is also true.
- Put differently, the truth of one proposition is a condition for the truth of another proposition.
- The proposition governed by the “if” is called the “antecedent”, while the proposition that follows from this “if proposition” is called the “consequent”.
–When symbolised P → Q, the antecedent always comes first and the consequent comes second.
true conditional
A) Whenever the antecedent is true, the consequent is also true.
B) The consequent may be true even though the antecedent is not true.
1. The truth of the antecedent is sufficient but not necessary for the truth of the consequent.
2. The truth of the consequent is necessary but not sufficient for the truth of the antecedent.
Conditional Arguments:
The four main argument forms that use conditional statements are
- Modus Ponens (valid)
- Affirming the Consequent (invalid)
- Denying the Antecedent (invalid)
- Modus Tollens (valid)
- Modus Ponens
Any argument with this form is always be valid
1.P → Q
2.P
∴Q
(P→ Q means the truth of P is sufficient for the truth of Q.)
- Affirming the Consequent
this argument form is not valid
1.P→ Q
2.Q
∴P
- [P → Q, Q ∴P] may (if it satisfies the relevant conditions) constitute a good abductive argument; however, it is never valid.
- always invalid because: can never know an antecedent with certainty on the basis of a conditional plus the consequent.
- Denying the Antecedent
•Denying the antecedent is invalid, because the antecedent is sufficient, but not necessary for the consequent.
[P → Q, ~P, ∴~Q] can be shown to be invalid:
1.If you play the guitar, then you play a musical instrument
2.You do not play the guitar
∴You do not play a musical instrument.
- Modus Tollens
Valid
[P → Q, ~Q , ~P]
•A conditional means that whenever P is true then Q must also be true.
•So, if Q is not true, then P cannot be true
• If it can be shown that a theory, position or hypothesis has an implication that is false, then that theory, position or hypothesis can be rejected
